Benders decomposition for bi-objective linear programs

In this paper, we develop a new decomposition technique for solving bi-objective linear programming problems. The proposed methodology combines the bi-objective simplex algorithm with Benders decomposition and can be used to obtain a complete set of efficient extreme solutions, and the corresponding set of extreme non-dominated points, for a bi-objective linear programme. Using a Benders-like reformulation, the decomposition approach decouples the problem into a bi-objective master problem and a bi-objective subproblem, each of which is solved using the bi-objective parametric simplex algorithm. The master problem provides candidate efficient solutions that the subproblem assesses for feasibility and optimality. As in standard Benders decomposition, optimality and feasibility cuts are generated by the subproblem and guide the master problem solve. This paper discusses bi-objective Benders decomposition from a theoretical perspective, proves the correctness of the proposed reformulation and addresses the need for so-called weighted optimality cuts. Furthermore, we present an algorithm to solve the reformulation and discuss its performance for three types of bi-objective optimisation problems.