Flow-Refueling Location Model

• Flow-Refueling Location Model (FRLM) is a path-based optimization approach ensuring vehicles can complete trips by placing refueling/charging facilities along preselected paths under range constraints.
• It utilizes set-covering formulations and MILP techniques to minimize costs or maximize flow coverage while accounting for budget, capacity, and spacing restrictions.
• Recent extensions incorporate multi-path routing, continuous candidate-site selection, and coupled transportation–power planning to enhance electric vehicle charging network design.

The Flow-Refueling Location Model (FRLM) is a path-based optimization model for siting refueling or charging facilities on a transportation network for vehicles with limited range. In the formulation studied in recent literature, the Fuel Refueling Location Problem is the same optimization goal and modeling framework as the FRLM introduced by Kuby and Lim (2005): candidate stations are selected so that vehicles traveling on preselected origin–destination paths, or on sets of feasible deviation paths, can complete trips under a budget on the number or cost of facilities (Sugishita et al., 2024). Classical FRLM is static and typically uncapacitated, but subsequent work has extended it to deterministic path-segment MILPs, continuous candidate-site spaces, multi-path and probability-weighted coverage, station sizing, bilevel reliability requirements, robust time-expanded charging networks, and coupled transportation–power planning for electric vehicles (Kastner et al., 2021).

1. Canonical model structure

In its classical form, FRLM is defined on a graph $G=(N,A)$ with candidate facility sites $N^s\subseteq N$, a set of OD flows or path flows, path-specific weights, and a vehicle range $R$. A station-opening variable $x_j\in\{0,1\}$ indicates whether a facility is placed at site $j$. The central feasibility logic is that the spacing between consecutive refueling opportunities, including the path ends, must not exceed $R$. One canonical set-covering formulation minimizes infrastructure cost while forcing every relevant subpath segment $B_{p,k}$ to contain at least one station: $$\begin{array}{ll} \text{minimize} & \displaystyle \sum_{j \in \mathcal{N}^s} o_j\, x_j \ \text{subject to} & \displaystyle \sum_{j \in B_{p,k}} x_j \ge 1 \qquad \forall p \in \mathcal{P},\ \forall k \in \mathcal{K}_p, \ & x_j \in \{0,1\} \qquad \forall j \in \mathcal{N}^s . \end{array}$$ A budget-limited variant instead maximizes covered flow: $$\begin{array}{ll} \text{maximize} & \displaystyle \sum_{p \in \mathcal{P}} w_p\, y_p \ \text{subject to} & y_p \le \displaystyle \sum_{j \in B_{p,k}} x_j \qquad \forall p \in \mathcal{P},\ \forall k \in \mathcal{K}_p, \ & \displaystyle \sum_{j \in \mathcal{N}^s} o_j x_j \le B, \ & x_j \in \{0,1\}\ \forall j \in \mathcal{N}^s,\quad y_p \in \{0,1\}\ \forall p \in \mathcal{P}. \end{array}$$ These formulations encode the classical FRLM “coverage logic” along paths by requiring every refueling-required segment to contain a facility (Godbersen et al., 2022).

A deterministic path-based refinement appears in the deterministic flow refuelling location problem (DFRLP), where a cyclic OD flow $f$ travels on a predetermined path, $N^s\subseteq N$0 indicates facility siting, $N^s\subseteq N$1 indicates whether the flow is covered, and $N^s\subseteq N$2 selects cycle segments between refueling opportunities. Its objective is

$N^s\subseteq N$3

subject to a station-count constraint $N^s\subseteq N$4, a driving-range constraint

$N^s\subseteq N$5

and linking constraints that enforce a valid origin-to-destination-and-back refueling sequence. In this formulation, explicit path segments and Big-$N^s\subseteq N$6 activation replace precomputed feasible station combinations by a single MILP (Kastner et al., 2021).

2. Coverage logic, detours, and route realism

Classical FRLM assumes deterministic flows on preselected paths, often one preferred path per OD pair, with full initial fuel at the origin and unlimited station capacity unless capacity is modeled explicitly. A central distinction in later work is between single-path FRLM, multi-path and deviation FRLM, and models that move from flow coverage to node-based or route-based coverage. The original objective is OD- or flow-based rather than node-based: FRLM maximizes covered flow, not mere proximity of nodes to stations (Sugishita et al., 2024).

Deviation and multi-path generalizations enlarge the feasible route set $N^s\subseteq N$7 for a demand $N^s\subseteq N$8. In the probability-based multi-path refueling location model, each OD pair may use multiple deviation paths, round-trip feasibility is modeled with fuel-state variables $N^s\subseteq N$9 and refueling variables $R$0, and the objective becomes expected node coverage,

$R$1

where $R$2 is the probability that origin $R$3 becomes a demand node and $R$4 is the fraction of destinations reachable from $R$5. Fuel conservation and capacity along a used path are enforced through Big-$R$6 equalities on consecutive arcs, and an OD is covered if at least one enumerated path is feasible. This shifts the objective from deterministic maximal coverage to maximal expected coverage (Li et al., 2018).

A more radical generalization removes the discrete candidate-site assumption. For fleet operations on a general network, stations may be located anywhere along edges, deviations from prescribed routes are allowed, and a route $R$7 is covered by a point $R$8 if a vehicle can deviate at $R$9, refuel, return to the route before the next unvisited stop, complete at least one full traversal, and return to the refueling station without violating the range $x_j\in\{0,1\}$0 or maximum deviation distance $x_j\in\{0,1\}$1. For off-route points, the coverage conditions are

$x_j\in\{0,1\}$2

$x_j\in\{0,1\}$3

This formulation treats refueling as a detour-aware coverage problem on dedicated closed routes rather than a shortest-path OD problem (Nayeem et al., 14 Sep 2025).

A frequent misconception is that FRLM is exhausted by static on-path spacing constraints. The literature shows a broader spectrum. Some models retain path coverage but allow deviations; some allow stations anywhere on edges; some replace path coverage by node-weighted “attractiveness”; and some enforce backup or reliability conditions on stations. These are related but not identical uses of the FRLM idea (Piedra-de-la-Cuadra et al., 31 Jan 2025).

3. Formulation theory and polyhedral strength

A major theoretical development concerns the relative strength of alternative FRLM formulations. Let $x_j\in\{0,1\}$4 denote the set of feasible routes for demand $x_j\in\{0,1\}$5, $x_j\in\{0,1\}$6 the siting decision, $x_j\in\{0,1\}$7 a path-coverage variable, and $x_j\in\{0,1\}$8 an aggregated OD-coverage variable. In the disaggregated path-based formulation,

$x_j\in\{0,1\}$9

subject to

$j$0

where $j$1 is the family of refueling-required segments on route $j$2. In the aggregated formulation,

$j$3

subject to

$j$4

where $j$5 is built by taking unions of one segment from each route in $j$6. The aggregated model compresses route-level feasibility into OD-level segment families (Sugishita et al., 2024).

The strength comparison is expressed through OD-wise LP value functions. For any fractional $j$7,

$j$8

If $j$9 contains a single path, then

$R$0

and if that path is a single, simple path, then

$R$1

The interpretation given in the literature is that the set-cover polyhedron on a single simple path has a totally unimodular structure, so the LP relaxation is integral in that case. By contrast, under a cardinality budget the aggregated relaxation strictly dominates the disaggregated relaxation, and the gap can be arbitrarily large (Sugishita et al., 2024).

The same line of work also identifies minimal segment sets as the relevant polyhedral objects. If $R$2 is the subfamily of minimal sets, then the polyhedron defined by all inequalities $R$3 for $R$4 is identical to that defined by the inequalities for $R$5, and each minimal-set constraint is facet-defining in the $R$6–$R$7 hull. Two tightening devices are emphasized under a cardinality budget: OD-specific feasible-site filtering, which replaces $R$8 by $R$9, and budget–noninterference facets,

$B_{p,k}$0

for suitable $B_{p,k}$1. These results clarify why aggregated branch-and-cut formulations are strong for deviation FRLM and why direct path disaggregation can yield weak LP bounds when multiple routes per OD are allowed (Sugishita et al., 2024).

4. Principal extensions of the FRLM family

The FRLM family now includes several distinct extensions that alter either the candidate-site space, the objective, the feasible-route definition, or the coupling with operational constraints.

Variant	Main modeling addition	Source
DFRLP, MCFRLP, LCFRLP, CFRLP, CMCFRLP	coverage quota, location-dependent construction costs, capacity-limited stations, simultaneous sizing and partial coverage	(Kastner et al., 2021)
Continuous-site FRLM with FDS	station can be anywhere on edges; finite dominating set $B_{p,k}$2 of endpoints $B_{p,k}$3 contains an optimal solution	(Nayeem et al., 14 Sep 2025)
Bilevel reinforced coverage	every opened station must have another station within $B_{p,k}$4; lower level maximizes node attractiveness under budget	(Piedra-de-la-Cuadra et al., 31 Jan 2025)
Probability-based multi-path refueling	node-specific probabilities $B_{p,k}$5, round-trip feasibility, and expected node coverage objective	(Li et al., 2018)
Modified CFRLM with SOCP service ability	heterogeneous PEV types, time-varying demand, AC power flow, and second-order-cone service constraints	(Zhang et al., 2017)
Diffusion-integrated FRLM	annual budgets from continuous-time adoption and supply curves, with Macro–Meso–Micro planning layers	(Luo et al., 2024)

The capacity and sizing extensions are particularly important for EV charging. In the capacity-limited station model, $B_{p,k}$6 gives the number of charging poles at site $B_{p,k}$7, $B_{p,k}$8 is the proportion of flow $B_{p,k}$9 covered, and the capacity restriction sums distance-weighted flow demand assigned to a station and bounds it by $$\begin{array}{ll} \text{minimize} & \displaystyle \sum_{j \in \mathcal{N}^s} o_j\, x_j \ \text{subject to} & \displaystyle \sum_{j \in B_{p,k}} x_j \ge 1 \qquad \forall p \in \mathcal{P},\ \forall k \in \mathcal{K}_p, \ & x_j \in \{0,1\} \qquad \forall j \in \mathcal{N}^s . \end{array}$$0. This allows simultaneous placement and sizing, and it permits partial coverage when capacity is insufficient. Location-dependent construction costs similarly replace a station-count budget by a budget on $$\begin{array}{ll} \text{minimize} & \displaystyle \sum_{j \in \mathcal{N}^s} o_j\, x_j \ \text{subject to} & \displaystyle \sum_{j \in B_{p,k}} x_j \ge 1 \qquad \forall p \in \mathcal{P},\ \forall k \in \mathcal{K}_p, \ & x_j \in \{0,1\} \qquad \forall j \in \mathcal{N}^s . \end{array}$$1, making the number of stations endogenous (Kastner et al., 2021).

The continuous-site extension replaces arbitrary discretization by an exact finite dominating set. For each route and edge, the Edge Scanning algorithm constructs refueling segments $$\begin{array}{ll} \text{minimize} & \displaystyle \sum_{j \in \mathcal{N}^s} o_j\, x_j \ \text{subject to} & \displaystyle \sum_{j \in B_{p,k}} x_j \ge 1 \qquad \forall p \in \mathcal{P},\ \forall k \in \mathcal{K}_p, \ & x_j \in \{0,1\} \qquad \forall j \in \mathcal{N}^s . \end{array}$$2, extracts their endpoints, and proves that for any point $$\begin{array}{ll} \text{minimize} & \displaystyle \sum_{j \in \mathcal{N}^s} o_j\, x_j \ \text{subject to} & \displaystyle \sum_{j \in B_{p,k}} x_j \ge 1 \qquad \forall p \in \mathcal{P},\ \forall k \in \mathcal{K}_p, \ & x_j \in \{0,1\} \qquad \forall j \in \mathcal{N}^s . \end{array}$$3 there exists an endpoint $$\begin{array}{ll} \text{minimize} & \displaystyle \sum_{j \in \mathcal{N}^s} o_j\, x_j \ \text{subject to} & \displaystyle \sum_{j \in B_{p,k}} x_j \ge 1 \qquad \forall p \in \mathcal{P},\ \forall k \in \mathcal{K}_p, \ & x_j \in \{0,1\} \qquad \forall j \in \mathcal{N}^s . \end{array}$$4 with $$\begin{array}{ll} \text{minimize} & \displaystyle \sum_{j \in \mathcal{N}^s} o_j\, x_j \ \text{subject to} & \displaystyle \sum_{j \in B_{p,k}} x_j \ge 1 \qquad \forall p \in \mathcal{P},\ \forall k \in \mathcal{K}_p, \ & x_j \in \{0,1\} \qquad \forall j \in \mathcal{N}^s . \end{array}$$5. At least one optimal solution therefore exists in the finite endpoint set $$\begin{array}{ll} \text{minimize} & \displaystyle \sum_{j \in \mathcal{N}^s} o_j\, x_j \ \text{subject to} & \displaystyle \sum_{j \in B_{p,k}} x_j \ge 1 \qquad \forall p \in \mathcal{P},\ \forall k \in \mathcal{K}_p, \ & x_j \in \{0,1\} \qquad \forall j \in \mathcal{N}^s . \end{array}$$6, and the final model is a set-cover problem

$$\begin{array}{ll} \text{minimize} & \displaystyle \sum_{j \in \mathcal{N}^s} o_j\, x_j \ \text{subject to} & \displaystyle \sum_{j \in B_{p,k}} x_j \ge 1 \qquad \forall p \in \mathcal{P},\ \forall k \in \mathcal{K}_p, \ & x_j \in \{0,1\} \qquad \forall j \in \mathcal{N}^s . \end{array}$$7

with $$\begin{array}{ll} \text{minimize} & \displaystyle \sum_{j \in \mathcal{N}^s} o_j\, x_j \ \text{subject to} & \displaystyle \sum_{j \in B_{p,k}} x_j \ge 1 \qquad \forall p \in \mathcal{P},\ \forall k \in \mathcal{K}_p, \ & x_j \in \{0,1\} \qquad \forall j \in \mathcal{N}^s . \end{array}$$8 obtained by grouping identical coverage sets (Nayeem et al., 14 Sep 2025).

The bilevel reinforced-coverage model departs further from classical FRLM. Its upper level minimizes the number of opened sites under conditional covering constraints

$$\begin{array}{ll} \text{minimize} & \displaystyle \sum_{j \in \mathcal{N}^s} o_j\, x_j \ \text{subject to} & \displaystyle \sum_{j \in B_{p,k}} x_j \ge 1 \qquad \forall p \in \mathcal{P},\ \forall k \in \mathcal{K}_p, \ & x_j \in \{0,1\} \qquad \forall j \in \mathcal{N}^s . \end{array}$$9

so that every node is within range $$\begin{array}{ll} \text{maximize} & \displaystyle \sum_{p \in \mathcal{P}} w_p\, y_p \ \text{subject to} & y_p \le \displaystyle \sum_{j \in B_{p,k}} x_j \qquad \forall p \in \mathcal{P},\ \forall k \in \mathcal{K}_p, \ & \displaystyle \sum_{j \in \mathcal{N}^s} o_j x_j \le B, \ & x_j \in \{0,1\}\ \forall j \in \mathcal{N}^s,\quad y_p \in \{0,1\}\ \forall p \in \mathcal{P}. \end{array}$$0 of some station and every opened station has a distinct backup station within $$\begin{array}{ll} \text{maximize} & \displaystyle \sum_{p \in \mathcal{P}} w_p\, y_p \ \text{subject to} & y_p \le \displaystyle \sum_{j \in B_{p,k}} x_j \qquad \forall p \in \mathcal{P},\ \forall k \in \mathcal{K}_p, \ & \displaystyle \sum_{j \in \mathcal{N}^s} o_j x_j \le B, \ & x_j \in \{0,1\}\ \forall j \in \mathcal{N}^s,\quad y_p \in \{0,1\}\ \forall p \in \mathcal{P}. \end{array}$$1. The lower level then maximizes node attractiveness $$\begin{array}{ll} \text{maximize} & \displaystyle \sum_{p \in \mathcal{P}} w_p\, y_p \ \text{subject to} & y_p \le \displaystyle \sum_{j \in B_{p,k}} x_j \qquad \forall p \in \mathcal{P},\ \forall k \in \mathcal{K}_p, \ & \displaystyle \sum_{j \in \mathcal{N}^s} o_j x_j \le B, \ & x_j \in \{0,1\}\ \forall j \in \mathcal{N}^s,\quad y_p \in \{0,1\}\ \forall p \in \mathcal{P}. \end{array}$$2 under a budget, with $$\begin{array}{ll} \text{maximize} & \displaystyle \sum_{p \in \mathcal{P}} w_p\, y_p \ \text{subject to} & y_p \le \displaystyle \sum_{j \in B_{p,k}} x_j \qquad \forall p \in \mathcal{P},\ \forall k \in \mathcal{K}_p, \ & \displaystyle \sum_{j \in \mathcal{N}^s} o_j x_j \le B, \ & x_j \in \{0,1\}\ \forall j \in \mathcal{N}^s,\quad y_p \in \{0,1\}\ \forall p \in \mathcal{P}. \end{array}$$3 representing the number of charging devices installed at $$\begin{array}{ll} \text{maximize} & \displaystyle \sum_{p \in \mathcal{P}} w_p\, y_p \ \text{subject to} & y_p \le \displaystyle \sum_{j \in B_{p,k}} x_j \qquad \forall p \in \mathcal{P},\ \forall k \in \mathcal{K}_p, \ & \displaystyle \sum_{j \in \mathcal{N}^s} o_j x_j \le B, \ & x_j \in \{0,1\}\ \forall j \in \mathcal{N}^s,\quad y_p \in \{0,1\}\ \forall p \in \mathcal{P}. \end{array}$$4. This is a reliability-conscious covering model rather than a path-spacing FRLM in the strictest sense (Piedra-de-la-Cuadra et al., 31 Jan 2025).

5. Exact algorithms, decomposition, and heuristics

Because FRLM variants combine siting, path feasibility, range restrictions, and often capacity or time dependence, solution methodology is a defining part of the literature. In the continuous-site setting, the Edge Scanning algorithm constructs the endpoint set $$\begin{array}{ll} \text{maximize} & \displaystyle \sum_{p \in \mathcal{P}} w_p\, y_p \ \text{subject to} & y_p \le \displaystyle \sum_{j \in B_{p,k}} x_j \qquad \forall p \in \mathcal{P},\ \forall k \in \mathcal{K}_p, \ & \displaystyle \sum_{j \in \mathcal{N}^s} o_j x_j \le B, \ & x_j \in \{0,1\}\ \forall j \in \mathcal{N}^s,\quad y_p \in \{0,1\}\ \forall p \in \mathcal{P}. \end{array}$$5 in polynomial time. Its complexity is $$\begin{array}{ll} \text{maximize} & \displaystyle \sum_{p \in \mathcal{P}} w_p\, y_p \ \text{subject to} & y_p \le \displaystyle \sum_{j \in B_{p,k}} x_j \qquad \forall p \in \mathcal{P},\ \forall k \in \mathcal{K}_p, \ & \displaystyle \sum_{j \in \mathcal{N}^s} o_j x_j \le B, \ & x_j \in \{0,1\}\ \forall j \in \mathcal{N}^s,\quad y_p \in \{0,1\}\ \forall p \in \mathcal{P}. \end{array}$$6, where $$\begin{array}{ll} \text{maximize} & \displaystyle \sum_{p \in \mathcal{P}} w_p\, y_p \ \text{subject to} & y_p \le \displaystyle \sum_{j \in B_{p,k}} x_j \qquad \forall p \in \mathcal{P},\ \forall k \in \mathcal{K}_p, \ & \displaystyle \sum_{j \in \mathcal{N}^s} o_j x_j \le B, \ & x_j \in \{0,1\}\ \forall j \in \mathcal{N}^s,\quad y_p \in \{0,1\}\ \forall p \in \mathcal{P}. \end{array}$$7, $$\begin{array}{ll} \text{maximize} & \displaystyle \sum_{p \in \mathcal{P}} w_p\, y_p \ \text{subject to} & y_p \le \displaystyle \sum_{j \in B_{p,k}} x_j \qquad \forall p \in \mathcal{P},\ \forall k \in \mathcal{K}_p, \ & \displaystyle \sum_{j \in \mathcal{N}^s} o_j x_j \le B, \ & x_j \in \{0,1\}\ \forall j \in \mathcal{N}^s,\quad y_p \in \{0,1\}\ \forall p \in \mathcal{P}. \end{array}$$8, and $$\begin{array}{ll} \text{maximize} & \displaystyle \sum_{p \in \mathcal{P}} w_p\, y_p \ \text{subject to} & y_p \le \displaystyle \sum_{j \in B_{p,k}} x_j \qquad \forall p \in \mathcal{P},\ \forall k \in \mathcal{K}_p, \ & \displaystyle \sum_{j \in \mathcal{N}^s} o_j x_j \le B, \ & x_j \in \{0,1\}\ \forall j \in \mathcal{N}^s,\quad y_p \in \{0,1\}\ \forall p \in \mathcal{P}. \end{array}$$9. After preprocessing, the remaining decision problem is a compact set-cover MIP on the reduced candidate set $f$0 (Nayeem et al., 14 Sep 2025).

In robust charging-network planning for metropolitan taxi fleets, the strategic problem is handled by a cutting-plane master problem

$f$1

with feasibility cuts

$f$2

while the operational feasibility problem is solved by branch-and-price. The column-generation master problem selects one charging-augmented route per vehicle, station capacities are enforced by route–time occupancy coefficients $f$3, and pricing is a resource-constrained shortest path problem on a time-expanded network with label updates

$f$4

The same work embeds this deterministic core in a robust framework based on adversarial sampling, with Full Scenario, $f$5-Scenario, and $f$6-Vehicle approaches (Godbersen et al., 2022).

For probability-based multi-path models, the mixed-integer formulation is NP-hard, and a genetic algorithm has been developed in which a binary siting vector $f$7 is the chromosome and fitness is evaluated by solving the remaining MILP with fixed $f$8. On the Sioux Falls network, the heuristic produced expected coverage values within $f$9–$N^s\subseteq N$00 of CPLEX optima for budgets $N^s\subseteq N$01, with CPU time reductions of up to $N^s\subseteq N$02 at larger $N^s\subseteq N$03 (Li et al., 2018).

When FRLM is coupled with service-level and power-flow constraints, exact solution methods move beyond MILP into conic optimization. A modified capacitated FRLM based on sub-paths is embedded in a stochastic mixed-integer second-order cone program, where station service ability is enforced by

$N^s\subseteq N$04

and radial AC power flow is imposed through SOCP constraints. The resulting MISOCP is solved by branch-and-cut in a commercial conic solver, with additional path-sharing equalities used as computational strengthening (Zhang et al., 2017).

6. Electric-vehicle planning, robustness, and current research directions

The most active contemporary use of FRLM is EV charging network planning, where static path coverage is no longer sufficient by itself. In metropolitan taxi-fleet planning, FRLM-like coverage logic is embedded in a time-expanded, state-of-charge-based operational model with station capacity, non-linear charging functions, partial recharges, and planner-selectable risk tolerance. On Munich-based instances with $N^s\subseteq N$05 vehicles and $N^s\subseteq N$06 potential charging station locations, the Full Scenario Approach yielded cost $N^s\subseteq N$07, $N^s\subseteq N$08, and $N^s\subseteq N$09, while allowing $N^s\subseteq N$10 vehicle infeasibility reduced cost by about $N^s\subseteq N$11 through the $N^s\subseteq N$12-VA solutions. The same study reports that increasing battery capacities yields up to $N^s\subseteq N$13 percentage points more vehicle feasibility than equivalent increases in charging speed, and that allowing depot charging dominates both options (Godbersen et al., 2022).

Coupled transportation–power formulations show that once capacity and electrical feasibility are modeled jointly, the FRLM problem becomes a siting-and-sizing problem on two interdependent networks. In a 25-node highway network refined to 93 transportation nodes and coupled to a 14-bus radial distribution system, the baseline case with $N^s\subseteq N$14 PEV/day required $N^s\subseteq N$15 stations and $N^s\subseteq N$16 charging spots, whereas $N^s\subseteq N$17 PEV/day required $N^s\subseteq N$18 stations and $N^s\subseteq N$19 spots. Cases that ignored power constraints or used DC power flow produced higher post hoc unmet demand or distorted investment decisions, indicating that transport-side coverage alone is insufficient when grid congestion is material (Zhang et al., 2017).

A different line of work embeds FRLM in a multi-year demand–supply framework based on innovation diffusion and fluid queues. There, annual station budgets are derived from a cumulative supply curve $N^s\subseteq N$20, demand follows Bass-type adoption dynamics, and charging-location planning is performed on Macro–Meso–Micro grids. In the Chicago sketch network, capacity constraints materially altered facility counts: at the macro level with $N^s\subseteq N$21 km range, the capacity-limited version selected $N^s\subseteq N$22 sites rather than about $N^s\subseteq N$23 without capacity; at the meso level it selected $N^s\subseteq N$24 sites; and at the micro level $N^s\subseteq N$25 sites. For $N^s\subseteq N$26 km range, the corresponding capacity-limited counts were $N^s\subseteq N$27, $N^s\subseteq N$28, and $N^s\subseteq N$29. This suggests that station capacity, EV range, and spatial resolution interact strongly in medium-term planning (Luo et al., 2024).

Across these strands, several limits of classical FRLM remain visible. The original model is static, path-based, and usually uncapacitated; it abstracts from queueing, charging duration, time windows, and endogenous route reassignment. Later work therefore adds station capacities, partial coverage, service-level approximations, robust scenario analysis, multi-path routing, continuous candidate sets, or bilevel public–private structure. A plausible implication is that FRLM is best understood not as a single fixed formulation but as a modeling core—range-constrained path coverage—around which distinct operational, economic, and infrastructure couplings are built in contemporary research.