A Refined Neutrosophic Over/Under Probability Model with Off-Norm Operators for Evaluating the Efficiency of AI-Based Digital Tools in Korean Language Teaching

This paper presents a new mathematical model based on refined neutrosophic theory to evaluate the efficiency of AI-based digital tools in Korean language teaching. The model uses the concept of Over/Under Probability and uses Off-Norm operators to capture uncertain, inconsistent, or extreme values often found in real classroom data. Instead of only measuring performance within the traditional range [0,1], our model allows values greater than 1 or less than 0 to represent over-performance or underperformance. We define precise equations to evaluate both the development and usage efficiency of educational tools. The model also includes inter-tool synergy using neutrosophic interaction rules. Two case studies are provided with full calculations to show how the model works in practice. Compared to traditional evaluation methods, our neutrosophic model gives deeper insight into tool performance, uncertainty, and collaboration.