Solving bilevel multi-objective programming problems is one of the hardest tasks facing researchers in the optimization community. Bilevel multi-objective programming problems is an optimization problem consists of two interconnected hierarchical multi-objective programming problems: upper-level problem and lower-level problem. Difficulty in solving bilevel multi-objective programming problems is the need to solve lower-level multi-objective programming problem to know the feasible space of the upper-level problem. The proposed algorithm consists of two nested artificial multi-objective algorithms. One algorithm is for the upper-level problem and the other is for the lower-level problem. Also, the proposed algorithm is enriched with a k-means cluster scheme in two phases. The first phase is before starting two nested algorithms to help the algorithm to start with more appropriates solutions to the bi-level problem. The second phase is within the two nested algorithms to guide the algorithm to the most preferred solutions to the upper-level decision-maker. The performance of the proposed algorithm has been evaluated on different test problems including low dimension and high dimension test problems. The experimental results show that the proposed algorithm is a feasible and efficient method for solving the bilevel multi-objective programming problem.

解决双层多目标规划问题是优化社区研究人员面临的最艰巨的任务之一。双层多目标规划问题是一个优化问题，由两个相互关联的分层多目标规划问题组成：上层问题和下层问题。求解双层多目标规划问题的难点在于需要求解下层多目标规划问题，知道上层问题的可行空间。所提出的算法由两个嵌套的人工多目标算法组成。一种算法用于上层问题，另一种算法用于下层问题。此外，所提出的算法在两个阶段丰富了 k-means 聚类方案。第一阶段是在启动两个嵌套算法之前，帮助算法从更合适的双层问题解决方案开始。第二阶段是在两个嵌套算法内引导算法为上层决策者提供最优选的解决方案。所提出算法的性能已经在不同的测试问题上进行了评估，包括低维和高维测试问题。实验结果表明，该算法是解决双层多目标规划问题的一种可行且有效的方法。所提出算法的性能已经在不同的测试问题上进行了评估，包括低维和高维测试问题。实验结果表明，该算法是解决双层多目标规划问题的一种可行且有效的方法。所提出算法的性能已经在不同的测试问题上进行了评估，包括低维和高维测试问题。实验结果表明，该算法是解决双层多目标规划问题的一种可行且有效的方法。