Normative Fish Swarm Algorithm (NFSA) is a novel variant of the Artificial Fish Swarm Algorithm (AFSA) proposed in 2019, and has been proven effective in solving single-objective optimization problems. Inspired by the potential of NFSA, this paper proposes an evolutionary multi-objective optimization algorithm, namely “Multi-Objective Normative Fish Swarm Algorithm (MO-NFSA)”. However, due to the fact that NFSA was originally modeled for single-objective optimization, a certain degree of transformation, modification, and additional strategies (related to multi-objective optimization features) must be integrated into the original NFSA as part of MO-NFSA modeling for further extension. This article adopts a total of 15 multi-objective optimization test cases in any category of fixed-dimensional, non-fixed-dimensional (ZDT set) or scalable multi-objective (DTLZ set) optimization types. These multi-objective optimization test cases are used to compare the performance of MO-NFSA with other comparative algorithms. However, performance evaluation of multi-objective optimization is a daunting task. Their performance results can only be digitized through quality indicators (i.e., performance metrics), which are mainly tested from three different aspects of high-quality approximation: convergence, uniformity, and spread. Here, quality indicators including generation distance (GD), spacing (S), and spread-delta (∆/∆*) are used to check the quality performance of each algorithm based on the corresponding approximation. In the work, 20 simulations are run, and hence, 20 sets of data are collected. The collected results prove that MO-NFSA is superior to other comparison algorithms in all aspects of high-quality approximation. MO-NFSA can solve different types of multi-objective optimization problems.

中文翻译：

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MO-NFSA 用于解决无约束多目标优化问题

Normative Fish Swarm Algorithm (NFSA) 是 2019 年提出的人工鱼群算法 (AFSA) 的一种新变体，已被证明可以有效解决单目标优化问题。受NFSA潜力的启发，本文提出了一种进化多目标优化算法，即“多目标规范鱼群算法（MO-NFSA）”。但是，由于 NFSA 最初是为单目标优化建模的，因此必须将一定程度的转换、修改和附加策略（与多目标优化功能相关）作为 MO-NFSA 的一部分集成到原始 NFSA 中建模以进一步扩展。本文共采用15个任意类别的定维多目标优化测试用例，非固定维度（ZDT 集）或可扩展多目标（DTLZ 集）优化类型。这些多目标优化测试用例用于比较 MO-NFSA 与其他比较算法的性能。然而，多目标优化的性能评估是一项艰巨的任务。它们的性能结果只能通过质量指标（即性能指标）进行数字化，主要从三个不同的高质量逼近方面进行检验：收敛性、均匀性和扩散性。在这里，质量指标包括生成距离（GD）、间距（S）和扩展增量（Δ/Δ*）用于根据相应的近似来检查每种算法的质量性能。在这项工作中，运行了 20 次模拟，因此收集了 20 组数据。收集的结果证明 MO-NFSA 在高质量逼近的各个方面都优于其他比较算法。MO-NFSA 可以解决不同类型的多目标优化问题。