Fuzzy Math Gets a Makeover: New Model Brings Fairer University Teaching Evaluations

Every year, universities around the world face the same stubborn problem: how do you turn something as slippery as teaching quality into a number you can trust? A new study published in Cluster Computing by Jawad Ali of Kohat University of Science and Technology and colleagues in Romania and Saudi Arabia offers one of the most mathematically sophisticated answers yet. The team has built a decision-making framework that combines a fresh way of scoring uncertain evaluations with a family of powerful averaging operators, all aimed squarely at the audit and evaluation of undergraduate teaching. The work, published on 28 August 2026, is technical to its core, but its ambition is simple: make high-stakes educational assessments less vulnerable to vagueness, bias, and the distortions that extreme opinions can inject into a panel’s final verdict.

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