^{1}

^{*}

^{2}

^{*}

^{2}

^{*}

Assume that a target is hidden or lost in one of several possible locations and is to be found by the unmanned aerial vehicle (UAV). A target can be either a hostile object or missing personnel in remote areas. Prior probabilities of target locations are known. Inspection operations done by the UAVs are imperfect, namely, probabilities of overlooking the hidden target and probabilities of false alarms exist for any possible location. The UAV has to sequentially inspect the locations so that to find the target with the minimum loss or damage incurred by the target before it is detected subject to a required level of confidence of target identification. A fast (polynomial-time) priority-based algorithm for finding an optimal search strategy is developed.

The need for searching for hidden or lost objects arises in many applications. Common civilian applications are policing, control of air/water/soil pollution, fire fighting, prevention of natural disaster like floods, and others. Common military applications are the battlefield surveillance, detection of enemy intrusion or hidden hostile targets, rescuing lost people or objects in the open sea, and many others. We refer the interested reader to Stone [

Suppose a target is hidden in some region. Until being found, it may cause loss or damage to the environment, the damage scale being dependent on the location of the target and the time needed to detect and neutralize it. The problem is to efficiently detect and remove or neutralize the target with the help of unmanned aerial vehicles (UAV). The UAV (also called drone) is an aircraft without a human pilot on board. It is controlled either autonomously by on-board computers or by an operator (or a group of operators) on the ground or in another aircraft. The UAVs are preferred for missions that are dangerous, complicated or just impossible for manned aircraft. Numerous examples of advanced practical applications of UAVs can be found in Valavanis [

A typical example arising in military logistics is a discrete-time search optimization problem in which a single automatic searcher (a military robot or an UAV) moves through a discretized 3-dimensional airspace and needs to find a static target located/hidden in a finite set of possible geographical locations. The searching automated device is subject to search-time constraints, fuel consumption and other factors. The typical objective of the search is to maximize the probability of detection or minimize the cost (or time) of the search (see, e.g., Dell et al. [

The choice of a search strategy strongly influences the search costs and losses incurred by the search as well as the probability of finding the target within a specific time limit. In military search-and-detection missions of the UAVs, the problem of finding good strategies has its own specificity because, unlike manned aircraft, these mobile agents need to operate in autonomous regime, without human advice and involvement. Under hostile or sabotage circumstances, like radio interference, two types of errors in search-and-detection operations usually occur: 1) the so-called “false-negative detection error”, wherein a target object is overlooked; and 2) the “false- positive detection error”, also called “a false alarm”, in which a decoy or a false target is wrongly classified as a correct target. Unfortunately, the problem of selecting the best search strategy is fundamentally hard due to its probabilistic nature, combinatorial structure and nonlinearity induced by the detection probability. In particular, looking twice into the same location by a searcher generally does not double the detection probability.

There exist hundreds of reviews, papers and texts providing a deep background and presenting wide applications of the discrete search problems (see, e.g., Benkoski et al. [

In the present work, we study a search-and-detection process subject to the false-negative and false-positive inspection outcomes. The general resource-constrained problem being NP-hard, we restrict our study to finding optimal strategies in a certain special scenario described below. The search is to minimize the losses incurred during the search. We organize the search process using a greedy strategy in which, at each step, the on-board computer used by the UAV computes a current search effectiveness for each location and sequentially searches for a next location with the highest current search effectiveness.

Complementary to the discrete search model developed by Chung and Burdick [

In the scenario studied in the present paper, the local-optimal (greedy) strategy yields a global optimal. Being attractive due to its simplicity and computational efficiency, such local-search strategy guarantees finding an optimal search sequence with a pre-specified confidence level of target identification. Several earlier known discrete search models, in particular, the search model with only false-negative outcomes (Kadane [

The remainder of the paper is organized as follow. The next section overviews the related work. Section 3 introduces the discrete search problem in more detail. Section 4 analyses the problem and introduces its solution algorithm. Section 5 presents numerical examples and computational results. Section 6 concludes the paper.

The discrete search problem is one of the oldest and most popular problems of Operational Research. Its pioneering study was made by Bernard Koopman and his team during the World War II to provide efficient methods for detecting submarines (Koopman [

For the sake of completeness, we start with a brief overview of related classic results obtained in the 1970s. Sweat [

Trummel and Weisinger [

When the location inspections are imperfect, such a problem type may need an infinite sequence of steps as its solution. Thus, the problem is to find an optimal strategy rather than an optimal sequence itself. The latter sequence depends upon a random instant when the searcher finds the target. A typical stopping rule for the random search process is the following: “Stop at the step when the target is found for the first time”. Obviously, this rule, being applicable for the case of no-positive-false tests is not valid if the positive-false error occurs. Other criteria for the termination of the search process are treated by Chew [

In recent decades, the operational research community has made great strides in the problem of sequential search under discrete time and space. Research results in Chung and Burdick [

In the present work, we continue and extend the results mentioned above, focusing on studying the search scenario in which the imperfect inspections can be either false-positive, or false-negative, or of the both types. A main contribution of our work is that, complementary to the works of Kress et al. [

We define the sequential discrete search problem as follows. A discrete search area of interest (AOI) is a complex geographical area which contains a set of N possible locations of a stationary target. The objective of the search is to detect and neutralize a hidden or lost hostile target. A target is to be found by a single searcher. Given the stationary nature of the target, it is assumed to be present throughout the entire duration of the search process, i.e., the target does not leave the search area and new targets do not appear. A target location in the search area is uncertain. In other words, an inspection of each location can be imperfect. This means that there given prior probabilities α_{i} of the false alarm (if the UAV’s sensors falsely announce that the target is found when the location i is clean, and, besides, there given prior probabilities β_{i} of overlooking the target-when actually location i contains a target,

A formal description of the problem is as follows. A system contains N modules

・ p_{i}―prior probability that location i contains a target;

・ α_{i}―prior probability of a “false alarm”, or a false-positive outcome; this is a conditional probability that sensors of the autonomous UAV, after inspecting location i, declare that the target is discovered in location i whereas, in fact, it is not in this location;

・ β_{i}―prior probability of overlooking, or a false-negative outcome; this is a conditional probability that the sensor of the UAV declares that target is not in location i whereas, in fact, it is exactly there;

・ t_{i}―expected time to inspect location i;

・ c_{i}―search cost rate per unit time when searching location i; this is the amount (in monetary or physical units) of loss or damage incurred during one time unit of search;

・ CL_{0}―a pre-specified confidence level, this is a measure of confidence that the target is identified correctly; this value is close to 1 and will be exactly defined below.

Each sequential inspection strategy is specified by an infinite sequence of steps:

where s[n] denotes the location number (further called the element number), more exactly, the number of the location which is inspected at the n-th step of sequence s, all

In what follows, the terms strategy and sequence are used interchangeably.

We suppose that the UAV makes its search moving in a certain sequence of steps s. In order to exactly define the stopping rule of the search process, we first introduce a notion of the confidence level CL. Without loss of generality, we suppose that the on-board computer, at each step of s, say i, may count and memorize the search history, that is, how many times, denoted h_{i}, the sensor has already observed that the location i contains the target. (Recall that, due to the presence of the false positive outcomes, we are not obliged to immediately stop searching in the case when the sensor claims for the first time that it discovered the target).

Notice that in the scenario studied by Kress et al. [_{[i,h]} that the examined location i indeed contains the target under the condition that, during the previous sequential inspections, the (imperfect) UAV’s sensors have already declared h times that the location i contains the target. Naturally, the conditional probability p_{[i,h] }increases with h growing, and we require that the p_{[i,h]} should be sufficiently high. More concretely, the p_{[i,h]} value should not be less than a pre-specified value that we called the confidence level and denoted by CL (for example, one can take CL = 0.95 or 0.99). Further we show that the on-board computer can me programmed to compute how many times, during their sequential inspections, the UAV’s sensors has to observe that the cell i does contain the target, in order to guarantee that p_{[i,h]} ≥ CL. A specific feature of this scenario is that the computer memorizes the entire history of the sensors’ observations during the UAV’s search.

The minimum positive integer h, such that the probability a_{[i,h] }exceeds or equals the confidence level CL is called the critical height H_{i}; this value will be precisely determined below from the non-linear Equation (1) below depending on the values of the input probabilities. Notice that all the values H_{i}, for each cell i, can be computed and memorized before the search process starts. In this scenario, the basic idea is that, instead of using the human involvement at each positive test, like in the Kress’s et al. [_{i}-th time, cell i is discovered to contain the target. It is clear that in this case the probability of the correct decision will be guaranteed to be sufficiently high, without the human-made additional testing. After arriving to such a case, the UAV carrying an explosive payload may hit a hostile target, or a human involvement, like sending a rescue team, is performed.

The number accumulated up to a certain step n, counting that last step, and associated with the location searched at step n in s is called the (current) height of the cell s[n] = i and denoted by h_{i}. As explained above, the conditional probability p_{[i,h]} is defined as “the probability of that the target is correctly identified under condition that the sensors have repeatedly declared a fixed number of times that the target is indeed in location i”. Clearly, the p_{[i,h]} values can be different for different location i, depending upon given probabilities p_{i}, α_{i} and β_{i}. The precise meaning of the term repeatedly declared a fixed number of times will be explained below in detail.

Let us summarize the said above. Without loss of generality, we suppose that the on-board computer of the UAV, in each step of the search sequence s, say n, counts and memorizes the entire search history, that is, how many times the UAV’s sensors had claimed, before the current step n, that different locations (indexed s[n] = i) contain the target. Such a number, accumulated up to step n and associated with any location searched at that step in s is called the height of the location s[n] = i at step n and denoted by h_{i} = h_{i}(n). For brevity, in the following we shall omit the index n in h_{i}(n) if this does not lead to confusion. Notice that, due to existence of the false alarms the search should not stop when at a step n the sensors declare for the first time that they discovered the target. We need to continue the search until the probability a_{[i,h]} reaches the pre-specified value CL.

Now we can give the exact definition: for a given n value, the confidence level CL_{i} at step n is defined as the desired lower bound for the conditional probability _{i} times (that is, in h_{i} previous steps) that the target is located in location i. When the h_{i}(n) value at each step n of s is known, there is a direct way to compute the corresponding probability _{i}, as the following claim states.

Claim 1. (a) The conditional probability _{i} = h_{i}(n) as follows:

(b) Probabilities _{i}.

The proof of Claim 1 is given in the next section.

Given the input probabilities, the optimal search scenario is specified by the following conditions:

1) the elements (locations) are inspected sequentially and independently of each other;

2) for any search strategy and any target location, the outcomes of inspections are independent;

3) the stopping rule is defined as follows: Given a required (permitted) value of the confidence level CL_{0}, say CL_{0} = 0.95, the search process stops when, at some step n for some location s[n] = i and the corresponding height h_{i}, the requested confidence level of identifying the target is achieved, that is:

Claim 2. If, for any location i, the probability 1 − β_{i} of “no-overlooking” exceeds the false-alarm probability α_{i} (that is, 1 − β_{i} ≥ α_{i}), then the probabilities _{i}, β_{i}).

The proof follows immediately from two following facts: (a) the height h_{i} of any location i increases with the growth of n in any search strategy s, and, (2) the function

is obviously increasing in h_{i}, whenever_{i} = h_{i}(n) and

Corollary. Given the probabilities p_{i}, α_{i} and β_{i}, for all i, and the required confidence level CL_{0}, one can solve the inequality (2) with respect to h_{i}. By virtue of Claim 2, the minimum integer h_{i} that satisfies (2) presents the minimum number of positive tests of location i guaranteeing the required confidence level of the search. We denote this minimum h_{i}-value by H_{i} and call it a critical height of location i.

The CL_{0} value being given in advance, all the H_{i} values can be computed from (2), for all locations, before the UAV starts its search process. Now we are able to formulate the stopping rule.

The stopping rule. The search process stops at the step at which the number of positive tests provided by the sensors for some location i^{*} reaches, for the first time, its critical height value, H_{i*}. Thus, our model does not require human operator interference at each positive test for defining the stop moment; indeed, the search is stopped automatically by the on-board computer program as far as the H_{i} values are computed from (2) for all i and for the given CL_{0} values, and the search history is known.

In the following analysis, for a given sequence s, we need the following notation:

・

n-th step of strategy s;

・ P_{s}_{[n]}―unconditional probability that the sensor has totally declared in H_{s}_{[n]} steps that the location s[n] contains the target, up to the n-th step of strategy s. (The values H_{s}_{[n]} and P_{s}_{[n]} depending on parameters α_{i} and β_{i}, and guaranteeing a required confidence level CL_{0} will be computed below).

・ F(s)―the expected total loss incurred by the target before the latter is found and neutralized.

In accordance with the above conditions (i) and (ii) of the considered search scenario, the expected (linear) total loss, F(s), is defined as follows:

In this notation, the stochastic infinite-horizon search problem is to find an optimal sequence s^{*} minimizing the expected total loss F(s) incurred by the target before the search is stopped.

We add the following notation:

Event B_{i} = {after a single inspection, the sensor declares that location i contains the target}.

Event C_{i} = {Location i is really contains the target}.

In terms of the events, the location probabilities p_{i} and probabilities of the errors of two types are expressed as follows:

The probability f_{i} that the sensor declares that location i is detected as containing the target is:

The conditional probability of the event that the location i contains the target under condition that the sensor has declared that the location i contains the target in a single inspection, is computed as follows:

Let us return to Claim 1 from Section 3. Recall that we need to prove that the conditional probability _{i} steps of the sequence s that the location I contains the target, satisfies the following relation:

Proof. Since the sequential inspections of locations made by the UAV are independent, we have that for any pair of indices

Then

Next, using the relation

we obtain:

Similarly,

Therefore,

Claim 1 is proved.

Consider in more detail the search strategy s which is defined as an infinite-horizon sequence of location numbers, where, at each step n, location s s[n] is inspected and tested whether or not it contains the target:

In this sequence, the initial sub-sequence denoted by S(0) starts the search process. It is defined in such a way that the search cannot stop during this initial sub-sequence. For instance, we selected S(0) in the following way

Let us compute the problem’s objective function F(s). For this aim, we need, first, to compute the unconditional probability P_{s}_{[n]} of the event that a location s[n] = i is detected by the sensor as a target-containing location exactly H_{i} times up to the nth step of strategy s (not necessarily successively). We introduce an auxiliary parameter s^{*}[n] as follows: For any given sequence s and the location s[n] inspected by the sensor on step n, let s^{*}[n] be the total number of inspections of location s[n] (not necessarily successively) up to the nth step of strategy s. Obviously, s^{*}[n] ≤ n, for all n, and H_{s}_{[n]} ≤ s^{*}[n], for all n in s.

Notice that the on-board computer of the UAV can easily count and store the s^{*}[n] value in its memory as soon as the sequence s is known up to the n-th step.

Claim 3. The probability P_{s}_{[n]} can be computed as follows:

for

This claim immediately follows from the above definitions of H_{s}_{[n]} and s^{*}[n], using the binomial distribution of H_{s}_{[n]} − 1 inspections of the location s[n] in which the sensor has revealed the target in s[n], within the total number s^{*}[n] of inspections of the location s[n] up to the n-th step of s.

All the components of the problem’s objective function F(s) can be now computed as follows.

・ The time T_{s}_{[n]} spent for the inspection of all the elements

・ The unconditional probability P_{s}_{[n]} of the event that a location s[n] is detected by the sensor as a target-containing location exactly H_{i} times up to the nth step of strategy s-as defined in Claim 3.

Our main result is the index-based greedy algorithm below that permits to find the search strategy. The following theorem completely describes the algorithm.

Theorem. Define ratios

The strategy s^{*} is optimal iff the ratios

The proofs of Lemma and Theorem are done straightforwardly by the standard interchange argument (the interested reader is referred, for example, to Levner [

The search procedure in Theorem is index-based and greedy, so far as at each step the searcher selects for inspection the next minimum-ratio location. When all the α_{i} = β_{i} = 0, for all i, we have a special case of the so-called search scheduling problem with perfect inspections well known in scheduling literature (see, for example, Kadane and Simon [_{i} = 0, but β_{i} ≠ 0, this is the case of false-negative inspections; if α_{i} ≠ 0, but all β_{i} = 0, this is the case of the false-positive inspections only

The following examples illustrate the validity and applicability of the suggested model.

Example 1. Consider a problem of searching for a lost target by a single automatic device in a stochastic setting. The model and application are motivated by those described in Chung and Burdick [_{i}, which, in turn guarantees the required confidence level in identification of the target. For simplicity, we consider an area with only three locations numbered 1, 2 and 3. The input data are given in

The results of intermediate computations are given in

The ratios Q_{i} in decreasing order and corresponding location numbers are presented in

The search process stops at the ninth step. The optimal strategy obtained up to the ninth step is: ^{*}) is 49.398. We compared the obtained minimum losses with the losses incurred by another strategy, s_{1}, in which the locations are visited periodically, being arranged in each period in decreasing order of location probabilities:_{1}) = 57.484, which implies that F(s^{*}) is essentially better.

Example 2. Consider another area of interest with N = 6 possible locations in one of which a target object is

Locations | 1 | 2 | 3 |
---|---|---|---|

p_{i} = P(C_{i}) | 0.1 | 0.15 | 0.75 |

0.04 | 0.06 | 0.12 | |

0.1 | 0.07 | 0.05 | |

_{i}t | 5 | 8 | 10 |

_{i}c | 1 | 1 | 1 |

Locations | 1 | 2 | 3 |
---|---|---|---|

H_{i} | 2 | 2 | 1 |

f_{i} | 0.126 | 0.190 | 0.743 |

Step no. in s | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

Ratio Q_{i} | 0.1527 | 0.0819 | 0.0384 | 0.0312 | 0.0257 | 0.0191 | 0.0141 | 0.0105 | 0.0089 |

No. of location | 1 | 2 | 1 | 2 | 3 | 3 | 3 | 3 | 2 |

hidden. In this example, we explore experimental data similar to those used in the failure search model developed by Levner [

The results of intermediate computations are given in

The ratios Q_{i} in decreasing order and corresponding location numbers are presented in

The optimal search strategy obtained up to the 13th step is:^{*}) is 79.912. In the both numerical examples the ratios Q_{i}, as well as the probability that the search stops in a wrong location caused by a false alarm, quickly decrease as the number of steps in the search sequence grows. We compared the obtained minimum losses with the losses incurred by strategy s_{2}, in which the locations are visited periodically, being arranged in each period in decreasing order of location probabilities:_{2}) = 85.143, thus, the optimal search strategy s^{*} is indeed advantageous.

In order to find the exact solutions of the considered problem on the random data, the developed algorithm was implemented using MATLAB v. 7.1 and executed on the following hardware: Dell PowerEdge R410 server, CPU-2x Intel Xeon X5660 @ 2.8 GHz (12 cores total), 64 GB RAM. The problem instances were generated randomly as shown in

A total of 10,000 different instances were tested by the MATLAB solver. For all experiments, the calculation time was limited to 0.01 CPU-hour per problem instance. For each instance, the solver provided a greedy-gen- erated solution and the best solution generated by the full search over 1000 randomly generated feasible strategies. The following phenomenon similar to the “80/20 rule” (also known as the Pareto rule, or “the law of the vital few”, see Rushton et al. [

Location | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

p_{i} = P(C_{i}) | 0.08 | 0.05 | 0.15 | 0.65 | 0.05 | 0.02 |

0.05 | 0.01 | 0.07 | 0.1 | 0.02 | 0.03 | |

0.06 | 0.05 | 0.1 | 0.01 | 0.07 | 0.03 | |

_{i}t | 5 | 3 | 7 | 13 | 3 | 2 |

_{i}c | 1 | 1 | 1 | 1 | 1 | 1 |

Locations | ||||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |

H_{i} | 2 | 1 | 2 | 1 | 2 | 2 |

f_{i} | 0.1212 | 0.057 | 0.1945 | 0.6785 | 0.0655 | 0.0484 |

No. of step | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|---|

No. of location | 1, 3, 5, 6 | 1 | 2 | 3 | 4 | 5 | 3 |

Ratio Q_{i} | 0.3421 | 0.3154 | 0.2865 | 0.2361 | 0.2101 | 0.1840 |

No. of step | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
---|---|---|---|---|---|---|---|

No. of location | 4 | 4 | 4 | 3 | 4 | 4 | 4 |

Ratio Q_{i} | 0.161 | 0.123 | 0.089 | 0.061 | 0.032 | 0.012 | 0.009 |

the yielded greedy solution.

The results for different average cost values (from $10 to $500) are presented in

In addition, we have observed that for the values of the confidence level CL varying in interval [0.8; 0.96], for different number of cells varying in interval [10; 30], the number of instances for which the greedy algorithm behaved better than all other strategies, slowly increased with the CL growing (see

In Figures 3-5, we present the average number K of steps before the search is terminated, as a function of the confidence level CL for 10,000 instances and different numbers of cells (N = 10; 20; 30).

It is worth noticing that the number of steps of the algorithm becomes approximately the same, for large values of the confidence level.

Data type | Range |
---|---|

Number of cells | Unif (10, 30) |

Expected inspection time (in Min) | Unif (1, 10) |

Location probabilities | Unif (0.04, 0.95) |

Probabilities of false alarms | Unif (0.03, 0.5) |

Probabilities of overlooking | Unif (0.04, 0.5) |

Search cost rate (in relative units) | Unif (10, 40) |

We have studied the sequential discrete search problem for a lost or hidden target and considered a practice- oriented case where the automatic device performs imperfect inspections subject to the false-negative and false- positive inspection outcomes. For optimizing the search process minimizing the expected loss or damage, we suggest a greedy strategy. Being attractive due to its computational efficiency and simplicity, such local search strategy guarantees finding an optimal (minimum-loss) search sequence, for a pre-specified confidence level.

We believe that the suggested approach can be applied for a wider range of search scenarios (e.g., with multiple mobile agents, multiple targets, resource―and precedence constraints, agents with―and without memory). Notice that in the considered scenario the on-board computer program uses only partial information about the history of all search outcomes. Incorporating more complete information into the search model (for instance, joint probabilities of several locations) leads to a more complicated dynamic setting, which is an attractive direction for further research. The suggested greedy method could be combined with more general and sophisticated solution methods such as dynamic programming, branch-and-bound, and biology-motivated algorithms.

Another prospective future research is to compare the computational efficiency of different sequential and non-sequential search methods using analytical and simulation tools. It is known that the min-cost and max- probability versions of the discrete search problem with general resource-and time constraints are NP-hard (Wegener [

BorisKriheli,EugeneLevner,AlexanderSpivak, (2016) Optimal Search for Hidden Targets by Unmanned Aerial Vehicles under Imperfect Inspections. American Journal of Operations Research,06,153-166. doi: 10.4236/ajor.2016.62018