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.

翻译：现实世界的若干优化问题涉及混合变量搜索空间，其中连续变量、有序变量和分类决策变量共存。然而，大多数基于种群的元启发式算法是为连续或离散优化问题设计的，无法自然处理异构变量类型。本文提出了一种适用于混合变量优化问题的改进萤火虫算法（FAmv）。所提方法依赖于改进的基于距离的吸引力机制，该机制将连续分量和离散分量整合到统一公式中。这种混合距离方法能够对异构搜索空间进行更适当的建模，同时保持探索与开发之间的平衡。所提方法在CEC2013混合变量基准测试集上进行了评估，该测试集包含单峰、多峰和复合函数。结果表明，与当前最先进的混合变量优化算法相比，FAmv算法取得了具有竞争力且通常更优的性能。此外，在工程设计问题上的实验进一步凸显了所提方法的鲁棒性和实际适用性。这些结果表明，将适当的距离公式引入萤火虫算法为求解复杂混合变量优化问题提供了一种有效策略。