This article is dedicated to the weight set decomposition of a multiobjective (mixed-)integer linear problem with three objectives. We propose an algorithm that returns a decomposition of the parameter set of the weighted sum scalarization by solving biobjective subproblems via Dichotomic Search which corresponds to a line exploration in the weight set. Additionally, we present theoretical results regarding the boundary of the weight set components that direct the line exploration. The resulting algorithm runs in output polynomial time, i.e. its running time is polynomial in the encoding length of both the input and output. Also, the proposed approach can be used for each weight set component individually and is able to give intermediate results, which can be seen as an “approximation” of the weight set component. We compare the running time of our method with the one of an existing algorithm and conduct a computational study that shows the competitiveness of our algorithm. Further, we give a state-of-the-art survey of algorithms in the literature.

本文致力于具有三个目标的多目标（混合）整数线性问题的权重分解。我们提出了一种通过Dichotomic Search解决双目标子问题的算法，该算法返回加权和标量化参数集的分解对应于权重集中的线探索。此外，我们提出了有关指导线探索的权重集组件边界的理论结果。所得算法以输出多项式时间运行，即，其运行时间为输入和输出的编码长度的多项式。同样，所提出的方法可以单独用于每个重量设定分量，并且能够给出中间结果，这可以看作是重量设定分量的“近似值”。我们将我们的方法的运行时间与现有算法之一进行比较，并进行了计算研究，证明了我们算法的竞争力。此外，我们对文献中的算法进行了最新的综述。