Planetary Defense Operations

WTA · Network Interdiction · MCLP · Security Games · POMDP

What if the most consequential operations research problems were not about supply chains or hospital scheduling — but about defending Earth from an extraterrestrial threat? This fictional domain maps 12 canonical OR problem families onto alien defense scenarios, from weapon-target assignment to Stackelberg security games. The mathematics is real. The aliens are not.

Every problem here has a direct real-world counterpart: WTA is used in missile defense, network interdiction in counter-terrorism logistics, MCLP in sensor network design, and security games in airport protection (ARMOR, IRIS). The alien framing makes the formulations memorable while the underlying OR is academically rigorous.

Weapon-Target Assignment (WTA) is one of the oldest problems in military OR, first formulated by Manne (1958). Given m weapons (interceptors) and n targets (incoming threats), each with a threat value Vˇ, and single-shot kill probabilities pᵢˇ for each weapon-target pair, assign weapons to targets to minimize expected surviving threat value. The objective ∑ˇ Vˇ · ∏ᵢ (1-pᵢˇ)ⁿᵢˇ is nonlinear — making this NP-hard even for identical weapons.

OR Mapping

Defense Domain	OR Element	Example
Orbital laser	Weapon i	Laser Alpha (3 shots)
Alien missile	Target j	Mothership Scout (V=100)
Hit chance	Kill probability pᵢˇ	0.8
Ammo count	Weapon capacity Wᵢ	3 shots
Survival chance	∏(1-p)ⁿ	0.04 (two hits at p=0.8)

MINIMIZE ∑ˇ Vˇ · ∏ᵢ (1 - pᵢˇ)^xᵢˇ subject to: ∑ˇ xᵢˇ ≤ Wᵢ ∀ i ∈ Weapons // weapon capacity xᵢˇ ∈ ℤ⁺ ∀ i,j // non-negative integer

Mini-Demo: Greedy WTA

★★☆ Heuristic

3 weapons × 4 targets. Greedy assigns the highest-kill-probability weapon to the highest-value remaining target.

See Full Application → 3 algorithms including Branch & Bound

Key References

Published Manne, A.S. (1958). “A Target-Assignment Problem.” Operations Research, 6(3), 346–351.

Published Ahuja, R.K., Kumar, A., Jha, K.C., & Orlin, J.B. (2007). “Exact and Heuristic Algorithms for the Weapon-Target Assignment Problem.” Operations Research, 55(6), 1136–1146.

Multi-Agent Path Planning (MAPP) coordinates multiple autonomous agents moving through shared space toward individual goals while avoiding collisions. The optimal control formulation is computationally intractable for more than a handful of agents. In practice, Reynolds flocking rules (1987) — separation, alignment, cohesion, and goal-seeking — provide a decentralized heuristic that produces emergent coordinated behavior without centralized optimization.

OR Mapping

Defense Domain	OR Element	Example
Defense drone	Agent i with position pᵢ	Drone Delta-3
Alien scout pod	Goal position gᵢ	Pod at (320, 180)
Collision zone	Safety distance d_safe	25 px
Steering force	Control input uᵢ	Reynolds weighted sum
Swarm behavior	Emergent coordination	No central controller

// Reynolds Flocking (what the demo actually implements): uᵢ(t) = w₁·separation(i) + w₂·alignment(i) + w₃·cohesion(i) + w₄·goal_seek(i) separation(i) = ∑ˇ∈Nᵢ (pᵢ - pˇ) / ||pᵢ - pˇ||² cohesion(i) = avg(pˇ for j ∈ Nᵢ) - pᵢ goal_seek(i) = (gᵢ - pᵢ) / ||gᵢ - pᵢ||

Mini-Demo: Reynolds Swarm Simulation

★☆☆ Educational Demo

8 drones pursue 4 alien scout pods using Reynolds flocking rules. This is a simulation, not an optimization solver.

See Full Application → adjustable flocking weights & velocity obstacles

Key References

Published Reynolds, C.W. (1987). “Flocks, herds and schools: A distributed behavioral model.” SIGGRAPH Computer Graphics, 21(4), 25–34.

Published Olfati-Saber, R. (2006). “Flocking for multi-agent dynamic systems.” IEEE TAC, 51(3), 401–420.

Shortest-path network interdiction is a bi-level optimization problem: a leader (defender) removes up to k arcs from a network to maximize the follower's (attacker's) shortest path from source to sink. First formalized by Israeli & Wood (2002), it models disrupting supply chains, communication networks, and logistics routes. The single-level MIP reformulation uses LP duality of the follower's shortest-path problem.

OR Mapping

Defense Domain	OR Element	Example
Alien supply base	Source node s	AlienBase
Earth outpost	Sink node t	EarthOutpost
Transit relay	Intermediate node	Relay1, Junction
Supply route	Arc (i,j) with cost cᵢˇ	AlienBase→Relay1 (cost 5)
Disruption budget	Interdiction budget k	k = 2 arcs

MAXIMIZE (leader) πₜ - πₛ // max shortest path after interdiction subject to: πˇ - πᵢ ≤ cᵢˇ + Mᵢˇ · xᵢˇ ∀ (i,j) ∈ A // dual + big-M ∑ xᵢˇ ≤ k // interdiction budget xᵢˇ ∈ {0,1} // binary // Mᵢˇ = longest simple s-t path length - cᵢˇ + 1 // NOT an arbitrary large number (Israeli & Wood 2002)

Mini-Demo: Network Interdiction

★★★ Exact (small k)

6-node network. Budget k=2. Click “Find Optimal” to enumerate all C(7,2)=21 arc pairs and find the interdiction maximizing shortest path.

This toy example is designed for educational clarity. Real network interdiction problems involve orders of magnitude more complexity and are subject to strict legal and ethical review.

See Full Application → interactive arc clicking & greedy heuristic

Key References

Published Israeli, E. & Wood, R.K. (2002). “Shortest-path network interdiction.” Networks, 40(2), 97–111.

Published Cormican, K.J., Morton, D.P., & Wood, R.K. (1998). “Stochastic network interdiction.” Operations Research, 46(2), 184–197.

All Planetary Defense Applications

12 interactive applications — click any to explore

Interceptor Assignment

Weapon-Target Assignment · WTA

Try It Live →

Drone Swarm Coordination

Multi-Agent Path Planning

Try It Live →

Supply Line Interdiction

Bi-Level Network Interdiction

Try It Live →

Defense Unit Task Allocation

Generalized Assignment · GAP

Try It Live →

Dynamic Defense Reallocation

Approximate Dynamic Programming

Try It Live →

Layered Defense Scheduling

RCPSP · Multi-Stage

Try It Live →

Infrastructure Fortification

Tri-Level Defender-Attacker-Defender

Try It Live →

Radar Network Placement

Maximal Covering Location · MCLP

Try It Live →

Unpredictable Patrol Routes

Stackelberg Security Game

Try It Live →

Reconnaissance Planning

POMDP · Partial Observability

Try It Live →

Resource Stockpiling

Newsvendor · Stochastic Programming

Try It Live →

Perimeter Defense Game

Stackelberg Game · Mixed Strategies

Try It Live →

Preparing for First Contact

Typically this section would encourage you to explore the GitHub repository, implement the algorithms, or contact us for collaboration opportunities. However, we find ourselves uncertain about the appropriate call to action for alien defense preparedness.

If the aliens arrive, we suspect you will not be visiting a GitHub Pages site. If they don't arrive, you have learned some excellent OR methods that transfer seamlessly to logistics optimization, sensor network design, project scheduling, and facility location — all of which are considerably more likely to be useful on a Tuesday afternoon.

We do recommend the Hungarian algorithm. It works on any planet.

Back to Normal OR Problems GitHub (for peaceful use only)

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Educational Fiction Disclaimer

This is a fictional educational scenario.

All “alien invasion” content exists purely to teach Operations Research concepts in an engaging way
All data, scenarios, and parameters are entirely fictional
No actual military applications are intended or endorsed
This content should NOT be used for any real-world defense planning
The author advocates for peace and opposes militarization
If actual extraterrestrials arrive, please consult astrophysicists and diplomats, not this website 🔭