A linear partitioning diversity metric for evaluation of permutation-based metaheuristic algorithms

Majid Shahbazi, Faramarz Safi-Esfahani, Sara Shahbazi, Seyedali Mirjalili

Abstract: Metaheuristic algorithms, implicitly or explicitly, typically aim to achieve a delicate balance between exploration and exploitation when improving a population of solutions for a given optimization problem. Diversity metrics are crucial indicators for both exploration and exploitation, which serves as a valuable tool for assessing the performance of metaheuristic algorithms. Although several metrics for diversity have been proposed, the majority of them are only suitable for problems with continuous space. However, many optimization problems are defined/modeled based on discrete variables, and permutation representation is widely used for modeling real-world problems. Many existing metrics are not applicable to permutation-based problems. Applying other existing metrics to such problems frequently encounters two primary drawbacks: computational cost and effectiveness to represent the actual diversity of the population. To address these drawbacks, a new exclusive diversity metric is proposed in this work for optimization problems modeled by permutations. This method calculates diversity by linearizing and partitioning the problem space based on the population distribution in the space and counting the number of individuals in the partitions. Compared to the eleven available metrics (including both permutation-based and usable non-permutation-based metrics) in the literature, the proposed method demonstrates superior performance, exhibiting acceptable accuracy, stability, sensitivity, and robustness, particularly in the presence of outliers and requiring minimal computational cost. The accuracy of the results of the method is, on average, 35% better than baseline metrics. The computational cost of the method is also nearly 50% better than the fastest available method. These characteristics make the metric a valuable mean for analyzing existing algorithms’ behavior or designing new ones.