Infrastructure Fortification

Tri-Level Defender-Attacker-Defender Game

Which power stations should you harden against alien EMP attacks? A defender fortifies, then an attacker strikes, then the defender operates with surviving infrastructure. This max-min-max structure requires three nested optimizations — one of the hardest problem structures in OR.

The Scenario

EMP Hardening

⚠ Educational context: This toy example is designed for educational clarity. Real infrastructure fortification problems involve orders of magnitude more complexity and are subject to strict legal and ethical review.

A city has 4 power plants serving 5 demand sectors. An alien EMP weapon can destroy one unfortified plant. The defender can fortify 1 plant (budget B_f=1) before the attacker chooses which remaining unprotected plant to strike (budget B_a=1). After the attack, the defender operates the surviving plants to satisfy as much demand as possible. The question: which plant to fortify?

Defense Domain	OR Element	Symbol	Example
Power station	Facility	i ∈ {A,B,C,D}	Plant-A (cap 30)
City district	Demand point	j ∈ {1,...,5}	Sector-1 (demand 15)
Hardening	Fortification	xᵢ ∈ {0,1}	Fortify Plant-A
EMP strike	Attack	yᵢ ∈ {0,1}	Destroy Plant-B
Power delivery	Operations	z	Serve 65 MW

Stage 1 — Defender fortifies (first mover): max_x [ Stage 2 value ] s.t. Σᵢ xᵢ ≤ B_f // fortification budget (= 1) xᵢ ∈ {0,1} Stage 2 — Attacker strikes (worst case): min_y [ Stage 3 value ] s.t. Σᵢ yᵢ ≤ B_a // attack budget (= 1) yᵢ ≤ 1 - xᵢ // cannot destroy fortified plant yᵢ ∈ {0,1} Stage 3 — Defender operates (surviving network): max_z Σˇ satisfied_demand(j) s.t. flow conservation, capacity of surviving plants // This is a max-min-max problem: three nested optimizations // Cannot be solved by standard MIP directly // Approaches: Benders decomposition, column-and-constraint generation // This demo uses brute-force enumeration (4 fortify × 3 attack = 12 combos)

★☆☆ Educational Demo

This page demonstrates the conceptual structure of a tri-level optimization problem. The brute-force enumeration shown here is only feasible because the instance has 4 facilities with budget 1 each. Real infrastructure fortification problems (Brown et al. 2006) have thousands of components and require specialized decomposition algorithms.

Interactive Solver

Enumeration Matrix

★☆☆ Educational Demo (brute-force)

4 Plants · 5 Sectors · B_f=1 · B_a=1

Plants: A (cap 30, serves S1,S2), B (cap 25, serves S2,S3), C (cap 20, serves S3,S4), D (cap 35, serves S4,S5,S1). Demands: S1=15, S2=20, S3=10, S4=25, S5=12. Total demand = 82.

References

Published Brown, G., Carlyle, M., Salmerón, J., & Wood, K. (2006). “Defending critical infrastructure.” Interfaces, 36(6), 530–544. — Original defender-attacker-defender framework for infrastructure protection.

Published Alderson, D.L., Brown, G.G., Carlyle, W.M., & Wood, R.K. (2011). “Solving Defender-Attacker-Defender Models for Infrastructure Defense.” Operations Research, 59(6), 1423–1442. — Decomposition algorithms for tri-level problems.

Preparing for First Contact

If the aliens arrive, we suspect you will not be visiting a GitHub Pages site. We do recommend the Hungarian algorithm. It works on any planet.

All Defense Applications GitHub

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Educational Fiction Disclaimer

This is a fictional educational scenario.

All data and parameters are entirely fictional
No military applications intended
The author advocates for peace