Pareto dominance-based multiobjective evolutionary algorithms use non-dominated sorting to rank their solutions. In the last few decades, various approaches have been proposed for non-dominated sorting. However, the running time analysis of some of the approaches has some issues and they are imprecise. In this paper, we focus on one such algorithm namely hierarchical non-dominated sort (HNDS), where the running time is imprecise and obtain the generic equations that show the number of dominance comparisons in the worst and the best case. Based on the equation for the worst case, we obtain the worst-case running time as well as the scenario where the worst case occurs. Based on the equation for the best case, we identify a scenario where HNDS performs less number of dominance comparisons than that presented in the original paper, making the best-case analysis of the original paper unrigorous. In the end, we present an improved version of HNDS which guarantees the claimed worst-case time complexity by the authors of HNDS which is \({\mathcal {O}}(MN^2)\).

基于帕累托支配的多目标进化算法使用非支配排序来对其解决方案进行排序。在过去的几十年中，人们提出了各种非支配排序方法。然而，一些方法的运行时间分析存在一些问题并且不精确。在本文中，我们重点研究一种这样的算法，即分层非支配排序（HNDS），该算法的运行时间不精确，并获得了显示最坏和最好情况下的支配比较次数的通用方程。根据最坏情况的方程，我们得到最坏情况的运行时间以及最坏情况发生的场景。基于最佳情况的方程，我们确定了一种场景，其中 HNDS 执行的优势比较次数少于原始论文中提出的数量，从而使原始论文的最佳情况分析不严格。最后，我们提出了 HNDS 的改进版本，它保证了 HNDS 作者声称的最坏情况时间复杂度，即\({\mathcal {O}}(MN^2)\)。