Abstract One way to reduce the computational burden of solving a bi-criteria optimization problem would be to obtain only a subset of its non-dominated points (NDPs) as representative for the entire collection. We introduce a new criterion (index) to evaluate the effectiveness of a representative subset of non-dominated points in the context of the Bi-criteria p-Median p-Dispersion problem (BpMD). A key difference between our proposed index and those previously introduced in the open literature is that in those indexes the distance between an arbitrary NDP and its nearest NDP in the representative subset is determined along the direction of both objective functions, while in our proposed index this distance is determined only along the objective function for which the representing NDP is worst than the represented NDP. Along this direction, the proposed index provides an upper bound on the relative gap between the representing NDP and any NDP that it represents. We then employ this index to design an algorithm to find a representative subset of NDPs for the problem BpMD and demonstrate its effectiveness through a computational experiment. Although we develop this index in the context of the problem BpMD, it is potentially applicable in the context of other bi-criteria optimization problems as well.

中文翻译：

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双准则 p 中值 p 色散问题中非支配点的代表性子集

摘要 减少解决双准则优化问题的计算负担的一种方法是仅获取其非支配点 (NDP) 的子集作为整个集合的代表。我们引入了一个新的标准（指数）来评估非支配点的代表性子集在双标准 p 中值 p 色散问题（BpMD）的背景下的有效性。我们提出的索引与之前在公开文献中介绍的那些索引之间的一个主要区别在于，在这些索引中，任意 NDP 与其在代表性子集中最近的 NDP 之间的距离是沿两个目标函数的方向确定的，而在我们提出的索引中，距离仅沿着表示 NDP 比表示的 NDP 最差的目标函数确定。沿着这个方向，提议的指数为代表的新民主党与其代表的任何新民主党之间的相对差距提供了上限。然后，我们使用该指数来设计一种算法，以找到问题 BpMD 的代表性 NDP 子集，并通过计算实验证明其有效性。尽管我们在问题 BpMD 的上下文中开发了该索引，但它也可能适用于其他双标准优化问题的上下文。