Surrogate-assisted evolutionary algorithms for a bilevel location and latency-oriented routing problem

Hierarchies among different stakeholders within a supply chain are common and should not be overlooked. In this study, we address both location and routing decisions within the supply chain framework. Specifically, we focus on a problem inspired by a real-life situation involving two stakeholders: one (the leader) responsible for determining the location and size of depots, and another (the follower) responsible for delivering products to customers. The leader aims to minimize costs, while the follower seeks to minimize latency, which is interpreted as the waiting time of customers along the routes. To address this hierarchical situation, we propose a novel bilevel optimization model. The complexity of this model, which includes both binary and continuous variables at each level and features high dimensionality due to a multi-level network accounting for modeling customers’ latency, precludes the use of a single-level reformulation. Therefore, we propose an evolutionary algorithm to solve the bilevel problem. Given the challenging nature of the follower's problem, a classical nested approach would be excessively time-consuming. Thus, we employ surrogate methods to approximate the latency-oriented routing decision process, integrating them into the evolutionary algorithm's framework. This approach provides an effective means of addressing the complexities while maintaining the feasibility of the bilevel solutions. The surrogate strategy is based on a committee of learning models trained on limited data from bilevel feasible solutions. Several variants are studied and compared against state-of-the-art surrogate algorithms, obtaining better results with less computational time for the problem under study.