A Firefly Algorithm for Mixed-Variable Optimization Based on Hybrid Distance Modeling

2026-03-25

AI Summary

This study addresses the challenge of optimizing mixed-variable problems involving continuous, ordinal, and categorical variables simultaneously—a scenario where existing swarm intelligence algorithms often struggle. To this end, the authors propose a Firefly Algorithm for Mixed Variables (FAmv), which introduces a novel unified mixed-distance model to naturally and cohesively handle heterogeneous variable types. By redefining the attraction mechanism based on this distance model, FAmv achieves a more accurate representation of the search space while maintaining an effective balance between exploration and exploitation. Comprehensive experiments demonstrate that FAmv either outperforms or is competitive with state-of-the-art algorithms on the CEC2013 mixed-variable benchmark suite and several real-world engineering design problems, thereby confirming its efficacy and practical applicability.

Abstract

Several real-world optimization problems involve mixed-variable search spaces, where continuous, ordinal, and categorical decision variables coexist. However, most population-based metaheuristic algorithms are designed for either continuous or discrete optimization problems and do not naturally handle heterogeneous variable types. In this paper, we propose an adaptation of the Firefly Algorithm for mixed-variable optimization problems (FAmv). The proposed method relies on a modified distance-based attractiveness mechanism that integrates continuous and discrete components within a unified formulation. This mixed-distance approach enables a more appropriate modeling of heterogeneous search spaces while maintaining a balance between exploration and exploitation. The proposed method is evaluated on the CEC2013 mixed-variable benchmark, which includes unimodal, multimodal, and composition functions. The results show that FAmv achieves competitive, and often superior, performance compared with state-of-the-art mixed-variable optimization algorithms. In addition, experiments on engineering design problems further highlight the robustness and practical applicability of the proposed approach. These results indicate that incorporating appropriate distance formulations into the Firefly Algorithm provides an effective strategy for solving complex mixed-variable optimization problems.

Problem

• mixed-variable optimization
• heterogeneous variables
• metaheuristic algorithms
• continuous and discrete variables
• optimization

Innovation

• Firefly Algorithm
• Mixed-variable Optimization
• Hybrid Distance Modeling
• Metaheuristic
• Heterogeneous Search Space