The foundation of the new approach is a branch of mathematics called fuzzy set theory, which was invented by Lotfi Zadeh in 1965 precisely to handle the kind of ambiguity that ordinary crisp numbers cannot capture. When an expert says a teacher’s classroom performance is ‘good to some degree,’ that statement is neither true nor false in a binary sense. Classical fuzzy sets assign a single membership grade to capture partial truth. Later refinements added a second number, a non-membership grade, to express how far the evaluation falls short. Intuitionistic fuzzy sets, Pythagorean fuzzy sets, and Fermatean fuzzy sets each progressively relaxed the constraints linking these two grades, allowing experts to express ever-wider bands of hesitation and uncertainty without breaking the underlying mathematics.

The framework adopted in the new paper is the p,q-quasirung orthopair fuzzy set, or pq-QOF set, which generalizes this entire lineage. In an orthopair, the membership and non-membership grades are constrained by a power relationship, and the quasirung formulation lets the exponents p and q take flexible values. The larger the exponents, the more room experts have to register strong degrees of support and opposition simultaneously. This flexibility matters in teaching audits, where reviewers may genuinely feel that an instructor excels in some dimensions and fails in others. Yet the authors identified a weakness: the score functions currently used to rank pq-QOF numbers, the devices that compress a pair of grades into a single comparable value, suffer from deficiencies that can distort rankings. Their first contribution is a novel score function designed to measure pq-ROF numbers more faithfully.

Scoring, however, is only half the battle. Once each expert has delivered fuzzy evaluations across many criteria, those judgments must be aggregated into a single overall assessment per candidate. The workhorse tools here are aggregation operators, and the paper introduces several new ones. The first pair builds on the Maclaurin symmetric mean, a classical mean rooted in an inequality that dates back to a 1729 letter by the Scottish mathematician Colin Maclaurin. Unlike a simple average, the Maclaurin symmetric mean captures interrelationships among multiple criteria at once, rewarding alternatives that perform consistently well across interconnected dimensions rather than those that shine on one criterion and collapse on another.

The crucial innovation is the word ‘partitioned.’ In real evaluation exercises, criteria naturally cluster into distinct groups. A teaching audit might bundle lecture delivery, student engagement, and clarity of explanation into one family of pedagogical criteria, while research supervision, administrative service, and curriculum development form entirely separate families. Relationships among criteria within the same family are strong, but criteria in different families are largely independent. Standard Maclaurin means wrongly assume all criteria interact. The new pq-QOF partitioned Maclaurin symmetric mean, and its weighted variant, apply the mean only within each partition and then combine the partition results, mirroring the actual structure of the evaluation problem. A companion operator family extends the same idea using the Bonferroni mean, another classical device for modeling pairwise interactions among criteria.

Even a well-partitioned average has an Achilles heel: extreme values. If one panelist submits an outlier evaluation, whether from an agenda, a bad day, or a genuine misreading, a conventional weighted average lets that outlier drag the result. The authors counter this with a power mechanism drawn from Ronald Yager’s power average operator, introduced in 2001. In a power average, the weight assigned to each piece of information depends on how much support it receives from all the other pieces. Values that cluster near the consensus of the group gain influence; isolated extremes are automatically down-weighted. By fusing this power-averaging logic with the partitioned Maclaurin symmetric mean inside the pq-QOF environment, the team produced the pq-QOF power partitioned Maclaurin symmetric mean and its weighted counterpart, operators that simultaneously respect criterion groupings, capture intra-group interactions, and suppress the corrosive effect of outlier judgments.

Aggregating evaluations requires knowing how much each criterion matters, and determining those weights is itself a decision problem. The paper extends the ordinal priority approach, or OPA, a relatively recent weight-derivation method introduced by Ataei, Mahmoudi, Feylizadeh, and Li in 2020. OPA is unusual in that it works entirely with ordinal information: decision-makers simply rank the criteria, rank themselves, and rank the alternatives, and a linear optimization model converts those ordinal priorities into precise weight vectors. This sidesteps the consistency problems that plague pairwise comparison methods such as the best-worst method or the full consistency method, which demand that experts make numerically consistent judgments across many comparisons. By extending OPA into the pq-QOF setting, the authors let the same fuzzy machinery that handles the evaluations also handle the weighting, producing a unified framework rather than a patchwork of mismatched techniques.

The full pipeline constitutes a multi-criteria group decision-making method: multiple decision-makers supply pq-QOF evaluations of teaching quality across partitioned criteria, the extended OPA derives the criterion weights, and the power partitioned Maclaurin operators fuse everything into a final score for each candidate. To demonstrate the framework, the authors applied it to university evaluation, drawing on institutional contexts in Khyber Pakhtunkhwa, Pakistan, including the Ghulam Ishaq Khan Institute, the University of Peshawar, Abdul Wali Khan University Mardan, Kohat University of Science and Technology, and the University of Haripur. The numerical illustration shows how the method converts heterogeneous, uncertain expert judgments into a defensible ranking of institutions or teaching programs.

What separates a proposed method from a trustworthy one is validation, and the paper devotes substantial effort to testing its framework. Sensitivity analysis probes whether small changes in weights or inputs produce wild swings in the ranking, a hallmark of fragile models. Comparative analyses pit the new operators against existing alternatives, including standard q-rung orthopair fuzzy Maclaurin means, Bonferroni means, and other aggregation families, to confirm that the partitioned and power-enhanced versions yield distinct and defensible advantages. The authors report that these comparisons illustrate and validate the effectiveness and utility of the developed model, supporting the claim that the extra mathematical machinery earns its keep rather than adding complexity for its own sake.

The broader significance extends well beyond teaching audits. Undergraduate teaching evaluation has become a global preoccupation, with systems in China and elsewhere undergoing scrutiny for problems of subjectivity and gaming, and prior studies have deployed neutrosophic and q-rung orthopair methods to tackle the same challenge. The new framework’s ingredients, a more faithful score function, partition-aware aggregation, outlier suppression, and ordinal weight derivation, are modular ideas that could transfer to any evaluation domain where criteria cluster naturally and experts disagree: hospital performance assessment, grant review, supplier selection, or site selection for infrastructure such as electric vehicle charging stations. The work was supported by funding from King Saud University through its Ongoing Research Funding Program and by King Khalid University through a Large Groups Project grant. As universities increasingly demand that quality assurance rest on transparent, reproducible computation rather than gut feeling, frameworks like this one point toward a future in which even our most human judgments are aggregated with mathematical care.

Subject of Research: A fuzzy multi-criteria group decision-making model for undergraduate teaching audit and evaluation

Article Title: Novel score function-based quasi-rung orthopair fuzzy OPA model with power-partitioned Maclaurin symmetric mean for undergraduate teaching audit and evaluation

Article References: Ali, J., Popa, I.-L., Hassan, M. M., Alkhamees, B. F., & Abdallah, S. A. O. (2026). Novel score function-based quasi-rung orthopair fuzzy OPA model with power-partitioned Maclaurin symmetric mean for undergraduate teaching audit and evaluation. Cluster Computing, 29(12), Article 701. https://doi.org/10.1007/s10586-026-06401-5

Image Credits: AI Generated

DOI: 10.1007/s10586-026-06401-5

Keywords: fuzzy sets, q-rung orthopair fuzzy, score function, Maclaurin symmetric mean, aggregation operators, ordinal priority approach, multi-criteria group decision-making, teaching evaluation, power average, uncertainty modeling, higher education assessment, Cluster Computing

The research, led by Yan Chen with Lei Xu and Lin Liu, all of Shenyang University of Technology in China, introduces a group decision-making framework built on picture fuzzy sets, a mathematical structure designed to capture the full spectrum of human opinion. Where classical logic demands that a statement be simply true or false, and even earlier fuzzy extensions allowed only membership and non-membership degrees, picture fuzzy sets add a third dimension: the refusal or neutral degree. An expert evaluating a candidate technology can simultaneously express support, opposition, and abstention, with the three values summing to one. This richer vocabulary matters in real committees, where ‘I am not sure’ is often the most honest answer on the ballot.

Yet the very richness that makes picture fuzzy sets expressive has historically made them unwieldy. When experts record their pairwise comparisons in picture fuzzy preference relations, checking whether those judgments are internally consistent becomes a computational ordeal. Consistency is not a pedantic concern. If an expert prefers option A over B, B over C, but then declares C preferable to A, the entire ranking built on those judgments is suspect. Traditional verification procedures for picture fuzzy matrices have been cumbersome enough to discourage practical adoption, and the new paper identifies this as one of three chronic defects in the existing literature, alongside the absence of an objective basis for weighting experts and the difficulty of quantifying how trustworthy the final decision actually is.

The first breakthrough is mathematical housekeeping of an unusually elegant kind. The authors deploy transformation functions that convert picture fuzzy preference relations into ordinary fuzzy complementary judgment matrices, the well-studied workhorses of classical decision theory. This conversion simplifies the data structure and dramatically reduces computational complexity, but the real innovation lies in what follows. By constructing a derived matrix from the transformed judgments, the team proves an additive consistency determination theorem, a criterion that allows researchers to verify consistency directly rather than iterating through the laborious checks that traditional methods require. In effect, a question that once demanded an elaborate auditing procedure can now be settled by inspecting a single algebraic condition.

The second pillar of the framework addresses a blind spot that has long troubled the field: how much should each expert’s opinion count? Conventional approaches often assign weights arbitrarily or rely on self-reported confidence, ignoring the social fabric in which real experts are embedded. Chen and colleagues turn to BrowseRank, an algorithm originally developed for ranking web pages by modeling user browsing behavior on a network of hyperlinks. In the decision-making context, the experts become nodes in a social network, and trust relationships between them become the links. Because real-world trust data is frequently incomplete, the authors first introduce a picture fuzzy trust propagation operator, which infers missing trust links from the chains of relationships that do exist, much as one might gauge a stranger’s reliability through a friend of a friend.

Once the trust network is complete, BrowseRank computes each decision-maker’s objective weight by integrating multiple network parameters, breaking free of the limitation of ignoring social attributes entirely. An expert who is widely trusted by other trusted experts accrues greater influence, not because of seniority or loudness in the meeting room, but because of their structural position in the web of professional confidence. This is a genuinely different philosophy of weighting. It treats expertise as a network property rather than an individual trait, echoing ideas from citation analysis and social network theory while grounding them in the fuzzy mathematics needed to handle uncertain trust assessments.

Aggregating individual opinions into a collective verdict is where many decision frameworks quietly fail. Even if every expert’s matrix is individually consistent, the merged matrix can drift into inconsistency, contaminating the final ranking. The proposed method guards against this with an aggregation strategy that combines the weighted geometric average operator with the Hadamard product, an element-wise multiplication of matrices. The authors show that this pairing ensures the consistency of the aggregated matrix, filling what they describe as a gap in the research literature. The practical consequence is that the group’s collective judgment inherits the logical soundness of its inputs, rather than requiring a fresh round of correction after aggregation.

Perhaps the most novel contribution is the framework’s answer to a question that decision scientists rarely ask with precision: how reliable is the result? The authors define decision-maker reliability using a picture fuzzy set similarity measure, comparing each expert’s judgments against the group consensus. Experts whose views align closely with the collective position score high reliability; outliers score low. This converts reliability from a vague qualitative impression into a quantitative evaluation, completing an evaluation system in which every component, from individual consistency to expert weight to final confidence, is computed rather than assumed. For organizations making high-stakes choices, such a number could become as standard a deliverable as the recommendation itself.

To demonstrate the framework in action, the team applied it to a problem with obvious contemporary urgency: selecting batteries for new energy vehicles. Battery choice is an ideal stress test for fuzzy decision theory because the criteria, energy density, cost, safety, lifespan, and environmental impact, are riddled with uncertainty, and the experts who assess them come from heterogeneous backgrounds with uneven trust relationships. Through simulation and comparative analysis against existing methods, the authors verified their approach’s advantages in consistency check efficiency, weight rationality, and result reliability. The comparisons showed that the BrowseRank-based weighting produced more defensible expert influence scores than methods lacking a social network foundation, while the additive consistency theorem streamlined a verification step that competing approaches handle far more slowly.

The broader significance of the work extends well beyond battery procurement. Group decision-making under uncertainty underpins technology selection, policy evaluation, medical diagnosis support, disaster risk analysis, and financial investment screening, all areas where picture fuzzy methods have recently gained traction. By fusing a web-ranking algorithm with judgment matrix consistency theory, the Shenyang team has built a pipeline in which fuzzy human opinions are tamed algebraically, social trust is measured rather than guessed, and the final answer arrives with a quantified confidence estimate. As decisions grow more complex and the experts consulted more numerous and dispersed, frameworks of this kind may become the quiet mathematical machinery ensuring that collective judgment is not just a vote, but a verifiable measurement.

Subject of Research: A picture fuzzy group decision-making method integrating the BrowseRank algorithm and additive consistency for expert weighting and reliability evaluation

Article Title: A picture fuzzy group decision-making approach based on BrowseRank algorithm and additive consistency

Article References: Chen, Y., Xu, L., & Liu, L. (2026). A picture fuzzy group decision-making approach based on BrowseRank algorithm and additive consistency. International Journal of Machine Learning and Cybernetics, 17(9), Article 452. https://doi.org/10.1007/s13042-026-03279-y

Image Credits: AI Generated

DOI: 10.1007/s13042-026-03279-y

Keywords: group decision-making, picture fuzzy sets, BrowseRank algorithm, additive consistency, trust propagation, social network analysis, fuzzy preference relations, expert weighting, decision reliability, new energy vehicle battery selection, aggregation operators, similarity measure