The hub location and pricing problem under a tree topology: a bilevel formulation and a co-evolutionary algorithm

Abstract

This paper introduces the bilevel tree-of-hubs location and pricing problem, a multiple-allocation hub location problem in which both the design and pricing of a tree-shaped hub backbone network are determined. Two types of decision-makers are considered: a leader, who defines the network structure and sets usage prices, and a follower, who selects the optimal way to use the network for shipping commodities. The leader aims to maximize profit, while the follower seeks to minimize transportation costs. A bilevel optimization formulation is proposed and reformulated into a single-level equivalent model. To solve it efficiently, we develop a Co-Evolutionary Algorithm (Co-EA) and test it on three well-known datasets adapted to our problem. The novelty of the proposed Co-EA lies in the interaction between its co-evolving populations: rather than representing leader and follower solutions separately, each population is associated with a subset of the leader's decision variables. For each leader decision, the follower's optimal response is computed, ensuring bilevel feasibility. Extensive computational experiments demonstrate the effectiveness of the proposed Co-EA compared with general-purpose solvers applied to the reformulated model. The best Co-EA solutions are also used as warm-starts for the exact solver, confirming the quality of the heuristic results. Finally, managerial insights are derived regarding how the number of hubs affects the backbone network configuration. The paper concludes with directions for future research on bilevel hub location and pricing models.