Quantifying the Explore-Exploit Trade-off in Metaheuristics via Normalised Positional Diversity

Abstract

A continuous development of metaheuristics necessitates a deeper understanding of internal search mechanisms, specifically the transition between exploration and exploitation, which is often overlooked in favour of final fitness values. This study introduces a standardized framework based on a Normalized Positional Diversity Index (D*) to quantify optimizer behaviour. We demonstrate that D* is a geometric generalization of existing measures, such as the one proposed by Hussain et al. [1], by replacing stochastic, path-dependent historical maximums with a fixed global upper bound anchored to the search space geometry. This transformation eliminates post-hoc bias and provides a scale-independent diagnostic. From D*, additional metrics including Area Under the Curve (AUC) and crossover point (txover) are derived to identify exact phase transitions. We evaluate five optimizers (GWO, DE, PSO, AGOA, and EES) across CEC 2020 benchmarks. Results confirm that D* offers consistent diversity monitoring with lower Coefficients of Variation compared to existing measures. Code and supplementary materials are available at: [https://github.com/saiddddd/D-Star-Metric-Optimization].