1. Contraband Detection and Retesting

2. The Inspection Problem A sensor is a device used to attempt to determine some truth about an object; we will assume that our objects fall into two categories (simply ‘good’ or ‘bad’).

3. Detection and False Alarms Each sensor will have a detection rate d and a false alarm rate f, which are the probabilities that a sensor says that an object is bad when it is bad or good, respectively. They are associated with some threshold t; a sensor will say an object is bad if the reading exceeds the threshold level. There is also a cost associated with using a sensor, which can lead to the use of a mixed strategy where they are not always used.

4. The ROC Curve (d vs f) We can plot d as a function of f for a given sensor and threshold. Using the sensor and threshold gives us a point (d 0,f 0 ). We can also clearly obtain the points (0,0) and (1,1) by manually inspecting nothing and everything, respectively. By using mixed strategies with suitable probabilities, it is possible to operate at any point along a line between any of the points we already have.

5. The C-d Curve (d vs C) Here, we wish to minimize (where C m is the cost of a miss and C f the cost of a false alarm): C total = C m (1-d) + C f f = C m + C f f – C m d For some of the threats we consider, C m is very large, but we cannot afford the inspection policy of manually checking everything which that would suggest. This cost is minimized by maximizing d, so we can plot the cost of inspection and harm to commerce against d, and find the optimum point for a given budget b.

6. One Example C-d Curve (using one of sensor A)

7. Multiple Sensors: A Decision Tree Suppose we have multiple stochastically independent sensors (all of type ‘A’). We can then set up many different decision trees using multiples of that sensor (one is pictured here). OK A A A X X

8. Single and Multiple Sensor Curves Side-by-Side Comparison

9. Ideas for Research The new version can take the performance of multiple sensors and aggregate them into a single sensor, which should increase the number of sensors we can simulate significantly. We wish to determine what the limit curve looks like as the number of sensors approaches infinity, to find a hypothesis that can be proven to be the limit mathematically.