B. Kantor (email@example.com) & Endre Boros (firstname.lastname@example.org)
This site provides a gateway to some demonstration software and the major software package, SNSRTREE developed at Rutgers University with support from the Domestic Nuclear Detection Office of the Department of Homeland Security under NSF Grant Number CBET-0735910 and DNDO Contract Number DHS#2008-DN-077-ARI003-02, in the program directed by Dr. Austin Kuhn.
It is well known that a real test or sensor does not give perfect information. In fact, the way in which is fails to be perfect cannot be summarized in a single number. This is because there are two fundamentally different kinds of mistakes (false negatives, and false positives). Their relative importance depends on how serious they are, as mistakes, and also depends on how likely it is that there are true positives to be detected.
Tests and sensors come in many types, and it is important to have the best mathematical representation of how they perform, in order to optimize cost, detection and other considerations. This is done using a concept called the ROC plot or ROC curve. The ROC curve is a very important tool which permits us to simultaneously consider all kinds of tests and sensors, which may be radiation detectors, or X-ray imaging, or the examination of documents.
The basic principle underlying the ROC curve is that any particular tool or test that we use gives us something that we will call a "reading". This reading might be a numerical score assigned on the basis of examining documents, or it may be the number of counts occurring in a particular energy (frequency) window in a radiation detector. Whatever form the reading may take, the ROC curve uses a fundamental rule from statistics called the Neyman-Pearson (NP) Lemma.
The NP Lemma gives us the unique optimal way to put the different possible readings in order, so that we are sure to consider the most suspicious cases first. Graphically, the result of the NP Lemma is summarized in a single curve which is called the ROC curve.
In the ROC curve the horizontal axis shows what fraction of the innocent or harmless cases will be flagged. The vertical axis shows what fraction of the harmful cases will be flagged. If flagged items are inspected by some perfect method, the difference between the y-value of the curve and the 100% level is the fraction of threats that will be missed. There is a well-developed statistical theory surrounding the ROC curve, and we have drawn on it in developing our dynamic programming algorithm as described below.
The software demonstrates three very important principles: mixing and deception; preserving information; and dynamic programming.
The first principle is that most strategies will have, for a specific number of incoming harmless and harmful packages, some precise cost and some precise detection rate. However, the real-world budget is unlikely to exactly match those. This can be dealt with by using mixed strategies, which have the inherent added value of being deceptive. Under a mixed strategy that mixes two policies P1 and P2 in the proportions a and 1-a , it can be shown that the detection rate is mixed in the same proportions, and so is the cost. Thus, if we are in a position to spend more than the cost of policy P1 than the cost of policy P2 we can actually select a specific intermediate point by randomly using the policies for some fraction a of the cases, and using the other policy for the rest.
The most striking advantage of using these mixed deceptive strategies occurs when the budget is low, and comparable to the cost of applying the test to everything. The first demonstration lets you experiment with the great benefits that can be achieved by mixing in this case. While a naive strategy would require that we test everything first and then inspect the most suspicious ones, there are several kinds of mixing strategies. The first, which we call simple mixing, can be applied when we have enough money to test everything but not to inspect everything suspicious. In this case we gain some benefits by using a mixed deceptive strategy to decide which ones to inspect.
However the really dramatic gains come from the realization that we can use a mixture of the strategy "test everything and inspect the most suspicious" with the strategy of "do not even test". This is shown in the more complex mixing which is the third example, and can produce substantial gains in detection at the same cost or, alternatively, if the detection level is deemed satisfactory the strategy can be used to reduce costs.
The second major principle underlying our work is that when there is more than one test, the information gained in the first test, even though it is imperfect, lets us "tune" the second test so that we get the best possible information gain from the second test.
This insight can be applied to generate a very large linear program, which can be solved by the column-generation technique. In application the available budget (and any other constraints due to equipment, port capacity, etc.) are imposed as constraints, and the optimization problem is solved to give maximum detection at that particular budget. For planning purposes it is necessary to solve the large problem over and over, for different possible budgets.
This strategy has been applied to generate substantial improvements over the case where the information is not retained.
Boros, E., Fedzhora, L., Kantor, P.B., Saeger, K., Stroud, P. (2009). A Large-Scale Linear Programming Model for Finding Optimal Container Inspection Strategies. Naval Research Logistics (NRL), 56 (5), 404-420. April 16, 2009. DOI 10.1002/nav.20349 (Subscription Required).
Boros, E., Fedzhora, L., Kantor, P.B., Saeger, K., & Stroud, P. (2006). Large scale LP model for finding optimal container inspection strategies. Technical Report. RUTCOR 26-2006.
The third key feature of our work applies to the important case of tests that are "stochastically independent". For this widely occurring situation, it is possible to not only solve more complicated problems involving larger number of sensors, but also to simultaneously solve the problem for every possible budget. This is done using a powerful mathematical technique called dynamic programming.
The details are quite technical, and we provide a link below to a technical report. However, intuitively it can be understood this way:
With this information stored in the computer, our program then asks "what if?": "What would happen if we prefixed each of the other possible sensors in turn to this best mixture?". In other words, supposed we used a particular sensor beforehand, and saved the information. The computation requires great care, as we know that if we are going to use a particular sensor in the very last step there is no benefit to applying it in the step before. And "buying the same information twice". The program keeps track of all these complications and the result is a complete composite curve showing the best strategies that are available using exactly two sensors.
The program then continues this process iteratively, prefixing one more sensor in each step, until finally there are no sensors left to be considered. The overall result is a grand curve of detection versus cost of operation, which can be used as a basis for planning and budgeting.
This tool can also be used in a very powerful kind of "what if" formulation. If a new kind of technology is available, and we have done some basic tests so that we know its performance, as expressed in an ROC curve, then the SNSRTREE code can be used to determine how much it would improve the overall cost and detection profile, if we proceed with production and purchase of these devices.
One could use this tool even before a new technology is developed (money spent). If the researchers suggesting a new type of sensor can give a reliable estimate of the ROC it will achieve, we can compute the marginal improvement that will result. One may also go from needs to technology. Decision makers may describe a "desired" ROC which would provide the needed marginal improvement, and pose that as a challenge to researchers, who would have to develop a corresponding physical device.
These alternatives provide a very good way to do prudent management of the interrelation between fundamental research, which continues to bring us more powerful tools, and the realities of having to defend the nation using a limited budget.