1. How to Fuse Independent Sensors Fast? (Nuclear Detection) Endre Boros MSIS & RUTCOR, Rutgers University Endre.Boros@rutgers.edu Joint work with Noam Goldberg, Paul B. Kantor and Jonathan Word

2. 2 The Problem: There are several tests that can be applied (document checks, passive and active sensors of several kinds). Find the “optimal” detection policy based on these tests! [Multiple branching + policy mixing!] Statement of Problem

3. Assumptions and Complexity Stochastically independent non-repeated sensors s sensors, k labels at each: >> 2 k s policies Futility of heuristic searches! 3 sensorslabelspolicies 42>> 2 16 45>> 2 625 82>> 2 256

4. 4 Move to a decision support model: Minimize total damage over all available policies Min P C(P) + πK(1- Δ(P)) Δ(P), C(P) - detection rate, and operating cost of policy P π (~ 0), K (~very large) - a priori probability of a “bomb”, and expected cost of false negative Max P {Δ(P) | C(P) ≤ B } mixing and domination of policies concave envelope of best policies 00.10.20.30.40.50.60.70.80.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Cost Detection

5. Sensor k Fusing sensor k on top of the given policies optimally is a multi-knapsack problem that can be solved by a modified greedy algorithm: Cost Detection Greedy Algorithm Dynamic Programming: Sensor Fusion

6. Sensor k Dynamic Programming: common concave envelope We then merge the given policies with the best combination of them with sensor k on top – and generate the common concave envelope of all these policies Cost Detection Greedy Algorithm

7. Dynamic Programming: Summary We build the concave envelope of best possible policies constructible from the given set of s sensors. We solve s2 s sensor fusion problems (for up to s ≤ 20) Each Sensor Fusion can be solved in O(P*B+Plog(P)) time, where B is the number of channels of the top sensor, and P is the number of pure strategies on the effective frontier.

8. 8 What about approximating the output in each step? Stroud and Saeger sensors (4 sensors, ~100 labels each) Time (sec)Number of PoliciesMax. Relative Error % 1440523190.005% 3.532040.05% 0.61330.5% 1.33 GHz Intel Atom processor

9. 9 Extremal frontier with 33 undominated policies

10. 2 R R RR R RR R II I I I I 13 3 4 1 4 1 41 4 Detection = 81.527% Cost = 0.1977826 units = $11.867 (< $13+)

11. Publications 1.Goldberg, N., Word, J., Boros, E. & Kantor, P. (2008). Optimal Sequential Inspection Strategies. Annals of Operations Research Vol. 187, 2011. 2.Boros, E., Fedzhora, P.B., Kantor, P.B., Saeger, K., & Stroud, P. Large Scale LP Model for Finding Optimal Container Inspection Strategies. Naval Research Logistics Quarterly, Vol. 56 (5), 404-420, 2009. 3. Kantor, P.B. & Boros E. Deceptive Detection Methods for Optimal Security with Inadequate Budgets: the Screening Power Index. Risk Analysis Vol. 30, 2010.