Two novel distance functions for interval-valued trapezoidal intuitionistic fuzzy numbers and their applications in decision-making

Despite the extensive application of interval-valued intuitionistic fuzzy numbers (IVIFNs) in modeling uncertainty within complex decision-making problems, distance measurement for their important subclass, interval-valued trapezoidal intuitionistic fuzzy numbers (IVTraIFNs), suffers from conceptual and computational limitations. Existing distance measures often rely on discrete features or neglect the continuous geometry and surface structure of these numbers, leading to information loss and inadequate discrimination between alternatives. This paper addresses this gap by introducing a novel surface-area-based distance measurement framework designed explicitly for IVTraIFNs. We propose two new distance functions, a Euclidean and a Manhattan measure, derived from the integration of areas under the left and right curves of the continuous membership and non-membership functions. A conceptual parametric framework for representing continuous IVTraIFNs is first established, and the mathematical properties of the proposed measures, including compliance with distance metric axioms and the triangle inequality, are formally proven. Explicit, practical formulas for these measures for IVTraIFNs are then developed. To demonstrate practical efficacy, the new measures are systematically integrated into two established multi-criteria group decision-making (MCGDM) methods: TOPSIS and CODAS. A comprehensive numerical example in the context of investment project selection, supported by sensitivity analysis on the weight parameter and standard validity tests, confirms the effectiveness, robustness, and advantages of the proposed approach compared to some existing methods. The results indicate that the surface-based distance measures provide more appropriate discrimination between alternatives while offering flexibility through a parameter that weights the membership and non-membership components, remaining consistent with existing decision-making schemes.