RT journal article
T1 The multi-commodity flow problem with outsourcing decisions
A1 Fernández Areizaga, Elena
A1 Ljubic, Ivana
A1 Zerega Oyarzun, Nicolás
A2 Estadística e Investigación Operativa
K1 Multicommodity flows
K1 Bilevel optimization
K1 Network optimization
K1 Outsourcing
AB We introduce a new prize-collecting problem involving the routing of commodities in a networkcomposed of predefined hub and non-hub nodes. In this setting, the service of non-hub nodesis outsourced to third-party carriers. This study is motivated by the growing prevalence of outsourcing practices in real-world transportation systems, where new entrants or local suppliersincreasingly take over parts of the transportation chain – enabling higher volumes and broadernetwork coverage. The problem is modeled as a Stackelberg game: a major firm (the leader)chooses a subset of commodities to serve. The leader aims to outsource the first and last legs ofthe transportation services to smaller carriers (the followers), assigning at most one carrier to eachnon-hub node. Each carrier seeks to maximize its own profit, which depends on the offers madeby the leader. The goal of the leader is to determine the optimal outsourcing fees, along with theallocation of carriers to the non-hub nodes, so that the profit from the routed commodities is maximized. The optimal response of the followers must be taken into account, as the followers mayreject assignments that do not align with their profit-maximizing goals. We also consider two alternative settings: one in which outsourcing fees are fixed, and another where carriers accept anyoffer yielding non-negative profit. We demonstrate that the set of feasible outsourcing fees canbe discretized and formulate the problem as single-level mixed-integer (non)linear program. Forall problem variants, we prove NP-hardness and conduct a computational study of various singlelevel MIP reformulations. Our benchmark instances cover various applications in transportation,such as air transportation, postal delivery and freight transportation. We assess the scalability ofproposed formulations and analyze the impact of different carrier reservation prices on solutionquality. Finally, by comparing the alternative problem settings, we extract valuable managerialinsights.
PB Elsevier
SN 0191-2615
YR 2025
FD 2025
LK http://hdl.handle.net/10498/38382
UL http://hdl.handle.net/10498/38382
LA eng
DS Repositorio Institucional de la Universidad de Cádiz
RD 07-nov-2026