A Fuzzy Difference Equation Framework for Water-Consumption Modeling Under Data Uncertainty

This paper develops a fuzzy difference equation model for analyzing long-term domestic water-consumption dynamics under parameter uncertainty. The proposed framework combines a difference equation with fuzzy parameter representations, allowing uncertainty to be incorporated directly into the modeling process through triangular fuzzy numbers and their associated α-cut intervals. Fundamental properties of the model are investigated, including the existence, boundedness, and nesting behavior of fuzzy solutions. The model is applied to annual domestic water-consumption data from Taiwan covering the period 2008–2024. Parameter estimates are obtained using ordinary least squares, and both deterministic and fuzzy solution trajectories are constructed. Numerical results show that the deterministic model captures the overall long-term behavior of the consumption series, while the fuzzy formulation provides uncertainty bands that contain 94.12% of the observed data. The results demonstrate that the proposed approach offers an interpretable framework for studying water-consumption dynamics while explicitly accounting for uncertainty in model parameters.