A bi-objective sub-path location model for the flow refuelling location problem

• This paper presents and solves a case of the Flow Refueling Location Problem (FRLP). • New formulation and four exact cover algorithms are presented and compared. • A New bi-objective algorithm, the Smoothest Descent Algorithm (SDA), is presented. • SDA was tested in 26 new instances derived from a real road transportation network. • Finds multiple optimal Pareto and evaluates efficiency with hyper-volumes and time. Strategic placement of alternative-fuel refuelling stations is a critical challenge for energy and transportation planners, who must navigate conflicting objectives, such as minimising capital costs and maximising service coverage. This paper presents a decision support system for the bi-objective Flow Refuelling Location Problem (FRLP) built upon the Sub-path Flow Refuelling Location Model (SPFRLM). This formulation distinguishes itself by enabling continuous facility siting along edges, managed through a dynamic separation procedure for sub-path constraints. To generate solutions efficiently, the system incorporates the Smoothest Descent Algorithm (SDA), a bi-objective method that approximates the Pareto front by dynamically switching between minimization and maximization strategies. The SDA relies on a Two-Phase Hybrid (TPH) algorithm that integrates Cut-and-Solve and Branch-and-Cut to solve the underlying sub-problems. We validate the system on a newly introduced library of 26 real-world test instances. The results demonstrate that the proposed approach captures approximately 97% of the optimal hyper-volume while requiring less than 10% of the computational time of exact methods. These findings confirm that the system is a powerful tool for stakeholders, providing rapid and accurate guidance for the strategic deployment of future energy infrastructure.