^{1}

^{*}

^{2}

^{*}

We consider the problem of searching for a target that moves between a hiding area and an operating area over multiple fixed routes. The search is carried out with one or more cookie-cutter sensors, which can detect the target instantly once the target comes within the detection radius of the sensor. In the hiding area, the target is shielded from being detected. The residence times of the target, respectively, in the hiding area and in the operating area, are exponentially distributed. These dwell times are mathematically described by Markov transition rates. The decision of which route the target will take on each travel to and back from the operating area is governed by a probability distribution. We study the mathematical formulation of this search problem and analytically solve for the mean time to detection. Based on the mean time to capture, we evaluate the performance of placing the searcher(s) to monitor various travel route(s) or to scan the operating area. The optimal search design is the one that minimizes the mean time to detection. We find that in many situations the optimal search design is not the one suggested by the straightforward intuition. Our analytical results can provide operational guidances to homeland security, military, and law enforcement applications.

Since World War II, search theory has provided many valuable tools to decision makers in both civilian and military operations including search and rescue (SAR), intelligence, surveillance, and reconnaissance (ISR), and enhanced situational awareness [

In this article, we consider a search problem as depicted in

We consider the situation where one or more searchers are deployed. Each searcher possesses an ideal sensor. The definite-range law, or cookie-cutter sensor is an ideal sensor which detects the target instantly once the target comes within distance R to the center of the sensor but cannot detect the target outside that range. The radius R is called the detection radius of the searcher. For the present discussion, we assume that the detection radius of the searcher is large enough to cover the full width of any route in the problems considered here. Under this assumption, if the target chooses to travel along a route that is monitored by a searcher, then the target will definitely be detected by the searcher along that route. Our goal here is to evaluate the performance of placing the searcher(s) at various routes/areas and obtain a search plan that minimizes the mean time to detection of the target.

The rest of this paper proceeds as follows. In Section 2, we consider a moving target which follows the constrained pathways defined above in the presence of one searcher and for which the probability of taking a given route is the same for both travel directions. Section 3 extends the results of Section 2 to the case where the target has different probability of taking a given route for the two travel directions. In Section 4 we further generalize the discussions in Section 3 to include two searchers. Section 5 presents a discussion of three or more searchers. Finally, Section 6 concludes the paper and points out avenues for future study.

We consider the situation where there is one searcher and the target behaves as follows.

・ The dwell time of the target in the hiding area is exponentially distributed with rate r_{f}, the forward rate of going from the hiding area to the operating area.

・ On its travels between the hiding area and the operating area, the target takes route k with probability p_{k}. In particular, the probability of choosing route k is the same for both directions. This assumption will be relaxed in the later discussion.

・ The dwell time of the target in the operating area is exponentially distributed with rate r_{b}, the backward rate of going from the operating area back to the hiding area.

・ The travel time between the operating area and the hiding area is negligible in comparison with the dwell times in the hiding area and the operating area. Mathematically, we treat the travel time along a route as zero.

After this setup, we have several options of placing the cookie-cutter searcher.

First, we investigate the situation where the searcher is placed on route k. In this case, the target is detected if and only it travels via route k. As mentioned above, we assume that the detection radius of the searcher is large enough to cover the full width of the route and the detection is instantaneous once the target enters into the detection area of the searcher.

Before the arrival of the searcher, the target jumps between two Markov states with forward rate r_{f} and backward rate r_{b}, as shown in

Once the searcher arrives at route k to commence monitoring there, the diagram of Markov transitions contains 3 states. As shown in

Let

with initial conditions (t = 0 being the time when the searcher arrives at route k)

The initial conditions (3) correspond to the equilibrium distribution of the target between the hiding area and the

operating area before the start of search. Equation (2) and

Both

where

The probability that the target is not detected by time t (i.e. the non-detection probability, also called the escape probability) has the same form which reads

where the coefficients c_{1} and c_{2} are calculated below and are contained in Equation (8).

The non-detection probability is constrained by two other conditions, namely,

where the second condition is derived from differential equations (2) and initial conditions (3). The two constraints in (7) allow us to calculate the coefficients c_{1} and c_{2} in (6) explicitly. After some algebra, we find the non-detection probability has the expression

where the coefficient

Note that

When

Therefore, when

which is approximately proportional to probability

In an effort to use one quantity to characterize the decay of the non-detection probability, we compute the mean time to detection. Let

The derivative of

Thus, the mean time to detection is a strictly decreasing function of

When

which is inversely proportional to

Now we consider the situation where the searcher looks over only the operating area. Since the searcher is committed to searching the operating area, the target can only be detected when it is in the operating area.

Here we do not specify how the searcher conducts search over the operating area, which may include random search, parallel sweeps (mowing the lawn), spiral-in or spiral-out paths. We consider the conditional probability of the non-detection given that the target is in the operating area. We model the conditional non-detection probability as a single exponential decay.

where

In addition to transitions between the hiding area and the operating area, there is now a new transition of the target going from the operating area to being detected. Upon the arrival of the searcher in the operating area, the Markov transitions of the target are depicted in the diagram shown in

For mathematical convenience, we represent the detection rate

where

The time evolution of probabilities

subject to initial conditions

Proceeding exactly as for Case 2A, we can succinctly express the non-detection probability as

where

The non-detection probability also satisfies two other conditions:

We solve for coefficients

where the coefficient

As discussed before, the non-detection probability contains two exponential modes: one decays with fast rate

From the results of these asymptotic expansions, we see that when

Accordingly, the non-detection probability takes the approximate form

Again, we calculate the mean time to detection from

Notice that the mean time to detection contains a constant part that does not decrease with increasing detection rate

tion can occur only when the target moves to the operating area. The constant term

elapsed time until the first arrival of the target in the operating area. This constant term gives the lower bound on the mean time to detection, which can be achieved when the detection rate in the operating area is much larger than other transition rates:

Combining the results of Case 2A and Case 2B, we can now determine the optimal placement of the searcher (on a particular route vs. in the operating area ) in order to attain the minimum mean time to detection of the target. We introduce the probability and identify of the route that is most likely to be travelled by the target:

We compare the mean times to detection in the two cases

If the former is smaller, we place the searcher on route

In this section, we consider the target which behaves the same as in Section 2 except that the probability of taking route k may have different values for the forward travel (from the hiding area to the operating area) and the backward travel (from the operating area to the hiding area). We use

・ On its travel from the hiding area to the operating area, the target takes route k with probability

・ On its travel from the operating area back to the hiding area, the target chooses route k with probability

We study the performance of placing the searcher on a route, which is analogous to Case 2A. Note that regardless of the target’s behavior in selecting which route to take, the results of Case 2B apply directly here, which is affected only by how frequently the target visits the operating area and how fast the target is detected while in the operating area.

Case 3A: The Searcher Is Placed on Route kTo analyze the case where the searcher is placed on route k, we observe that the Markov transitions are illustrated in

being detected). The transition rate from the hiding area to being detected is

The time evolution of probabilities

with initial conditions

The non-detection probability is of the form

where

The non-detection probability needs to meet two more conditions:

With the help of these two constraints, we can determine the coefficients

with the coefficient

As in the earlier discussion, the non-detection probability is a linear combination of two exponential modes: one decreases with fast rate

When both

It therefore follows that when

which is approximately proportional to probability

Next we find the mean time to detection. Using the formula (23), we compute the mean time to detection:

The most important thing to notice about this result is that the mean time to detection is not just a function of

An Example: Let us consider a target traveling between the hiding area and the operating area. The transition rates and the probabilities of taking individual routes in each travel direction are listed below.

The small value for

Intuitively, the optimal route for placing the searcher seems to be

It is clear that route 4 has the smallest mean time to detection, and thus, is the optimal route for placing the searcher. This example highlights that when the probability of taking a given route has different values for the two travel directions, the optimal placement of the searcher is not the one suggested by straightforward intuition.

We study the effect of two searchers where each searcher acts and detects independently. We assume the same setup for the target as in Section 3. The difference here is that there are two cookie-cutter searchers deployed in search for the target. We consider several options of placing the two cookie-cutter searchers.

Since we assume that one searcher is capable of covering the full width of a route, there is absolutely no benefit for placing more than one searchers on any one route. Thus, the two searchers are to be placed on two different routes. In this case, the mathematical formulation is exactly the same as in Case 3A in Section 3 except that probability

Then, the non-detection probability is given by

where eigenvalues

Recalling (25) obtained in Case 3A of Section 3, one finds the mean time to detection in the form

where function

The optimal routes for placing the two searchers are selected by minimizing the mean time to detection:

It is attempting to simplify the selection process by using the optimal route for a single searcher plus the second optimal route for a single searcher. The example below demonstrates that this intuitive approach does not necessarily give us the optimal routes for placing the two searchers.

An example: We consider a target traveling between the hiding area and the operating area over 5 routes. The transition rates and the probabilities of taking route k in the two travel directions are given below for the target

When only a single searcher is present, the mean times to detection for individual routes are calculated from (25)

The optimal route and the second optimal route for placing a single searcher are, respectively, route 2 and route 1. It seems reasonable to conclude that in the case of two searchers, the optimal routes for placing the two searchers are routes 2 and 1. However, this is not true. When the two searchers are placed respectively on routes 1 and 2, the mean time to detection is then obtained from (28) as

In comparison, when the two searchers are placed respectively on routes 1 and 3, the mean time to detection becomes

which is still not yet the optimal. It turns out that the smallest mean time to detection among all possible pairs of routes is achieved by placing the two searchers respectively on routes 3 and 4:

In this example, the optimal pair of routes for placing the two searchers are routes 3 and 4, not the collection of optimal and second optimal routes for a single searcher.

When both searchers are put to search only the operating area, the mathematical formulation is exactly the same as in Case 2B of Section 2 except that the detection rate

Based on the results in Case 2B of Section 2, the mean time to detection can be written as

where function

The last mathematical formulation is for the case where one searcher is put on a pathway while the other scans the operating area. After the arrival of one searcher at route k and the other in the operating area, the target evolves stochastically according to the Markov process shown in

For mathematical convenience, we express the detection rate

These probability functions satisfy the initial conditions

Once again, one finds that the non-detection probability has the form

where

The coefficients c_{1} and c_{2} can be derived by imposing two constraints on the non-detection probability:

From these two constraints, one derives that the non-detection probability is given by

where coefficient

The mean time to detection is calculated from the non-detection probability

This result lies at the heart of everything else that is going on and it includes all of these as special cases. In particular, function

We are now ready to summarize what we have found so far.

When the two searchers are placed on two routes k and j (one searcher on each route), the shortest mean time to detection is given by

When one searcher is placed on route k and the second searcher is placed in the operating area, the smallest mean time to detection is similarly defined by

When both searchers are placed in the operating area, the mean time to detection is

By examining these cases, we obtain the overall minimum mean time to detection over these three options:

where function

We now turn to the case where three cookie-cutter searchers are deployed in the search for the target. Probabilistic independence is assumed for each searcher. Suppose the target behaves exactly the same as described in Sections 3 and 4. We apply function

When all three searchers are placed on three routes (one searcher on each route), the shortest mean time to detection is found to be

Similarly, when two searchers are placed on two routes (one searcher on each route) and the third searcher is placed in the operating area, the shortest mean time to detection is

When one searcher is placed on route k and the two other searchers are placed in the operating area, the smallest mean time to detection is equal to

Finally, when all three searchers are placed in the operating area, it follows that the mean time to detection is

We conclude that the overall minimum mean time to detection over these options can be computed by comparing the results in these cases

It is also evident that this method can be extended to finding the optimal placement for any number of searchers.

This paper presented a mathematical approach to investigate the search problem of detecting a mobile target that travels between a hiding area and an operating area over a collection of fixed pathways in the presence of single or multiple searchers with cookie-cutter sensors. Both the dwell time of the target in the hiding area and the dwell time of the target in the operating area were assumed to be exponentially distributed and were modeled mathematically by Markov transition rates. The travel time of the target on a pathway was assumed to be short in comparison with the dwell times in the hiding area and the operating area. The target took a route to and from the operating area based on a probability distribution. Under these assumptions, a mathematical formulation was developed and solved analytically.

The main results can be summarized as follows.

1) When only one searcher is present, one can compute the mean time to detection, respectively, when the searcher is placed on route k or when the search is placed in the operating area. By comparing the mean times to detection, we can decide whether to put the searcher on a certain route or in the operating area.

2) In a similar fashion, when multiple searchers are deployed, we can calculate the mean times to detection for various scenarios and thereby obtain the overall minimum time to detection and the optimal placement of searchers.

It was discovered that in many cases the optimal search design was not necessarily the one indicated by straightforward intuition.

There are a few possible future research directions. For example, extending current study to include time- dependent transition rates and detection rates would provide a more sophisticated modeling. Multiple targets could also be considered in the future studies.

Hong Zhou would like to thank Naval Postgraduate School Center for Multi-INT Studies for supporting this work. Special thanks go to Professor Jim Scrofani and Deborah Shifflett. The views expressed in this document are those of the authors and do not reflect the official policy or position of the Department of Defense or the U.S. Government.