Aggregation of Fuzzy Nutrient-Level Evaluations Using a Parametric T-Norm: Mathematical Properties, Calibration, and a Proof-of-Concept Analysis

Source abstract

Backgroumd/Objectives: Since 1995, the intake of individual nutrients has been evaluated using fuzzy logic, with aggregation of the resulting fuzzy sets enabling the weighing of one nutrient against another, and the evaluation and optimization of overall nutrient intake. The aggregation operator used in earlier work, defined as the product of the minimum and harmonic mean operators (MIN×H), performed well in practice but had undesirable mathematical properties: it is not associative, model extensions can shift results even when all conditions are fully satisfied, and the resulting evaluation field contains discontinuities inconsistent with biological behavior. Methods: We therefore reviewed the family of parametrized T-norms for an alternative and identified the Schweizer (3) operator, fitting its parameter to best match the existing operator using 246 real-world dietary records from 123 participants (a sustainability study cohort of predominantly students and young adults; mean age 25.3 ± 16.4 years; 54 male, 69 female). Results: Strong agreement was achieved at p = 6 (R2 = 0.9949; RMSE = 0.0187), outperforming p = 4 (R2 = 0.9852) while remaining more balanced than p = 8 (systematic intercept offset). The new operator satisfies all required mathematical properties and produces a smooth, simply structured evaluation field that reveals synergistic and substitution effects among foodstuffs, as well as the essentiality of certain foodstuffs under specific conditions. The resulting evaluation field shows a wide, high-dimensional plateau rather than a sharp peak, interpretable as stability, error tolerance, and a high degree of diversity, offering a new, mathematically robust perspective on healthy nutrition and the interactions among foodstuffs. Conclusions: This work proposes and illustrates a mathematical aggregation framework; its nutritional and clinical validity remains to be established in future studies.