Exact solution algorithms for biobjective mixed integer programming problems

Abstract

We consider criterion space algorithms for biobjective mixed integer programs. The algorithms solve scalarization models in order to explore predetermined regions of the objective space called boxes, defined by two nondominated points. When exploring, the algorithm exploits information on its corner points and chooses the scalarization problem accordingly so as to detect line segments quickly, without having to solve many scalarizations. We propose three algorithms: The first one creates new boxes immediately when it finds a nondominated point, whereas the second algorithm conducts additional operations after obtaining a nondominated point by the Pascoletti–Serafini scalarization. The third algorithm is another variant that uses the computational advantage of dichotomic search whenever possible. Our computational experiments demonstrate the computational feasibility of the algorithms and show that the number of mixed integer linear programming models is significantly lower compared to similar approaches in the literature. The results further validate the utilization of Pascoletti–Serafini scalarization, aimed at enhancing the representativeness of solutions under time and cardinality limits. We observe that the third variant is particularly effective in finding a representative subset of the nondominated solutions under such limits.