Bi-level multi-objective combinatorial optimization using reference approximation of the lower level reaction

Abstract Bi-level optimization has gained a lot of interest during the last decade. This framework is suitable to model several real-life situations. Bi-level optimization problems refer to two related optimization tasks, each one is assigned to a decision level (i.e., upper and lower levels). In this way, the evaluation of an upper level solution requires the evaluation of the lower level. This hierarchical decision making necessitates the execution of a significant number of Function Evaluations (FEs). When dealing with a multi-objective optimization context, new complexities are added and imposed by the conflicting objectives and their evaluation techniques. In this paper, we aim to reduce the induced complexity using approximation techniques in order to obtain the lower level optimality. To this end, ideas from multi-objective optimization have been extracted, improved, and hybridized with evolutionary methods to build an efficient approach for Multi-objective Bi-Level Optimization Problems (MBLOPs). In this work, three techniques are suggested: (1) a complete lower level approximation Pareto front procedure, (2) a reference-based approximation selection procedure, and (3) a sub-set reference-based approximation selection one. The proposed variants are applied to a new multi-objective formulation of a well-known combinatorial problem integrating two systems in the supply chain management, namely, the Bi-level Multi Depot Vehicle Routing Problem (Bi-MDVRP). The statistical analysis demonstrates the efficiency of each algorithm according to a set of metrics. Indeed, a large number of savings are detected which confirms the efficiency of our proposals for solving combinatorial optimization problems.