In this article, we use the hypervolume indicator as a scalarizing function for biobjective combinatorial optimization problems. In particular, we describe a generic solution approach that determines the nondominated set of a biobjective optimization problem by solving a sequence of hypervolume scalarizations with appropriate choices of the reference point. Moreover, this solution technique can also provide a compact representation of the efficient set that is a $(1-1/e)$-approximation to the optimal representation in terms of the hypervolume in an a priori manner. We illustrate these concepts for a particular variant of the biobjective knapsack problem and for a biobjective shortest path problem. Numerical results are presented for the former problem.

在本文中，我们使用超体积指标作为双目标组合优化问题的标量函数。特别地，我们描述了一种通用的解决方法，该方法通过用适当的参考点选择解决一系列超体积标化来确定双目标优化问题的非支配集。此外，这种求解技术还可以提供有效集合的紧凑表示，即$(1-1/\mathrm{电子})$- 以先验方式逼近超体积方面的最佳表示。我们为双目标背包问题的特定变体和双目标最短路径问题说明了这些概念。给出了前一个问题的数值结果。