A Novel Co-Evolutionary Algorithm for Solving a Bilevel Pricing and Hubs Location Problem under a Tree Topology

TL;DR

The paper tackles the Bilevel Tree-of-Hubs Location Problem with Prices (BTHLPwP), where a leader designs a tree-shaped hub backbone with $p$ hubs and sets prices $\pi$ to maximize profit, while followers route commodities to minimize costs, potentially using a third-party service. It develops a bilevel MILP and an equivalent single-level reformulation, but given computational intractability, introduces a novel Co-Evolutionary Algorithm that partitions the leader's decisions into two subpopulations (tree design and pricing) with nested follower resolution via shortest-paths. Computational experiments on adapted AP, CAB, and TR datasets demonstrate that the Co-EA delivers high-quality solutions rapidly and can effectively warm-start the exact reformulation to improve solver performance. The approach advances pricing-enabled hub location under tree topologies and highlights the value of nested co-evolution for bilevel problems in logistics networks.

Abstract

This paper introduces the Bilevel Tree-of-Hubs Location Problem with Prices (BTHLPwP). The BTHLPwP is a multiple-allocation hub location problem in which, in addition to determining the nodes and links of a tree-shaped hub backbone network, the prices for using this network must also be set. We assume that two different types of agents make decisions in this problem. On the one hand, one agent (the leader) determines the structure and sets the prices for using the hub backbone network. On the other hand, the other agent (follower) decides on the optimal usage of the network. The leader seeks to maximize its profit, while the follower aims to minimize the costs incurred for using the network to ship their commodities. We present a bilevel optimization formulation for this problem, followed by an equivalent single-level reformulation. Then, we propose a novel Co-Evolutionary Algorithm (Co-EA) to solve three well-known datasets of instances adapted for our problem. The main novelty of the proposed Co-EA lies in the way the co-evolving populations are considered. While traditionally one population focuses on the leader's solutions and the other on the follower's, in our approach, each population is associated with a subset of the leader's decision variables. Consequently, the follower's optimal reaction is obtained for a specific decision made by the leader, resulting in bilevel feasible solutions. We then analyze the results obtained from extensive computational experimentation using the proposed Co-EA.

Figures (9)

• Figure 1: Illustrative example showcasing the decision-making process in the BTHLPwP.
• Figure 2: Flowchart for the proposed Co-EA
• Figure 3: Example with 5 potential sites and 3 hubs to be activated.
• Figure 4: Example of a pricing solution: Each entry represents the price of using a specific arc. For instance, if the follower uses the arc (3,2), then a cost of 8 is incurred.
• Figure 5: Illustration of the evolutionary operators applied to the tree-structured networks with $p=3$.