Cost-Based Filtering for Determining the Pareto Frontier

Real life problems involve seldom only one criterion, but multiple criteria should be optimised at the same time. The criteria can even be conflicting each other, so the classical techniques used in single criteria optimization cannot be simply reused. Many ideas have been presented in the literature for addressing multiple criteria optimization problems; one of the ideas is to provide the so-called Pareto frontier, i.e., the set of solutions that are not dominated by other feasible solutions. The Pareto frontier provides a lot of information to the decision-maker, that could be used to select the best preferred solution. On the other hand, finding the Pareto frontier is a time-consuming task. In single-criteria optimization, integration of Constraint Programming and Operations Research techniques has often been a successful approach to tackle difficult problems. Information derived by a solver that handles the linear relaxation of the original problem (like lower bounds, reduced costs, dual solution) has been used to improve the bounding and pruning capabilities of a Constraint Programming solver. In this paper, we propose an approach that integrates linear programming in Constraint Programming to speedup the search process in multiple-criteria optimization. We extend a Constraint Programming algorithm for finding the Pareto frontier, integrate it with a linear solver that provides bounds and reduced costs. Promising preliminary experiments show the effectiveness of the approach.