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

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.

Figures (15)

• Figure 1: Overall performance of MVO-based methods on CEC benchmark functions: baseline algorithms (left) and FA-based variants (right). The x-axis shows the number of results statistically similar to the best, and the y-axis shows the number of best results. Algorithms in the top-right achieve the best performance. See Tables\ref{['tab:Results_CEC2013']} and \ref{['tab:Ablation_Results_CEC2013']} for details.
• Figure 2: Performance in terms of the absolute error (the lower the better) scored by the seven mixed-variable optimization methods on the CEC benchmark for the unimodal functions.
• Figure 3: Performance in terms of the absolute error (the lower the better) scored by the seven mixed-variable optimization methods on the CEC benchmark for the multimodal functions.
• Figure 4: Performance in terms of the absolute error (the lower the better) scored by the seven MVO methods on composite functions in the CEC benchmark.
• Figure 5: Performance of MVO methods on the three engineering design problems (BEAM, CSD, and Vessel), measured by absolute error (top) and convergence curves (bottom). Lower values indicate better performance.