Flow Refuelling-Location Model (FRLM)

• FRLM is a combinatorial optimization framework that models the strategic placement of alternative-fuel stations to ensure vehicles with limited range can complete designated trips.
• It integrates various formulations—including deterministic, multi-path, continuous-space, and capacitated models—to address operational constraints and cost-coverage trade-offs.
• The model employs mixed-integer linear programming and advanced algorithms such as edge-scanning, branch-and-cut, and metaheuristics to enhance scalability and resilience.

The Flow Refuelling-Location Model (FRLM) is a class of combinatorial optimization models formulated to guide the strategic placement of alternative-fuel refueling stations (AFSs) in transportation networks. The paradigm ensures that vehicles—typically with limited driving range—have refueling opportunities sufficient to complete designated trips, subject to operational and demand-side constraints. FRLM variants underpin planning for electric vehicle (EV) charging, hydrogen fueling, and other alternative-fuel infrastructure, and encompass a spectrum of mathematical, algorithmic, and network-geometric generalizations.

1. Canonical Formulation: Problem Definition and Model Structure

The classical FRLM, as originally proposed by Kuby and Lim (2005) and developed in subsequent works, is defined on a spatial network $G=(V,E)$, where $V$ is a set of nodes (e.g., intersections, candidate sites) and $E$ is a set of edges (e.g., road segments). The principal objective is to site the minimum number or cost of refueling stations such that a prescribed collection of vehicle flows, usually represented as origin–destination (O–D) pairs or looped routes, are "covered": that is, every trip along each flow can be partitioned into legs, none exceeding a technology-determined driving range $R$, with refueling available at the endpoints of each leg.

There are several canonical FRLM problem classes depending on how coverage, flow structure, and capacity are modeled:

• Deterministic FRLM (DFRLM): Maximizes the volume of flows covered or minimizes the number/cost of stations, assuming homogeneous driving range and fixed candidate locations (Kastner et al., 2021).
• Multi-path and Deviation FRLM: Allows vehicles to utilize multiple feasible paths (including non-shortest or detour paths) between O–D pairs, often precomputing several $k$-shortest paths per pair (Li et al., 2018, Nayeem et al., 14 Sep 2025).
• Continuous-space FRLM: Permits station siting at any point on an edge, subsequently reduced to a finite dominating set (FDS) via coverage geometry (Nayeem et al., 14 Sep 2025).
• Capacitated FRLM: Incorporates hard capacity constraints on station throughput, usually requiring joint location–sizing optimization (Kastner et al., 2021, Nordlund et al., 2023, Luo et al., 2024).
• Stochastic and Probability-weighted FRLM: Weights O–D demand by adoption or risk probability, explicitly maximizing expected system-wide coverage (Li et al., 2018).

Key variables typically include binary station-placement decisions at candidate sites, flow-covering indicators for each O–D pair or path, and, in capacitated versions, integer variables for the number of refueling units per station.

2. Coverage Constraints, Geometric Generalizations, and the FDS Approach

Coverage in the FRLM is enforced by constraints ensuring no segment of a covered path exceeds the specified range $R$ without access to a station. In classical models, this induces a set-cover or path-covering structure: all feasible traversals must be "interdicted" by station placements at intervals no longer than $R$.

A significant generalization is provided by the continuous-space FDS approach (Nayeem et al., 14 Sep 2025), where station locations are initially modeled as lying anywhere on the network (not just discrete nodes). For a fixed set of dedicated closed routes $H=\{U_1,\dots,U_h\}$, each with route length $\ell(U_t)$ and daily flow $f(U_t)$, coverage of $V$0 by a candidate station $V$1 is feasible if:

1. The total round-trip length, including detour to $V$2, does not exceed $V$3.
2. Deviation from the route to $V$4 does not exceed threshold $V$5.

Mathematically, define $V$6 (deviation point on the route) and characterize the feasible "refueling set" $V$7 of locations $V$8 on each edge $V$9. The union over all route–edge pairs is finite (≤4 segments per edge & route), and, by Theorem 1 and 2 in (Nayeem et al., 14 Sep 2025), a solution exists among the finite set $E$0 of endpoints—this is a finite dominating set (FDS). Thus, the original infinite candidate space is reduced to a tractable discrete model without loss of optimality.

3. Mathematical Programming and Algorithmic Developments

The majority of FRLM variants are formulated as mixed-integer linear programs (MILP or MIP):

• Vertex covering: $E$1, subject to $E$2 for all covered flows ($E$3 binary station-open variables; $E$4 coverage indicator) (Nayeem et al., 14 Sep 2025, Kastner et al., 2021).
• Flow conservation & path feasibility: For multi-path/flow models, variables for refueling at each site along feasible paths, subject to range and flow continuity constraints (Li et al., 2018, Luo et al., 2024).
• Capacity constraints: Upper bounds on the flow or number of vehicles serviced at each selected station, resulting in additional knapsack or assignment constraints (Kastner et al., 2021, Nordlund et al., 2023, Luo et al., 2024).
• Reinforced/redundant coverage: Bilevel models integrating public-sector minimization of uncovered nodes and private-sector maximization of utilization, with redundancy constraints (e.g., each station has a backup within range) (Piedra-de-la-Cuadra et al., 31 Jan 2025).

Solution strategies include:

• Edge-Scanning (ES) algorithms: Polynomial-time procedures for computing FDS endpoints in continuous-space variants (Nayeem et al., 14 Sep 2025).
• Branch-and-cut and branch-cut-and-price: Advanced decomposition algorithms for solving capacitated and large-scale FRLM models, integrating separation of coverage and capacity cuts, column generation, and dual resource-constrained path pricing (Nordlund et al., 2023).
• Metaheuristics: Genetic algorithms for large, NP-hard multi-path or stochastic FRLMs, using fitness functions based on MILP relaxations (Li et al., 2018).

Empirically, computational runtimes are typically polynomial in network size for unconstrained models, but increase by 1–2 orders of magnitude for tight capacity or redundancy requirements (Nordlund et al., 2023). For moderately sized instances (hundreds of nodes), optimal solutions are usually attainable; for very large networks, hybrid or metaheuristic algorithms provide near-optimal results.

4. Extensions: Capacity, Cost, Coverage, and Redundancy

Several research strands have introduced problem enrichments reflecting practical and policy-driven considerations:

• Capacity-constrained FRLM: Realistically models limited throughput at stations (charging rate, server count), introducing the need for joint location-and-sizing optimization. Key constraints ensure that the sum of flows assigned to a station does not exceed its capacity. Empirical results emphasize the necessity of carefully distributing capacity to high-traffic nodes to avoid underutilization (Kastner et al., 2021, Nordlund et al., 2023, Luo et al., 2024).
• Minimum coverage and cost-coverage trade-offs: The minimum coverage FRLM seeks to minimize station installations while guaranteeing fractional flow coverage. Results demonstrate diminishing returns beyond ~70–80% coverage, with total station count growing sharply if absolute coverage is mandated (Kastner et al., 2021).
• Location-dependent construction costs: Incorporating heterogeneous installation costs (urban/rural, local incentives) produces nontrivial spatial patterns in optimal siting solutions, sensitive to cost differentials and budget constraints (Kastner et al., 2021).
• Redundancy (reinforced coverage): Bilevel models ensure each station has backup within range, thus ensuring resilience to single-station failure and enhancing network robustness. Leader–follower structures permit explicit trade-offs between public viability (minimal stations) and private sector utilization (flow-weighted demand) (Piedra-de-la-Cuadra et al., 31 Jan 2025).
• Stochasticity and probabilistic demand: Weighting nodes or flows by probability of EV adoption enables expected-value optimization, reflecting spatial heterogeneity in uptake (Li et al., 2018). Embedding probabilities can shift optimal allocations toward high-adoption areas, with substantial differences observed in empirical studies.

5. Practical Applications and Computational Experience

FRLMs have been applied to synthetic benchmarks and real networks, including the Sioux Falls, San Antonio, and Chicago grids (Nayeem et al., 14 Sep 2025, Li et al., 2018, Luo et al., 2024). Notable applications include:

• Urban and inter-city planning: Multi-scale (macro–meso–micro) hierarchical grid frameworks enable tractable and robust planning for large metropolitan regions, supporting city- and corridor-level deployment (Luo et al., 2024).
• Public transportation and fleet operations: Dedicated-loop FRLMs serve transit and logistics networks operated by AFVs; detour-permitting models account for allowable off-route refueling (Nayeem et al., 14 Sep 2025).
• Rollout policy analysis: Minimum coverage and cost-coverage trade-off studies inform regulatory and subsidy decisions by quantifying marginal coverage gains against infrastructure investment (Kastner et al., 2021).

Empirical findings repeatedly demonstrate that small increases in vehicle range or allowable detour distance can strongly reduce required station counts and increase solution overlap between routes (Nayeem et al., 14 Sep 2025). Station utilization rates and coverage efficiency depend crucially on judicious sizing and siting, with diminishing returns for excessive infrastructure in low-demand areas.

6. Limitations, Model Generalizations, and Future Research Directions

Several research fronts remain active in the FRLM domain:

• Class heterogeneity: Most models assume homogeneous vehicle range $E$5 and deviation $E$6. Multi-class models, where $E$7 and $E$8 vary across vehicle types, require constructing class-specific or combined FDSs, substantially increasing model complexity (Nayeem et al., 14 Sep 2025).
• Station capacity and queueing: Many analyses assume unlimited capacity; incorporating finite capacities and endogenous delays yields larger, potentially nonlinear programs, with challenging implications for computational tractability (Kastner et al., 2021, Nordlund et al., 2023, Luo et al., 2024).
• Multiple detour and dynamic deviation models: Most current models permit at most one allowed detour per refueling; real-world dynamic or repeated deviation strategies remain underexplored (Nayeem et al., 14 Sep 2025).
• Temporal and uncertainty modeling: Integration with time-expanded networks and dynamic adoption forecasts (e.g., using ODE-based diffusion models) is emerging as a vital area, especially for infrastructure staged over multi-year horizons and under uncertainty (Luo et al., 2024).
• Scalability: For very large networks and strict capacity constraints, hybrid decomposition and advanced column-generation/routing heuristics are essential to keep computation tractable (Nordlund et al., 2023).
• Redundancy and resilience: Systematic consideration of multi-station redundancy, cascading failures, and repair processes is an open domain with significant policy implications (Piedra-de-la-Cuadra et al., 31 Jan 2025).

An emphasis on rigorous geometric generalization and scalable discrete-continuous reductions (via FDS) remains central for bridging practical requirements with global optimality guarantees, particularly in time-sensitive and real-time operational settings (Nayeem et al., 14 Sep 2025).