An inner approximation method to compute the weight set decomposition of a triobjective mixed-integer problem

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.