Multiobjective Decomposition of Positive Integer Matrix: Application to Radiotherapy

We consider the following problem: to decompose a positive integer matrix into a linear combination of binary matrices that respect the consecutive ones property. The positive integer matrix corresponds to fields giving the different radiation beams that a linear accelerator has to send throughout the body of a patient. Due to the inhomogeneous dose levels, leaves from a multi-leaf collimator are used between the accelerator and the body of the patient to block the radiations. The leaves positions can be represented by segments, that are binary matrices with the consecutive ones property. The aim is to find a decomposition that minimizes the irradiation time, and the setup-time to configure the multileaf collimator at each step of the decomposition. We propose for this NP-hard multiobjective problem a heuristic method, based on the Pareto local search method. Experimentations are carried out on different size instances and the results are reported. These first results are encouraging and are a good basis for the design of more elaborated methods.