Healthcare systems face significant challenges and financial burdens due to patient no-shows, highlighting the need for accurate and interpretable predictive models. This study evaluated the efficacy of twelve classification algorithms that can generate self-explanatory results illustrations across four categories: Rule Set classifiers, Rule List classifiers, Rule Tree classifiers, and Algebraic Models, using a real-world dataset, “Brazilian Medical Appointment No Shows”. Analysis across multiple performance metrics revealed significant differences among the algorithms. Advanced models like Tree- Generalized Additive Model (GAM), Fast Interpretable Greedy-Tree Sums (FIGS), Tree Alternating Optimization (TAO) Tree, and RuleFit demonstrated superior predictive capabilities using Over-Sampling and feature selection, achieving an accuracy of 87.53%, AUC 0.87, and F1-score of 0.86, compared to basic tree algorithms like Greedy Tree and C4.5. While Tree-GAM showed high accuracy, it had a significantly longer runtime of approximately 101 seconds. FIGS and TAO Tree offered compelling alternatives with comparable accuracy but significantly reduced computational demands, with runtimes under 1 second. These findings highlight the trade-offs between predictive power, computational efficiency, and practical implementation in healthcare settings. The study also revealed the value of flexible, adaptive architectures in capturing nuanced factors influencing patient no-shows. Overall, these advanced algorithms present accurate and interpretable solutions for forecasting patient no-shows, with FIGS and TAO Tree emerging as particularly effective choices that offer a good balance between predictive insight and practical viability. These insights aim to guide health systems in optimizing patient access and reliability while addressing the complex issue of no-shows, underscoring the importance of considering multiple performance metrics when selecting algorithms for real-world applications.