New Fuzzy Distance Metric Handles Heterogeneous Data Scales

Decision-making in complex systems often involves data that is imprecise or uncertain, frequently represented using fuzzy sets, particularly Triangular Fuzzy Numbers (TFNs). A critical component of many fuzzy methods is the ability to accurately measure the distance between these TFNs. However, existing distance measures often assume that all values operate on the same scale, necessitating a separate normalization step when dealing with heterogeneous attributes that have different units or scales. This research proposes a new metric, the Triangular Fuzzy Rescaling Distance (d_TR), specifically designed to overcome this challenge. The d_TR uniquely incorporates Linear Rescaling (LRE) directly into its distance calculation, thereby performing normalization as part of the comparison process. The paper formally proves that d_TR satisfies all properties of a metric, including non-negativity, identity, symmetry, and the triangle inequality. Furthermore, d_TR is demonstrated to be bounded, scale-invariant, and origin-invariant. These robust properties, combined with the ability to incorporate a weighting vector for prioritizing different dimensions, make d_TR exceptionally well-suited for applications involving diverse fuzzy data. This includes areas such as constructing synthetic indicators, developing distance-based machine learning algorithms, and aiding in multicriteria decision-making.