Solving combinatorial bi-level optimization problems using multiple populations and migration schemes

In many decision making cases, we may have a hierarchical situation between different optimization tasks. For instance, in production scheduling, the evaluation of the tasks assignment to a machine requires the determination of their optimal sequencing on this machine. Such situation is usually modeled as a Bi-Level Optimization Problem (BLOP). The latter consists in optimizing an upper-level (a leader) task, while having a lower-level (a follower) optimization task as a constraint. In this way, the evaluation of any upper-level solution requires finding its corresponding lower-level (near) optimal solution, which makes BLOP resolution very computationally costly. Evolutionary Algorithms (EAs) have proven their strength in solving BLOPs due to their insensitivity to the mathematical features of the objective functions such as non-linearity, non-differentiability, and high dimensionality. Moreover, EAs that are based on approximation techniques have proven their strength in solving BLOPs. Nevertheless, their application has been restricted to the continuous case as most approaches are based on approximating the lower-level optimum using classical mathematical programming and machine learning techniques. Motivated by this observation, we tackle in this paper the discrete case by proposing a Co-Evolutionary Migration-Based Algorithm, called CEMBA, that uses two populations in each level and a migration scheme; with the aim to considerably minimize the number of Function Evaluations (FEs) while ensuring good convergence towards the global optimum of the upper-level. CEMBA has been validated on a set of bi-level combinatorial production-distribution planning benchmark instances. The statistical analysis of the obtained results shows the effectiveness and efficiency of CEMBA when compared to existing state-of-the-art combinatorial bi-level EAs.