1The decision hierarchy
There is no single firefighting-UAV optimization problem. There are three, they live on different timescales, and the constraint that binds at one layer is a parameter at the next.
Trying to write one program covering base placement, fleet routing, and rotor thrust is the most common modeling error in this space: the result is a mixed-integer nonconvex problem with a fire-spread PDE in the constraints that no solver will touch. The productive structure is a hierarchy where each layer abstracts the layer below into a cost coefficient or a feasibility test.
| Layer | Timescale | Decision variables | Class | Passes down | Passes up |
|---|---|---|---|---|---|
| I — Strategic base siting, fleet mix | season / pre-incident | Where to put hangars; how many and which UAVs at each | MILP (facility location, p-median, maximal covering) | Available fleet , depot set , response-time floors | Expected area burned, coverage % |
| II — Tactical routing, allocation, coverage | minutes–hours, replanned | Who goes where, in what order, when, at what speed; who rendezvouses with whom | MILP / MINLP / submodular maximization / CE-TSP | Waypoint sequence, arrival windows, speed setpoints | Tour feasibility, energy shortfall, revisit times |
| III — Operational trajectory & control | 0.02–10 s, receding horizon | : the actual flight path and rotor commands | NLP via collocation, QP via flatness, NMPC | Actuator commands | Realized arc cost , realized energy , infeasibility flags |
The single most important coupling
Layer II needs arc costs and arc energies . Those are outputs of Layer III. If you compute them from Euclidean distance and a constant power draw, your tactical plan will be infeasible in wind, in a plume, and with a partially drained tank. The honest options are (a) precompute from a fitted model with a wind field, (b) make speed a Layer-II decision variable so the energy–speed coupling is explicit, or (c) iterate: solve II, simulate III, update , re-solve. Option (b) is the one with the best published evidence behind it[1].
2Notation and decision variables
One consistent symbol set for the whole document.
| Symbol | Meaning | Where it lives |
|---|---|---|
| , | UAV fleet; index of a vehicle. | all layers |
| Depots / bases / launch points (Layer I output) | I → II | |
| Refuel / battery-swap / refill nodes (may be mobile) | II | |
| Targets: firespots, perimeter cells, ignition points, drop points | II | |
| Vehicle traverses arc | II | |
| Node is visited / served (prize-collecting) | II | |
| Task assigned to vehicle | II | |
| , | Service start time at ; MTZ position potential | II |
| Energy remaining on arrival at node ; = usable capacity | II | |
| or | Speed on arc / on sub-tour — a decision variable, not a constant | II |
| Position, velocity, attitude, body rates | III | |
| Per-rotor thrust, body torque, control vector | III | |
| Total mass; is the (draining) agent payload | III (and II via ) | |
| , | State of charge as an augmented state | III |
| , | Fire front; level-set field with | exogenous, estimated |
| , | State-estimate covariance; innovation covariance | coupling II↔III |
| , | Communication-graph Laplacian; algebraic connectivity | II |
Constraint tags used throughout: hard violation is unsafe or infeasible · soft penalize with slack · convex preserves tractability · nonconvex needs SCP, MI, or heuristic · integer introduces binaries
3Platform and energy constraints
The binding constraint on almost every wildfire UAV mission. Get this layer wrong and nothing downstream matters.
3.1 Endurance — three ways to write it, in increasing fidelity
(a) Duration budget. The crudest and most common form — a flat cap on flight time:
hard linear Used by Bailon-Ruiz et al.[2]: "the flight duration must be lower than the maximum flight endurance." Correct only if power draw is roughly constant — i.e. constant speed, constant mass, no wind. Fails for any mission with a payload release.
(b) Energy as a resource propagated along arcs. The MILP-standard form, with big- resource propagation and a full reset at refuel nodes:
hard big-M The lower-bound pair can be dropped when energy is a feasibility resource only, halving the constraint count. Partial recharge replaces the reset with , , plus a charge-time term in the time propagation. Sundar & Rathinam[3].
(c) State of charge as a continuous augmented state. At Layer III:
hard nonconvex in is the regulatory/operational return margin — typically 20–30% of . Do not set it to zero even in simulation; it is what makes the plan flyable.
3.2 The power model — where the nonconvexity enters
Every energy constraint is only as good as . Three forms are in circulation, and they are not interchangeable:
Zeng, Xu & Zhang[4] — the rotary-wing model used throughout the communications and routing literature. U-shaped with an interior minimum-power speed ; energy per metre, , is minimized at the strictly larger maximum-range speed .
Consequence you must encode
Because has an interior minimum, "fly as fast as possible" and "fly as slow as possible" are both wrong. A planner that fixes speed at is throwing away range. Diller & Han measured 11.9–47.1% improvement from adaptive speed, and — more tellingly — 93.4% mission feasibility with adaptive speed versus 69.3% at [1]. Empirically, Cabuk et al. found the minimum-power speed moves with mass: ≈6 m/s at 1102 g, ≈10 m/s at 1418 g, ≈12.5 m/s at 1734 g for a small quadrotor[5].
3.3 Speed as a decision variable — the resulting MINLP
Arc energy is , so total energy is a product of a binary and a nonconvex function of a continuous variable. Diller & Han's MT-OTM formulation makes this explicit and tractable by inverting the maximum-reachable-distance function and fitting the inverse:
hard nonconvex The adaptive-speed constraint. Read it plainly: the longer the sub-tour, the slower the UAV must fly for the battery to last. This is the single most important structural constraint in energy-aware UAV routing, and it is not exactly linearizable[1].
Standard linearizations, if you need a pure MILP: (i) discretize speeds into levels with , , and precomputed — costs variables; (ii) SOS2 piecewise-linear outer approximation of ; (iii) McCormick envelopes on the bilinear terms.
3.4 Refuel, swap, and rendezvous
Battery swap with a moving ground vehicle is the case that matters operationally (an engine or crew buggy on a fire road cannot stop and wait). It requires a synchronization constraint that destroys separability:
The refuel-site selection subproblem is a set cover embedded in the routing problem — with the sites from which target is fuel-reachable round-trip:
3.5 Platform capability bounds
| Constraint | Form | Typical values |
|---|---|---|
| Endurance | 32–59 min small multirotor; 22–30 min heavy-lift loaded; 90–180 min VTOL fixed-wing; 18 h heavy-fuel fixed-wing; effectively unbounded tethered | |
| Speed envelope | 14.7–25 m/s multirotor; fixed-wing = stall, typically 11–17 m/s cruise | |
| Payload | 385 g–6 kg ISR multirotor; 15 kg Alta X; 45 kg hybrid; 2,722 kg unmanned helicopter | |
| Service ceiling / density altitude | 4,000–7,000 m nominal — but see §6.3, hot air moves this down | |
| Wind limit | 12 m/s sustained typical multirotor; 14 m/s cont. / 18 m/s gust VTOL fixed-wing | |
| Turn radius (fixed-wing) | Sets the Dubins abstraction; ≈ 30–50° | |
| Consumable inventory | 450 spheres IGNIS II, 225 IGNIS III Mini, ≤120 spheres/min drop rate | |
| Temperature envelope | −20 to +50 °C airframe; Li-ion degradation accelerates above ~45–55 °C |
4Payload and variable-mass constraints
The constraints that distinguish a firefighting UAV from a delivery drone. Almost entirely absent from the published optimization literature.
4.1 Capacity and inventory
At Layer II this becomes a capacitated-VRP structure: a sortie can serve a set of drop points only if their combined demand fits the tank, and a refill node resets exactly as a refuel node resets . The coupling that has no VRP analogue is that appears in — the vehicle gets cheaper to fly as it discharges, so arc costs are path-dependent, not arc-dependent. A standard CVRP model is silently wrong here.
4.2 Endurance–payload coupling
From momentum theory, in hover and , so
Adding 5 kg of water to a 10 kg airframe costs , a 46% endurance loss on the induced-power term alone — before lifting the battery that supplies it. This exponent is why multirotor direct suppression is confined to spot work, and why tethered and hose-fed architectures exist.
4.3 Thrust margin through the release
hard A 10 kg multirotor carrying 5 L changes TWR by a factor of two through a full dump. The lower bound matters as much as the upper: at low mass a multirotor can hit (rotors cannot be commanded below idle) and lose descent authority.
4.4 Constraints that only exist because the payload is liquid
| Constraint | Form | Why it binds |
|---|---|---|
| Slosh–crossover avoidance | must avoid the attitude-loop crossover , or discharge fast enough to sweep through it | As the tank drains, : the slosh mode sweeps downward through crossover. Resonance during partial discharge is guaranteed, not accidental. Constrain so the dwell near is short, or add baffles and raise . |
| CG-offset torque budget | Asymmetric release (one of two tanks) produces a constant roll disturbance the attitude loop must have authority to trim while retaining maneuvering margin. | |
| Jet-recoil authority | , | A 6 mm nozzle at 0.8 MPa gives ≈45 N — comparable to the whole weight of a 4.5 kg aircraft. Either bound , or use tilting rotors / a steerable-nozzle allocation QP. |
| Nozzle allocation feasibility | , , | The Dragon Firefighter distributes the desired wrench across four rotating nozzles by a QP at 100 Hz — structurally identical to rotor control allocation, with jets substituted for rotors[9]. |
| Hose tether tension | ; flow-induced flutter | The pressurized hose is a pipe conveying fluid: the term is a compressive follower load that goes unstable above a critical internal flow velocity[10]. |
| Water-hammer transient | , peaks 2–3× steady state | Valve actuation. Either rate-limit the valve (a constraint on ) or size the recoil budget for the transient, not the mean[11]. |
4.5 Drop accuracy as a constraint
For a released compact payload the aim-point error budget is dominated by release altitude, since every term scales with or :
hard nonconvex Enforced as , a chance constraint (§9.3).
The design conflict to state explicitly in your formulation
Accuracy wants low release altitude. Plume updraft, thermal exposure, and terrain clearance want high release altitude. Measured updrafts of 6–10 m/s at ~115 m AGL inside prescribed-fire plumes[12] mean this is not a soft trade — at low altitude, itself grows, so the "accuracy improves as falls" assumption breaks down somewhere. Nobody has published where. This is a genuinely open modeling question and a defensible contribution.
5Dynamics and actuation constraints
Layer III. These are the constraints that make a trajectory flyable; everything above treats them as a feasibility oracle.
5.1 Dynamics as equality constraints
How the dynamics enter depends entirely on the transcription:
| Method | Dynamics appear as | Jacobian | Notes |
|---|---|---|---|
| Single shooting | Eliminated by integration — not constraints | Dense | Sensitivity grows as ; poor for long horizons |
| Multiple shooting | Block-banded | Parallelizable; the structure acados/ACADO exploit | |
| Hermite–Simpson collocation | Sparse banded | Path constraints hold only at nodes — inter-sample violation is real | |
| Pseudospectral (LGR) | Dense but small | Spectral accuracy for smooth solutions; costate estimate . Used by the wildfire risk-aware planner[13] | |
| Differential flatness | Eliminated exactly — no dynamic constraints at all | — | Turns the problem into a QP in polynomial coefficients[14] |
Practical note on collocation
Bounds enforced at collocation points can be violated between them. Two fixes: mesh refinement (-adaptive), or — cleaner — express segments in the Bernstein basis and constrain the control points, which by the convex-hull property gives sufficient conditions valid for all :
, and similarly for acceleration. The same trick makes corridor membership a linear constraint.
5.2 Actuator and allocation constraints
The last line is the important one: eliminating turns actuator saturation into a linear constraint on the commanded wrench, which is what lets a flatness-based planner check actuator feasibility without simulating the mixer[14]. For , use or a constrained QP.
5.3 Attitude, tilt, and derivative bounds
hard SOC-representable The right-hand form is the tilt constraint written purely in flat outputs — a second-order cone constraint, so it upgrades a minimum-snap QP to an SOCP without leaving convexity. This matters for a UAV carrying liquid: excessive tilt sloshes the tank and spills the drop.
Derivative bounds , , , become meaningful through the flat map , , , with involving snap.
5.4 Robustness to wind and unmodeled forces
Tube MPC is the principled form when disturbances are bounded, . With nominal state , ancillary gain , and robust positively invariant :
Terminal set for recursive feasibility — the condition most UAV MPC papers omit and then wonder why the horizon matters:
Common choice: with from the DARE of the linearization. Mayne et al.[17].
5.5 Fixed-wing and VTOL specifics
| Constraint | Form | Note |
|---|---|---|
| Stall / minimum airspeed | Load-factor dependent — tightens in a turn, which is exactly when you are orbiting a fire | |
| Bank / turn rate (Dubins) | The abstraction underlying nearly all fire-monitoring path planning[2],[18] | |
| Flight-path angle | Dubins-airplane extension for terrain | |
| Wind-relative kinematics | Bailon-Ruiz et al. state exactly this[2] — wind must be in the kinematics, not a disturbance, when is not small | |
| Loiter radius vs. coverage | , requires | A fixed-wing cannot stare; its "coverage" is an annulus. Patrinopoulou et al.[19] |
| VTOL transition | sweeps 15°–90°; quasi-steady coefficients invalid | Highest-risk phase; usually excluded from the optimization and handled as a fixed maneuver |
6Environmental and fire-specific constraints
The constraints with no analogue in delivery, inspection, or agricultural UAV problems. Almost none of them appear in the published wildfire optimization models.
6.1 Thermal and plume exclusion
The simplest form is a keep-out set around the front, which is what the two wildfire papers that address it at all actually use:
A sharper form ties the standoff to fire intensity rather than geometry — with Byram intensity and radiative flux falling as :
An intensity-scaled standoff. A 10 MW/m fireline demands a far larger exclusion radius than a 300 kW/m backing fire, and a geometric that ignores this is either unsafe or needlessly conservative. This form does not appear in the reviewed literature.
6.2 Plume updraft as a state-dependent disturbance
Unlike wind gusts, the plume field is a function of position and therefore both constrainable and feedforward-compensable:
Morton–Taylor–Turner similarity. Updraft decays only as — a small UAV at 100–150 m over an active front sits in a broad, persistent core, not a transient gust.
Three ways to enter it into a program:
- As a constraint: — never plan through updraft exceeding a fraction of your climb-rate authority. With measured 6–10 m/s at 115 m AGL[12] and typical small-multirotor descent limits of 3–6 m/s, this constraint is frequently active.
- As a cost term: add to the running cost, giving soft plume avoidance that degrades gracefully.
- As MPC preview: feed predicted from the fire estimate into the NMPC as a known disturbance. This is the coupling flagged as the field's most concrete unexploited opportunity — and it is a modeling choice, not an algorithm, so it costs nothing to adopt.
6.3 Density altitude — the constraint everyone evaluates at the wrong point
hard ISA 288 K → 333 K (60 °C near a front) costs 13.5% of density; 288 K → 400 K (in-plume) costs 28%. Required shaft power rises as exactly when battery capability is degrading. Evaluate thrust margin and at fire-environment density altitude, not sea-level ISA.
6.4 Time-validity of observation and action
A wildfire target is only worth visiting during a window, and this is what makes the problem not a static orienteering problem:
Bailon-Ruiz et al. impose exactly this[18]. The equivalent for suppression is narrower still: a drop is useful only while the fire at that point is within a suppressible intensity band, which opens and closes as the front passes.
Two idioms encode the decaying value of a stale observation:
Utility proportional to rate of spread, with spatial-correlation spillover, and consumed once observed — Bailon-Ruiz et al.[2]. The consumption is what makes the objective submodular in flavour.
Islam & Hu's importance metric[21]: "the new fire spread distance in the direction of a cell between when the cell was last visited and when it is projected to be re-visited." This is the cleanest published statement of why revisit time matters for a spreading process — the cost of staleness is proportional to .
6.5 Fire propagation as an exogenous constraint or an embedded model
The fire enters the program in one of four ways, in increasing order of fidelity and cost:
| Treatment | Form | Cost / caveat |
|---|---|---|
| Static obstacle | Fixed polygon keep-out | Wrong within minutes; only defensible for a single short sortie |
| Prescribed forecast | precomputed by FARSITE / ELMFIRE / CA, then treated as a time-varying parameter | The dominant approach[2],[22]. Decouples cleanly; ignores that observations should update the forecast |
| Embedded dynamics | as an equality constraint in the OCP | Correct, but puts a PDE in the constraint set — only tractable on a coarse grid or via a surrogate |
| Estimated in the loop | EKF/PF state with covariance ; planner drives | The right structure. Seraj et al.[22] is the reference instance |
Why the process noise is the whole story
Because — the fire keeps spreading whether or not you are watching — the estimate covariance regrows on any unattended segment. This single fact is why a static information-maximization objective (§12.3) is inadequate here and why revisit-time and covariance objectives are the same objective viewed from two sides: a trace-of-covariance cost induces a revisit schedule endogenously rather than requiring you to impose one.
7Sensing and estimation constraints
What the camera can actually see, and what the estimator needs in order not to diverge.
7.1 Geometric sensing constraints
| Constraint | Form | Note |
|---|---|---|
| Field of view | Pham et al.[23]. Couples altitude to footprint — the source of the resolution/coverage trade | |
| Ground sample distance | Sets a ceiling on altitude for a given detection task (e.g. 2 in spot fire) | |
| Slant range / LRF | 3–1,200 m (H20T), 3–3,000 m (H30T) | |
| Altitude-dependent noise | Popović et al.[24]. The mechanism that makes altitude a genuine decision variable rather than a fixed parameter | |
| Straight-segment-only observation | Utility accrues only on non-turning segments | Bailon-Ruiz et al.[2] — a fixed-wing in a turn observes nothing usable. Directly changes the routing graph |
| Smoke / occlusion | Rarely modeled; smoke is where LWIR beats RGB and where the sensor-model constraint should be modality-dependent | |
| Perception-aware pointing | in the cost | PAMPC[25] — keeps the target in frame as a soft cost rather than a hard constraint |
7.2 Estimator non-divergence as a constraint — the sharpest idea in the wildfire literature
Seraj, Silva and Gombolay define a quality-of-service guarantee as a bound on the growth of innovation covariance across a revisit cycle. This is the only formulation found in this review that guarantees something about the monitoring rather than about the optimizer:
hard In words: for every prioritized firespot, predicted uncertainty after the worst-case inter-visit interval must not exceed its value at the last observation. A URR above one means uncertainty is growing and the UAV cannot supply online state estimates. Paired with , [22].
The constraint becomes a routing budget through analytically derived temporal upper bounds, one per fire regime:
Stationary firespots (C1) and moving firespots (C2). Note the implicit feasibility side condition in C2: bounds how many firespots a single UAV may own — which is exactly how a QoS requirement turns into a fleet-sizing constraint. A third closed form covers moving-and-spreading fronts.
7.3 Covariance propagation as a planning constraint
The general machinery, with the pose-dependence that makes it plannable:
depends on UAV pose — altitude, range, viewing angle — which is precisely what makes covariance a controllable quantity. Constrain it () or minimize it (§12.3).
A subtlety worth knowing before you pick a criterion
(D-optimal / entropy / mutual information) is submodular, so greedy inherits the guarantee. (A-optimal, variance reduction) is not submodular in general. If you want a theoretical guarantee on a greedy or sequential-allocation planner, use the log-det form; if you want an objective that is easier to interpret operationally, use the trace and accept that you have no bound.
8Communication and connectivity constraints
A wildfire fleet that spreads out to cover the perimeter loses its mesh exactly when relaying observations matters most. This trade-off is explicit in the networking literature and absent from the fire-specific literature.
8.1 Four levels of rigour
| Form | Constraint | Character |
|---|---|---|
| Pairwise distance | Local, cheap, fully decentralized — but it fixes the topology rather than optimizing it. Pham et al. use exactly this[23] | |
| Algebraic connectivity | The standard tool. Nonconvex in positions, but has a closed-form supergradient and a barrier surrogate [26] | |
| -connectivity | Sufficient (via ) for tolerance to vehicle losses. The right constraint if a UAV can be lost to the fire | |
| Rate / SNR chance constraint | , | Correct when the actual requirement is offloading imagery, not graph connectivity. Fink et al.[27] |
8.2 Connectivity as a weighted utility rather than a hard constraint
Devaraju, Ihler and Kumar's connectivity-aware pheromone model is the cleanest published example of multiplicatively discounting a coverage utility by connectivity, so the constraint binds only when the network is thin:
is coverage attractiveness — low pheromone means unvisited; is a connectivity discount that saturates once the UAV has at least distance-weighted neighbours. Evaluated on 6 km × 6 km with 20–40 UAVs at 1 km transmission range[28]. Note they measure connectivity by component count and average neighbour count, not .
The pheromone map itself is a legitimate soft coverage-state constraint, with evaporation and diffusion :
8.3 Data-volume and latency constraints specific to fire imagery
- Onboard storage: — a 1280×1024 radiometric thermal stream at 30 fps fills storage faster than the battery drains, so a mission may be storage-limited rather than energy-limited.
- Age of Information: , where is the generation time of the freshest delivered sample. Distinct from sensing latency because it includes upload and relay delay. AoI minimization collapses to a shortest-Hamiltonian-path problem when the offload is instantaneous[29].
- Relay geometry: if one UAV is a relay, it is removed from the sensing fleet — a constraint that reduces available for coverage, and the correct place to evaluate whether a tethered aircraft is cheaper.
9Safety, deconfliction, and risk constraints
9.1 Inter-vehicle separation and obstacles
| Form | Constraint | Tractability |
|---|---|---|
| Pairwise separation | nonconvex reverse-convex; typical 10–50 m | |
| Ellipsoidal (wake / downwash) | nonconvex anisotropic — vertical separation must exceed horizontal | |
| Safe flight corridor | convex the key trick — decompose free space into polytopes and the avoidance constraint becomes linear[30] | |
| Signed distance field | , or penalty | soft gradient-based local planners[31] |
| Control barrier function | ; discrete | linear in a QP safety filter; multi-robot pairwise [32],[33] |
| Big-M disjunctive | , | MILP exact but exponential in practice[34] |
Terrain adds with typically 30–60 m over timber, and canopy penetration is why LiDAR-derived surfaces beat bare-earth DEMs here.
9.2 Recovery-margin and abort constraints
The constraint that most academic formulations omit and every operational one has:
hard Always be able to reach a landing site. The Spanish helicopter-planning model states it explicitly as "enough remaining time to return to base"[35]. In a receding-horizon setting the equivalent is a terminal safe set: FASTER's construction requires a backup trajectory ending in a known-free stopping region at every replan[36].
9.3 Chance constraints and risk measures
The fire boundary is uncertain, so a deterministic keep-out is either optimistic or wasteful. The standard treatment converts a probabilistic requirement into a tightened deterministic one:
SOC Exact for Gaussian state and affine constraint — the tightened half-space is a second-order cone constraint. Blackmore, Ono & Williams[37].
Aggarwal et al. apply precisely this to a spreading wildfire, with a random offset on each polygon face:
" is the respective risk threshold for the violation of obstacle boundaries." Solved by Legendre–Gauss–Radau pseudospectral transcription in GPOPS-II + IPOPT, with a constrained Delaunay triangulation supplying the homotopy-class initial guess[13]. This is the reference risk-aware wildfire trajectory formulation.
Joint constraints need risk allocation: is conservatively enforced by , , with the themselves optimized (iterative risk allocation). CVaR is the coherent alternative and is what you want when tail losses matter — a burnover is not "one violation among many":
CVaR dominates the chance constraint () and has an epigraph LP/QP representation under sample-average approximation[38].
10Regulatory and operational constraints
These are not soft. They determine whether a formulation describes something anyone can fly.
| Constraint | Form | Source / value |
|---|---|---|
| One pilot, one aircraft | 14 CFR 107.35. NASA's prescribed-fire CONOPS names this the blocker for multi-drone burns; waivers need 90 days' notice[39]. Every multi-UAV formulation in §17 assumes this away. | |
| Crew ratio | NASA CONOPS: 2 pilots + 1 analyst + 2 maintenance per 10-drone swarm, +1 backup pilot per additional 10[39] | |
| Visual line of sight | Binds unless a BVLOS waiver or public-aircraft COA is held. Part 108 NPRM published Aug 2025, comment period reopened Jan 2026 — not a final rule | |
| TFR / airspace exclusion | 14 CFR 91.137 over incidents; deconfliction with crewed aviation is the operational reason agency integration is conservative | |
| Duty cycle / crew rest | Max working time triggers mandatory rest; start/end/rest transition binaries | Constraints (8)–(13) of Ginzo-Villamayor et al.[40]; constraints (19)–(23) of the helicopter model[35] — legally mandated and a major driver of model size |
| Prescription window (ignition only) | mph at 5–10 ft AGL; °F; % | NASA CONOPS[39]. A hard, exogenous, time-varying feasibility window on the whole mission |
| Burn-plan geometry | Line spacing 82.5 ft; altitude 271.2 ft; drops inside the control perimeter and unburnt area | NASA CONOPS[39]; Beachly et al. enforce the containment-polygon membership explicitly[41] |
| Resource cap per target | Zhu et al. score exceeding four UAVs per fire point as mission failure[20] |
11Combinatorial structure constraints
The bookkeeping that makes a routing or allocation model well-posed. Cheap to state, expensive to get wrong.
11.1 Subtour elimination — pick deliberately
MTZ[71] is compact ( constraints) but its LP relaxation is weak — generic solvers stall around 40–60 nodes. DFJ is exponential but strong, added as lazy cuts; this is what lets branch-and-cut handle thousands of nodes. For anything beyond a toy fleet, use DFJ-as-lazy-constraints, not MTZ.
11.2 Flow, assignment, and coverage
When you cannot visit everything — always true for a spreading fire — the correct base model is the team orienteering problem: prizes collected at visited nodes, subject to a per-vehicle budget. It replaces "visit all targets" with "visit the most valuable reachable subset", which is what a fire-monitoring fleet actually does.
11.3 Scheduling constraints
Where multi-UAV MILPs go to die
Synchronization constraints are the single biggest source of intractability in multi-UAV planning: they couple otherwise-independent vehicle subproblems and destroy the block-diagonal structure that Dantzig–Wolfe and branch-and-price rely on. The standard workaround is logic-based Benders — master decides assignment plus synchronization pattern, per-vehicle subproblems check route feasibility and return no-good cuts. If your fire model needs simultaneous multi-angle observation or a coordinated drop, budget for this from the start.
11.4 Allocation structure — and the one-line change that makes it NP-hard
Adding a capacity/endurance budget destroys total unimodularity and makes the problem contain bin packing. This one line separates "assign UAVs to fires" (easy) from "assign UAVs to fires subject to endurance" (hard) — the unconstrained case is the classical Hungarian assignment[72] — and the second is the real problem.
Gerkey and Matarić's taxonomy tells you immediately which regime you are in[42]:
| Class | Equivalent problem | Complexity | Wildfire instance |
|---|---|---|---|
| ST-SR-IA | Optimal assignment (LAP) | P, | Assign each available UAV to one hotspot, right now |
| ST-MR-IA | Set partitioning | NP-hard, not approximable | A fire point needs a coalition of ≥2 UAVs |
| MT-SR-IA | Set covering | NP-hard; greedy | One UAV services several nearby targets per sortie |
| ST-SR-TA | Scheduling / multi-vehicle routing | NP-hard | The realistic case: who goes where, in what order, over a shift |
11.5 Decentralized allocation: CBBA and the DMG condition
When there is no central planner and comms are intermittent — the wildfire norm — the consensus-based bundle algorithm targets
Non-separable: depends on the whole bundle, which is what makes this a routing problem in disguise.
DMG guarantees conflict-free assignment after consensus, convergence in iterations ( = network diameter), and a 50% optimality bound[43]. Time-discounted reward satisfies DMG — which is exactly the right score for a decaying fire observation. Makespan-type scores generally do not.
For heterogeneous fleets (an ISR multirotor plus an ignition aircraft plus a relay), coalition formation adds componentwise capability sufficiency whenever coalition is assigned to task — a set-packing problem with columns, tractable only by column generation or by capping [44].
12Cost-function library
Grouped by what you are actually buying. The choice of objective determines the tractability class more often than the constraints do.
12.1 Trajectory-level (Layer III)
| Objective | Form | When it is right |
|---|---|---|
| Minimum time | Initial attack, transit through a spreading field[13]. Nonsmooth in practice; MPCC/progress-variable surrogates are the tractable route | |
| Minimum snap | Quadrotor drop runs and perimeter following. Snap differential rotor-thrust command, so it is a control-effort surrogate in flat coordinates[14] | |
| Minimum jerk | When the input interface is body-rate + collective thrust rather than torque. Also the right choice with a liquid payload — jerk is what excites slosh | |
| Minimum energy | Endurance-limited monitoring; couples directly to the Layer-II arc costs | |
| Control effort | Baseline regularizer; weights per-rotor authority | |
| Tracking (LQ) | NMPC workhorse; from the DARE for stability[17] | |
| Perception-aware | Keeps the fire front in frame during aggressive maneuvers[25] | |
| Plume-exposure penalty | Soft plume avoidance. Not in the literature; proposed here as the graceful-degradation alternative to a hard exclusion zone |
12.2 Route- and schedule-level (Layer II)
| Objective | Form | Character |
|---|---|---|
| Min-sum | Efficient, linear, strong LP bound — but can leave one vehicle overloaded | |
| Min-max (makespan) | The right objective for time-critical fire ISR. One extra variable, but a much weaker LP bound and badly degenerate for identical vehicles — break symmetry with | |
| Latency / cumulative | Responsiveness — the travelling-repairman objective. Right when early information beats complete information | |
| Prize-collecting | The honest model when you cannot visit everything. from rate of spread or estimator covariance | |
| Lexicographic | minimize makespan, then total cost s.t. makespan | Avoids min-max degeneracy. Skorin-Kapov et al. use a three-level hierarchy: hit the target water flow, then max-min fairness across areas, then total water output[45] |
| Mission time with speed | Diller & Han's MT-OTM — battery-swap time plus sub-tour durations, with a variable[1] |
12.3 Information and monitoring objectives
These are the objectives that actually describe fire monitoring, as opposed to fire visiting. The canonical statement is subject to — maximize information subject to a travel budget[73].
Popović et al. use the A-optimal reduction in an information-per-cost objective s.t. [24]. E-optimal is the minimax choice — right when one poorly-observed segment of perimeter is what kills you.
Alamdari et al.[48]; Cassandras, Lin & Ding[49]. In the fire case is the local uncertainty-growth rate — fast at the active front, ≈0 in the burned interior — so the persistent-monitoring formulation maps onto a spreading front with almost no modification. Its structural result is valuable: the optimal trajectory reduces to a finite vector of switching locations and dwell times, solved by gradient descent with infinitesimal perturbation analysis.
Cortés et al.[50]. The bridge to fire: set to front intensity or to posterior variance . That one substitution converts static coverage into a sensing-driven, time-varying objective, and the move-to-centroid law stays distributed.
12.4 Fire-specific objectives that appear in the literature
| Objective | Form | Source |
|---|---|---|
| Consumed observation utility | , utility initialized from RoS and consumed after observation | Bailon-Ruiz et al.[2] |
| Inverse-distance coverage | Bit-Monnot et al.[51] — submodular in flavour: an observation's marginal value drops if a nearby one exists | |
| Boundary-weighted resolution | , with | Pham et al.[23] — harmonic mean of per-UAV image resolution, deliberately up-weighting low-intensity cells so the fleet tracks the boundary, not the burned interior |
| Staleness × spread rate | , | Islam & Hu[21] |
| RL shaped reward | , , , | Julian & Kochenderfer[52] — front-seeking, FoV-efficiency, bank-angle effort, inter-aircraft repulsion |
| Discharged water | Rodríguez-Veiga et al.[53] — maximize water delivered, big- penalize unmet targets and unattended fronts, tie-break on distance | |
| Cost + damages (C+NVC) | Gallego Arrubla et al.[54]; Ginzo-Villamayor et al.[40]. The canonical wildfire-management objective form | |
| Expected area burned | over Monte Carlo scenarios | Hulse et al.[55] — asset placement, solved with DIRECT because the simulation objective is noisy and non-differentiable |
| Protected landscape value | s.t. | Nguyen et al. fuel-break prioritization[56] — the cleanest budget-constrained fire MIP in the literature |
13Handling multiple objectives
Firefighting is irreducibly multi-objective — coverage, latency, energy, risk, and connectivity all matter and are not commensurable. Four honest treatments:
- Weighted sum: . Only recovers the convex hull of the Pareto front — it cannot find solutions in a nonconvex region no matter how you tune . Heyns et al. explicitly rejected scalarization for tower siting for this reason and solved for the front[57].
- -constraint: s.t. . Recovers nonconvex Pareto points, and is usually the more natural statement anyway — "maximize coverage subject to never exceeding 20 minutes of staleness" is what an incident commander actually means.
- Lexicographic: optimize in priority order with the previous optimum fixed as a constraint. The right structure when there is a genuine safety-then-effectiveness hierarchy[45].
- Constrained MDP, when the model is an MDP/POMDP rather than a program: s.t. . The occupation-measure form is a linear program, and a structural fact worth knowing is that the optimal CMDP policy is generally stochastic, mixing at most deterministic policies[74] — so a monitoring policy that randomizes its revisits is not a bug.
- Pareto front by evolutionary search: NSGA-II is what the fire literature actually uses when the objective is a simulation output[20],[57]. Appropriate when objectives are non-differentiable, not when a MILP exists.
Normalization matters more than the weights
Rodríguez Barreiro et al. normalize each of six objective terms by a precomputed upper bound before weighting[35]. Without this, weight tuning is meaningless because the terms differ by orders of magnitude, and a "sensitivity analysis on " is really a sensitivity analysis on your units.
14Layer I — strategic: siting and fleet mix
Where to put bases and what to put in them. Solved once per season, not per incident.
The structure is facility location. Two canonical forms, both well-solved by CPLEX/Gurobi at realistic sizes:
— the sites that can reach demand point in time. This is the model behind airtanker basing and watchtower siting, and it transfers directly to UAV hangar placement.
Verified instances in the wildfire literature:
| Study | Decision variables | Objective | Method |
|---|---|---|---|
| Zell & Fügenschuh[58] | Which hangar sites to open, of which type; UAV types/quantities per site | Minimize response time subject to a guaranteed on-station monitoring time — i.e. an endurance floor as an -constraint | MILP, CPLEX |
| Hulse, Mbaye & Davies[55] | Air-base coordinates (continuous) | — expected percent area burned | DIRECT global search on a Monte Carlo simulation; box bounds only |
| De la Fuente et al.[59] | Site multiple technology types (towers, balloons, drones) and route drones among facilities | Maximize terrain coverage | MILP + 6-component matheuristic; authors note the MILP alone "can only solve small instances" |
| Heyns et al.[57] | 20 tower sites from 741,813 candidates | Two competing viewshed-coverage objectives; Pareto front | NSGA-II, multi-resolution; 70 runs → 72 non-dominated solutions |
Constraints characteristic of this layer: budget ; site eligibility (slope < 12°, ≤100 m from a road, in Heyns et al.); forced-open incumbents (six existing towers); minimum on-station endurance; and a fleet-composition constraint .
15Layer II — tactical: a reference formulation
Putting §3, §7, §8 and §11 together into one statement for a fire-monitoring fleet.
A prize-collecting, energy-and-speed-coupled, QoS-constrained team orienteering problem. It is a MINLP; nothing about it is convex. Solve it the way the field does: relax the adaptive-speed constraint by speed discretization to get a MILP, or partition-first/route-second with a per-cluster IP.
Reading the formulation
Three lines carry all the fire-specific content: the observation window (a target is worth visiting only while burning), the estimator non-divergence constraint (which converts "coverage" into a guarantee), and the intensity-scaled standoff. Delete those three and you have a generic multi-UAV persistent-monitoring problem — which is exactly what most published "wildfire" formulations reduce to.
16Layer III — the trajectory problem
The Bolza statement, and what the firefighting terms add to it.
The firefighting NMPC that follows from §4–§6, stated concretely:
Five things distinguish this from a stock quadrotor NMPC: mass as a state, energy as a state, the density correction on available thrust, the plume field as a known (estimated) disturbance in the cost, and a soft fire-standoff with slack so the problem stays feasible when the front moves faster than forecast.
Solve it with the real-time iteration scheme — one SQP iteration per sample, split into a preparation phase (integration and sensitivity propagation before the measurement arrives) and a feedback phase (substitute into the condensed QP), which reduces control delay to a single QP solve[62]. ACADO and acados implement this and run multirotor NMPC at 50–500 Hz[63].
When you can stay in a QP
Three conditions together: differential flatness with polynomial flat outputs (dynamics eliminated, not constrained); fixed time allocation ; and feasible sets expressed linearly — safe corridors plus Bernstein control-point bounds. Then s.t. is a convex QP. Adding the tilt cone makes it an SOCP; adding obstacle disjunctions makes it a MIQP; freeing the times makes it an NLP. Losing convexity is a decision, so make it deliberately.
17What the wildfire field actually optimizes
A survey of the papers that state an explicit program, with objective, binding constraints, and method. This is the honest picture of the state of the art.
| Study | Objective | Key constraints | Method |
|---|---|---|---|
| Bailon-Ruiz et al. 2022[2] | Max plan utility; utility from RoS, spatially spilled over, consumed on observation | Dubins-in-wind kinematics; return-to-base; endurance; flight windows; observation only on straight segments | Variable Neighborhood Search over four neighborhoods; replans against a live forecast |
| Bit-Monnot et al. 2018[51] | Flight windows; Dubins feasibility; return-to-base | VNS with neighborhood-dependent utility | |
| Seraj, Silva & Gombolay 2022[22] | (i) minimize number of UAVs subject to the QoS bound; (ii) minimize cumulative uncertainty residual | URR ≤ 1 on EKF innovation covariance; ; implicit cap on firespots per UAV | Adaptive EKF + FARSITE; Close-Enough TSP with Steiner zones; -means partitioning; VNS + -opt |
| Pham, La et al.[23] | Min harmonic-mean image-resolution cost weighted by — boundary-seeking | FoV cone; comm range; inter-UAV separation; ground clearance | Distributed gradient descent + artificial potential field. No central solver |
| Aggarwal et al. 2022[13] | Chance constraints on uncertain fire-polygon half-spaces; Dubins dynamics; turn-rate bounds | LGR pseudospectral → NLP; GPOPS-II + IPOPT; Delaunay triangulation for the homotopy class | |
| Islam & Hu 2024[21] | Max covered importance, | Motion restricted to perimeter cells; convex-fire assumption; onboard compute | Real-time greedy: reverse direction iff . Deliberately myopic |
| Patrinopoulou et al. 2024[19] | Max priority-weighted coverage (1.0 at boundary, 0.2 intermediate, 0.0 elsewhere) | Fixed-wing kinematics; ; thermal standoff | Three virtual forces; eight gains tuned by GA (200 generations); RVO for deconfliction |
| Zhu et al. 2025[20] | Path: flight time + turn angle + altitude variance + threat. Allocation: efficiency vs. resource use | Constant speed; fixed payload, no resupply; 3-D no-fly zones (heat, smoke, RF-shadow); time windows; ≤4 UAVs per fire point | Multi-population Grey Wolf Optimizer for paths; NSGA-II for the allocation Pareto front |
| Julian & Kochenderfer 2019[52] | Discounted return with a four-term shaped reward | Coordinated-turn dynamics; ; m/s; | Deep Q-learning; multi-agent by summing pairwise Q-contributions |
| Beachly et al. 2018[41] | Cover unburned area with a grid ignition pattern; minimize flight time | Fire-safety standoff (reject a segment if it would ignite before arrival); drops inside the control perimeter and unburnt area; 30 min endurance; 200 spheres | Greedy sequential ignition-line selection with an elliptical fire model in the safety check |
| Gallego Arrubla et al. 2014[54] | Deployment + relocation + operating cost + fire damages | Joint chance constraint bundling containment capability, risk threshold, response-fraction cap and logic, satisfied w.p. | Probabilistically constrained stochastic IP → big-M 0–1 MIP; CPLEX 12 |
| Rodríguez-Veiga et al. 2018[53] | Max discharged water (weighted by route suitability × capacity), big- penalties on unmet targets | One route per aircraft; group size per route; water-point simultaneous-use capacity; per-front allocation share; refuel slots and fuel stock | Two ILPs in AMPL + Gurobi. Mean ≤2 min, worst ≈2 h |
| Ginzo-Villamayor et al. 2023[40] | Variable + selection costs + hectares affected | Fireline production ≥ perimeter; containment per period; travel before work; enough time to return; legal duty-cycle rules; min/max per group | Four decompositions compared: augmented Lagrangian, branch-and-price, Benders, fixed-activity Benders |
| Skorin-Kapov et al. 2024[45] | Lexicographic: hit target water flow → max-min fairness across areas → max total output | Resource allocation and timing across areas and periods | MILP for small instances; randomized greedy + SA with LNS moves for large |
| Nguyen et al. 2024[56] | — landscape value protected across sampled fires | Budget ; containment logic requiring every bounding break treated | CPLEX 12.8; all instances to 0% gap in ≤1.8 h |
What this table shows
Only three of these use a general-purpose optimizer on a stated mathematical program (Aggarwal, Gallego Arrubla, Rodríguez-Veiga, plus Nguyen on the fuels side). The UAV-side work is dominated by metaheuristics — VNS, GA, GWO, greedy — and by RL. That is not laziness: it reflects that the coupled problem is a nonconvex MINLP with a moving, uncertain boundary. But it does mean almost nothing in the UAV wildfire literature comes with an optimality bound, and comparisons across papers are impossible because every one builds its own simulator with a different fire model.
18Tractability — what you can actually solve
Know which regime you are in before you write the model, not after the solver has run for six hours.
| Problem | Status | Practical exact limit |
|---|---|---|
| Assignment (ST-SR-IA) | P — totally unimodular | Thousands; Hungarian |
| Bottleneck (min-max) assignment | P | Threshold + matching |
| Generalized assignment (with endurance) | NP-hard | Contains bin packing |
| TSP, DFJ branch-and-cut (Concorde) | NP-hard | Thousands of nodes routinely |
| TSP with MTZ in a generic solver | NP-hard | ~40–60 nodes before the weak bound kills you |
| VRPTW, branch-price-and-cut | NP-hard | ~100 classically; 200–1000 with ng-routes + subset-row cuts |
| Team orienteering, exact | NP-hard | ~100 nodes |
| Flying-sidekick TSP MILP | NP-hard | ≈10 customers — heuristics mandatory beyond |
| Speed-coupled routing (MT-OTM MINLP) | NP-hard MINLP | ≈30 waypoints, ≈5,139 s at [1] |
| Same, -means + IP heuristic | Polynomial | in ≈0.74 s, 3.8% gap — a ~7,000× speedup for 3.8% |
| Aerial-resource assignment ILP | NP-hard | 15 aircraft / 6 water points / 3 fronts: mean ≤2 min, worst ≈2 h[53] |
| Coalition structure generation | NP-hard, inapproximable | — cap or use column generation |
| Submodular max, cardinality constraint | NP-hard; greedy | Tight under P≠NP |
| Submodular max, single matroid | greedy ; continuous greedy | Partition matroid = per-UAV action budget |
| Submodular orienteering (routing-constrained) | in quasi-polynomial time | The regime informative path planning actually lives in[64] |
| Multi-robot IPP by sequential allocation | degradation vs. single-path guarantee | Plan robot against the residual information[65] |
| Dec-POMDP, finite horizon | NEXP-complete | Not solvable; use sequential allocation or decentralized RL |
The submodularity ladder — check which rung you are on
Claims of "near-optimal multi-UAV informative path planning" are only meaningful relative to one of four regimes: for cardinality; greedy or continuous-greedy for a matroid; quasi-polynomial once the constraint is a path; and a further for sequential multi-robot allocation. Routing constraints are where the guarantees collapse, and routing constraints are exactly what a UAV has.
18.1 Decomposition patterns that work
- Partition-first, route-second. -means on targets, per-cluster TSP as an IP with lazy subcycle cuts, then assign speeds post hoc from . 3.8% gap at ~7,000× speedup[1]. The pattern to copy for multi-UAV energy-aware planning.
- Column generation / branch-and-price. Set-partitioning master + ESPPRC pricing. Energy enters as an extra monotone resource in the labeling dominance rule — this is the cleanest way to make battery constraints free rather than expensive.
- Logic-based Benders. Master decides assignment and synchronization; per-vehicle subproblems check route feasibility and return no-good cuts. The answer to §11.3.
- CMSA. Sample solution components, merge into a reduced sub-instance, solve that exactly with a MILP solver, age out unused components. Right when you have a good model but too large an instance[66].
- Sequential convex programming for the trajectory layer: linearize about the previous iterate, add virtual control with an penalty to remove artificial infeasibility, and a trust region . Converges to a KKT point under standard assumptions[67].
- Lossless convexification. The nonconvex annulus relaxes to , and under controllability conditions the relaxation is exact[68]. Directly applicable to multirotor collective-thrust bounds and the tilt cone.
19Modeling pitfalls and open problems
19.1 Pitfalls
- Euclidean arc costs. and is the default and it is wrong in wind, in a plume, and with a draining tank. At minimum, compute from a fitted ; better, make speed a variable.
- Fixed cruise speed. Costs 12–47% of range and drops mission feasibility from 93% to 69%[1]. And the optimum moves with payload mass[5], which is the regime an ignition or suppression UAV lives in.
- MTZ by default. Compact and comfortable, and it stalls at 40–60 nodes. Use DFJ as lazy cuts.
- Constant mass. A CVRP-style capacity constraint plus a constant-power energy model double-counts nothing and captures nothing: the vehicle gets cheaper to fly as it discharges, so arc costs are path-dependent.
- Weighted-sum scalarization. Cannot reach nonconvex Pareto points regardless of weights. And without normalizing each term by a computed upper bound, weight tuning is unit arithmetic[35].
- Zero energy reserve. A plan that lands with 0% is not a plan. and the recovery-margin constraint are what make a formulation flyable.
- Assuming the swarm away. 14 CFR 107.35 permits one aircraft per remote PIC. Every multi-UAV formulation in §17 needs a waiver. Say so, or model the crew constraint.
- Path constraints at collocation nodes only. Bounds can be violated between nodes. Use Bernstein control-point conditions for sufficiency.
- Trace-of-covariance with a submodularity claim. is not submodular; is. Pick one and make the matching claim.
- Silent truncation. If your planner caps at top- targets or samples the scenario set, report what was dropped. A bounded-coverage result reads as complete coverage otherwise.
19.2 Open problems worth a paper
| Gap | What is missing | Why it is tractable now |
|---|---|---|
| Variable-mass routing | No formulation couples the drop schedule to to the route. Payload is a capacity, never a state | The pieces exist: CVRP structure + a validated [5] + variable-mass dynamics. Nobody has assembled them |
| Plume-in-the-loop control | Fire estimate and flight controller are separate subsystems in essentially all reviewed work | The disturbance is state-dependent and therefore feedforward-compensable; measured magnitudes exist[12]. It is a modeling choice, not an algorithm |
| Coverage ∧ connectivity | Fire papers optimize coverage; networking papers optimize connectivity. No fire-specific paper does both | CAP[28] supplies the mechanism; supplies the rigorous version |
| Ignition-pattern optimization | No paper treats the ignition geometry (dot/strip/ring, spacing, sequence) as a decision vector inside an objective coupled to a fire model. Beachly et al.[41] is greedy; QUIC-Fire work optimizes a fireline width scalar[69] | A physics-constrained QUIC-Fire emulator with 3–5× speedup now exists[70] — the natural inner loop. Ignition pattern is currently an input to it |
| UAV-scale drop model | No validated coverage-level model for a 1–20 L release; airtanker correlations are 2–4 orders of magnitude away in volume and Weber number | Would make suppression optimization meaningful rather than notional |
| A shared benchmark | Every paper builds its own simulator with a different fire model. Cross-comparison is impossible | SimFire/SimHarness is the plausible base; it is mitigation-oriented and would need a sensing/UAV layer |
If you are picking one
Variable-mass routing is the most self-contained: every ingredient is published and verified, the gap is real and easy to demonstrate (a constant-mass plan versus a mass-aware plan on the same instance), and it sits precisely at the Layer-II/Layer-III coupling that this document argues is where the field's modeling error lives. Plume-in-the-loop control is higher-risk and higher-reward — it needs plume-field data you may have to generate, but it is the structurally distinctive feature of the domain and nobody has exploited it.
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