Mathematical Methods of Operations Research

, Volume 68, Issue 2, pp 257–276

Approximately solving multiobjective linear programmes in objective space and an application in radiotherapy treatment planning

• Lizhen Shao
• Matthias Ehrgott
Original Article

Abstract

In this paper, we propose a modification of Benson’s algorithm for solving multiobjective linear programmes in objective space in order to approximate the true nondominated set. We first summarize Benson’s original algorithm and propose some small changes to improve computational performance. We then introduce our approximation version of the algorithm, which computes an inner and an outer approximation of the nondominated set. We prove that the inner approximation provides a set of $${\varepsilon}$$-nondominated points. This work is motivated by an application, the beam intensity optimization problem of radiotherapy treatment planning. This problem can be formulated as a multiobjective linear programme with three objectives. The constraint matrix of the problem relies on the calculation of dose deposited in tissue. Since this calculation is always imprecise solving the MOLP exactly is not necessary in practice. With our algorithm we solve the problem approximately within a specified accuracy in objective space. We present results on four clinical cancer cases that clearly illustrate the advantages of our method.

Keywords

Multiobjective linear programming Radiotherapy treatment planning $${\varepsilon}$$-efficient solution

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