Employing Fixed-Point Theory for Fuzzy Regression Analysis: Methodology and Empirical Application

<ns3:p>Background Traditional fuzzy regression approaches, such as Tanaka’s fuzzy minimum method and fuzzy least squares, often lack theoretical guarantees of existence, uniqueness, and numerical stability. These limitations are of paramount importance in engineering applications involving uncertainty, such as predicting the compressive strength of concrete. This study addresses these issues by presenting a mathematically rigorous fuzzy regression model, based on fixed-point theory, formulated within the full metric space of trapezoidal fuzzy numbers using the metric scale d ∞ . Methodology We define a shrinkage coefficient on the trapezoidal fuzzy coefficient vector space and prove, using Banach’s fixed-point theory, the existence and uniqueness of the regression solution. An iterative algorithm is constructed to estimate the coefficients, using alpha computation and Lipschitz continuity to ensure convergence. The University of California, Irvine concrete compressive strength dataset was amplified using ASTM and ACI-based uncertainty coefficients, and the proposed fixed-point model was evaluated against the Tanaka method and least-squares fuzzy regression. Performance was assessed by mean squared error (MSE), coefficient ambiguity, convergence behavior, and toughness under ±5% noise. Results The proposed method demonstrated consistent geometric convergence with an average of 12.3 iterations and zero divergence across all experiments. It also reduced the overall mean squared error by an average of 12.5%, and by up to 25.1% at best, compared to the comparator methods. Coefficient ambiguity—measured by ambiguity width—was reduced by 18.3% compared to the Tanaka method and by 13.3% compared to ambiguity squares. Under noise perturbation, the model exhibits a significantly smaller increase in the mean error (+6.2%) compared to the Tanaka approach (+24.7%) and the LS-based approach (+18.3%), indicating a substantial improvement in robustness. Conclusions Incorporating fuzzy regression within a fixed-point theoretical framework helps resolve the stability, existence, and singularity challenges that have long plagued classical fuzzy regression models. The proposed approach offers a mathematically consistent, computationally stable, uncertainty-aware regression tool suitable for engineering applications involving imprecise measurements. Future work includes extending the model to nonlinear fuzzy structures, Gaussian/LR fuzzy representations, and broader applications in data-driven prediction under uncertainty.</ns3:p>