This paper explores the combination of a classic mathematical function named “hyperbolic tangent” with a metaheuristic algorithm, and proposes a novel hybrid genetic algorithm called NSGA-II-BnF for multi-objective decision making. Recently, many metaheuristic evolutionary algorithms have been proposed for tackling multi-objective optimization problems (MOPs). These algorithms demonstrate excellent capabilities and offer available solutions to decision makers. However, their convergence performance may be challenged by some MOPs with elaborate Pareto fronts such as CFs, WFGs, and UFs, primarily due to the neglect of diversity. We solve this problem by proposing an algorithm with elite exploitation strategy, which contains two parts: first, we design a biased elite allocation strategy, which allocates computation resources appropriately to elites of the population by crowding distance-based roulette. Second, we propose a self-guided fast individual exploitation approach, which guides elites to generate neighbors by a symmetry exploitation operator, which is based on mathematical hyperbolic tangent function. Furthermore, we designed a mechanism to emphasize the algorithm’s applicability, which allows decision makers to adjust the exploitation intensity with their preferences. We compare our proposed NSGA-II-BnF with four other improved versions of NSGA-II (NSGA-IIconflict, rNSGA-II, RPDNSGA-II, and NSGA-II-SDR) and four competitive and widely-used algorithms (MOEA/D-DE, dMOPSO, SPEA-II, and SMPSO) on 36 test problems (DTLZ1–DTLZ7, WGF1–WFG9, UF1–UF10, and CF1–CF10), and measured using two widely used indicators—inverted generational distance (IGD) and hypervolume (HV). Experiment results demonstrate that NSGA-II-BnF exhibits superior performance to most of the algorithms on all test problems.

中文翻译：

---

精英开发：数学概念和EMO方法的组合，用于多目标决策

本文探索了经典的数学函数“双曲正切”与元启发式算法的结合，并提出了一种新颖的混合遗传算法NSGA-II-BnF，用于多目标决策。最近，已经提出了许多元启发式进化算法来解决多目标优化问题（MOP）。这些算法展示了出色的功能，并为决策者提供了可用的解决方案。但是，它们的融合性能可能会受到某些具有精心设计的Pareto前沿的MOP的挑战，例如CF，WFG和UF，这主要是由于人们对多样性的忽视。为了解决这个问题，我们提出了一种采用精英开发策略的算法，该算法包括两部分：首先，我们设计了一种有偏见的精英分配策略，通过拥挤基于距离的轮盘，将计算资源适当地分配给人口精英。其次，我们提出了一种自我指导的快速个体开发方法，该方法基于对称双曲正切函数，通过对称的开发算符来引导精英生成邻居。此外，我们设计了一种机制来强调算法的适用性，该机制允许决策者根据自己的喜好调整利用强度。我们将我们提出的NSGA-II-BnF与其他四个改进版本的NSGA-II（NSGA-IIconflict，rNSGA-II，RPDNSGA-II和NSGA-II-SDR）以及四种竞争性且广泛使用的算法（MOEA / D）进行了比较-DE，dMOPSO，SPEA-II和SMPSO）处理36个测试问题（DTLZ1-DTLZ7，WGF1-WFG9，UF1-UF10和CF1-CF10），并使用两个广泛使用的指标进行测量-逆世代距离（IGD）和超体量（HV）。实验结果表明，在所有测试问题上，NSGA-II-BnF的性能均优于大多数算法。