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Optimization of Air Defense System Deployment Against Reconnaissance Drone Swarms | TUP Journals & Magazine | IEEE Xplore

Optimization of Air Defense System Deployment Against Reconnaissance Drone Swarms

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Abstract:

Due to their advantages in flexibility, scalability, survivability, and cost-effectiveness, drone swarms have been increasingly used for reconnaissance tasks and have pos...Show More

Abstract:

Due to their advantages in flexibility, scalability, survivability, and cost-effectiveness, drone swarms have been increasingly used for reconnaissance tasks and have posed great challenges to their opponents on modern battlefields. This paper studies an optimization problem for deploying air defense systems against reconnaissance drone swarms. Given a set of available air defense systems, the problem determines the location of each air defense system in a predetermined region, such that the cost for enemy drones to pass through the region would be maximized. The cost is calculated based on a counterpart drone path planning problem. To solve this adversarial problem, we first propose an exact iterative search algorithm for small-size problem instances, and then propose an evolutionary framework that uses a specific encoding-decoding scheme for large-size problem instances. We implement the evolutionary framework with six popular evolutionary algorithms. Computational experiments on a set of different test instances validate the effectiveness of our approach for defending against reconnaissance drone swarms.
Published in: Complex System Modeling and Simulation ( Volume: 3, Issue: 2, June 2023)
Page(s): 102 - 117
Date of Publication: 20 June 2023

ISSN Information:


SECTION 1

Introduction

With the rapid improvement of functionality and performance, drones, i.e., Unmanned Aerial Vehicles (UAVs), have been widely used in various applications, especially in areas that are too dangerous or inaccessible to humans[1], e.g., disaster sites and battlefields. By dynamically grouping a number of drones that interact with one another to complete common tasks cooperatively and autonomously, a drone swarm can have significantly higher flexibility, scalability, survivability, and cost-effectiveness than the simple accumulation of individual drones[2]. In recent years, drone swarms have demonstrated great superiority in modern battlefields, such as in Syria, Nagorno-Karabakh, and Ukraine.

The development of military attack and defense technologies is similar to the relationship between a spear and a shield. The aforementioned advantages of the “spears” of drone swarms have posed great challenges to the “shields” of their opponents, who need to protect against the invasion of drone swarms by reducing their mission capabilities as much as possible. However, current research on countermeasures against drone swarms, such as Global Positioning System (GPS) spoofing, electromagnetic interference, false target jamming[3], [4], and anti-aircraft weapons[5], is still limited and relatively simple. Compared to these spoofing and interference measures, anti-aircraft weapons are more active and aggressive in causing damage to drones and are, therefore, more appealing to the opponents of drones. However, the flexibility and survivability of drone swarms make it difficult or impossible for a single air defense system to bring substantial damage to a swarm at a time. Typically, if one or several drones are damaged by an attack from the air defense system, the remaining drones can quickly escape from the attack area and reform the swarm that still has a majority of the original mission capability. Therefore, how to deploy available air defense systems to effectively defend against drone swarms is a challenging problem, which is important for the safety and security of the defending side.

In this paper, we study an optimization problem for deploying air defense systems against enemy reconnaissance drone swarms. Given a set of available air defense systems, the problem determines the location of each air defense system in a predetermined region where a reconnaissance drone swarm will pass through. Whenever the swarm is threatened/attacked by an air defense system, it replans the path to the target, as illustrated by the flowchart in Fig. 1. Therefore, we need to optimize the deployment of air defense systems, such that the cost for the enemy drone swarm to pass through the region will be maximized. We calculate the cost based on a counterpart drone path planning problem. To solve this adversarial problem, we first present an iterative search algorithm, which can produce exact optimal solutions to small-size problem instances; then, we propose an evolutionary framework for obtaining optimal or near-optimal solutions to large-size instances. We implement the evolutionary framework with six popular evolutionary algorithms, namely Genetic Algorithm (GA) with variable mutation rate[6], adaptive Particle Swarm Optimization (PSO) using comprehensive learning and self-adaptive parameters[7], Ecogeography-Based Optimization (EBO)[8], Quantum-inspired Tabu Search (QTS)[9], dual-strategy Differential Evolution (DE)[10], and Water Wave Optimization (WWO)[11]. We conducted computational test on a set of problem instances, the results of which validate the effectiveness of the method for defending against reconnaissance drone swarms. Among the six algorithms, EBO and WWO exhibit more competitive performance than the other ones. The main contributions of this work can be stated as follows:

  • It presents a problem of deploying air defense systems against drone swarms, which, to our best knowledge, has not been addressed before.

  • It proposes an exact iterative algorithm and an evolutionary framework with operators adapted for the problem, which are efficient for solving small- and large-size problem instances, respectively.

  • It conducts tests to validate the effectiveness of the proposed approach for improving the defensive abilities against reconnaissance drone swarms.

In the remainder of this paper, Section 2 reviews related work, Section 3 formulates the air defense system deployment optimization problem based on a basic drone path planning problem, Section 4 presents the exact iterative search algorithm, Section 5 describes the evolutionary framework and its typical implementations, Section 6 presents the computational experiments, and finally Section 7 concludes this paper.

Fig. 1 - Basic flow of the invading drone swarm and the opponent for defending against the swarm.
Fig. 1

Basic flow of the invading drone swarm and the opponent for defending against the swarm.

SECTION 2

Related Work

The considered air defense system deployment problem can be classified as a facility location problem which is known to be NP-hard[12]. Exact algorithms are only applicable to small-size instances[13], [14]. Many recent studies have been conducted on heuristic and evolutionary algorithms for finding near-optimal solutions. Teran-Somohano and Smith[15] used a bi-objective evolutionary strategy algorithm to solve a semi-obnoxious and multiple capacitated facility location problem that often arises in public planning. Zheng et al.[16] presented a master-slave evolutionary algorithm for a problem of integrated civilian-military emergency supply pre-positioning. Wang et al.[17] presented a dual-population evolutionary algorithm to solve a facility location problem with two objectives on reliability and coverage under the uncertainty of facilities. Vansia and Dhodiya[18] utilized non-dominated sorting genetic algorithm and modified self-adaptive multi-population elitism Jaya algorithm for a multi-objective transportation-p-facility location problem that minimizes the overall transportation time, cost of transportation, and carbon emission. Zhang et al.[19] proposed an enhanced group theory based evolutionary algorithm to solve the uncapacitated facility location problem. Eriskin et al.[20] studied a robust multi-objective model for the location of hospitals during pandemics. Karatas et al.[21] surveyed the facility location models and solution techniques for military organizations.

The present problem aims to maximize the effectiveness of defense against enemy targets. In this sense, the problem is similar to a subclass of Weapon Target Assignment (WTA) problems[22] that aim to maximize the total expected damage or total cost of the targets. Observing that exact algorithms[23], [24] are only applicable to small-size WTA instances, most recent research efforts have been devoted to heuristic and metaheuristic algorithms, including very large neighborhood search[23], [25], ant colony optimization[26], [27], GA[28], PSO[29], eminent domain[30], etc., to efficiently solve medium- and large-size WTA instances. Moreover, our problem considers that a drone swarm can dynamically reform and replan after being attacked and hence is closer to dynamic WTA[31] which uses a sequence of decisions to tackle the dynamic change of targets and is much more complex than its static countcrpart[32], [33].

Although there are many research work on general WTA problems, studies focusing on air defense system deployment are relatively limited. Han and Shi[34] proposed a simulated annealing algorithm for optimizing the deployment of air defense missile systems, where the combat effectiveness of defense systems is evaluated based on the stochastic service system theory. Wang and GuO[35] studied the disposition of air defense systems at the company level in two scenarios: one using a single system to protect multiple targets, and the other using multiple systems to protect a single target. They developed linear programming and dynamic programming methods to solve the problems. Yu et al.[36] proposed an artificial neural network approach to weapon system configuration, which first uses a backtracking network to approximate the system effectiveness, and then translates the problem to the traveling salesman problem which is solved by Hopfield network. Wang and Pan[37] considered an air defense disposition problem with uncertainties and risks measured by fuzzy entropy, and they presented a hybrid intelligent algorithm for the problem. Han et al.[38] proposed two air defense system configuration models based on integer programming, one for firing range covering and the other for firing angle covering, and they proved that the layout solution obtained with the firing angle concept is more efficient.

Currently, few studies have been conducted on air defense system deployment against drone swarms, which is much more difficult than traditional anti-aircraft air defense system deployment because drone swarms are much more flexible in reforming and replanning than traditional aircraft. To explore drone vulnerability to deceptive GPS signals, Kerns et al.[39] established necessary conditions for drone capture via GPS spoofing and explored the spoofer's range of possible post-capture control over drones. Considering a problem of using a team of mini drones acting as a cooperative defensive system against enemy unmanned aerial systems, Castrillo et al.[5] studied sensing, mitigation, and command and control technologies to realize such a defense system. Su et al. [40] studied a problem of false target jamming against UAVs, where each false target jamming solution is evaluated based on its adversarial effects on possible VA V detection solutions. To the best of our knowledge, the problem of air defense system deployment against drone swarms has not been addressed before.

SECTION 3

Problem Formulation

In this section, we formulate the air defense system deployment optimization problem, which aims to maximize the cost for enemy reconnaissance drones to pass through a predetermined region. We first describe a basic drone path planning problem from the viewpoint of the holders of drones, and then describe the adversarial defense problem from the viewpoint of the opponents of drones.

3.1 Drone Path Planning

First consider the basic drone path planning problem in three-dimensional space[41], [42], where a swarm of drones needs to traverse a predetermined region. According to the terrain of the region, a total number n of waypoints \boldsymbol{\Psi}=\{\psi_{1},\psi_{2},\ldots,\psi_{n}\} are marked, where each waypoint \psi_{i} is characterized by three-dimensional coordinates (x_{i},y_{i},z_{i})(1\leqslant i\leqslant n). The aim of the problem is to find a path \boldsymbol{P}=\{S,P_{1},P_{2}, \ldots,T\} that starts from the entry point S, passes through a subset of waypoints \{P_{1},P_{2}, \ldots\}\subseteq\boldsymbol{\Psi}, and finally reaches an exit point T, such that the total cost of the path is minimized. For the convenience of expression, we denote S=P_{0}, T=P_{n_{P}}, where n_{P} is the number of waypoints (including T but excluding S) on the path \boldsymbol{P}. Here, we measure the cost of a path based on the following four criteria:

• Length of the Path

The longer the path, the higher the cost. We calculate the length cost as the ratio of the total length to the Euclidean distance between S and T, \begin{equation*} L(\boldsymbol{P})=\frac{1}{\vert S,T\vert }\sum_{i=0}^{n_{P}-1}\vert P_{i},P_{i+1}\vert \tag{1} \end{equation*}View SourceRight-click on figure for MathML and additional features.

• Height of the Path

The higher the path, the higher the cost. This is because flying at a low flight height can improve the reconnaissance efficiency and reduce the risk of being discovered. We calculate the height cost as the ratio of the total height to (z_{\max}-z_{\min}), and the difference between the maximum height and minimum height is as follows:\begin{equation*} H(\boldsymbol{P})=\frac{1}{z_{\max}-z_{\max}}\sum_{i=0}^{n_{P}-1}(\overline{z}(P_{i},P_{i+1})-z_{\min}) \tag{2} \end{equation*}View SourceRight-click on figure for MathML and additional features. where \overline{z}(P_{i},P_{t+1}) denotes the average flight height of the segment [P_{i},P_{i+1}]. For a simple straight segment, the height can be directly calculated as the mean of the heights of P_{i} and P_{i+1}. For a segment comprising complex parts, the height can be obtained by equally dividing the segment into a sufficient number N_{e} of parts and calculating the mean of the heights of the N_{e} parts.

• Excessive Horizontal Rotation of the Path

If a horizontal rotation angle is larger than a predefined threshold \widehat{\omega}, the drones need to decelerate to facilitate the change of direction and then re-accelerate to the normal speed, which would reduce the flight efficiency. We calculate the excessive horizontal rotation cost as follows:\begin{equation*} R(\boldsymbol{P})=\frac{1}{n_{P}\widehat{\omega}}\sum_{i=0}^{n_{P}-2}\max(\omega(P_{i},P_{i+1},P_{i+2})-\widehat{\omega},0) \tag{3} \end{equation*}View SourceRight-click on figure for MathML and additional features. where \omega(P_{i},P_{i+1},P_{i+2}) denotes the horizontal angle between segments (P_{i},P_{i+1}) and (P_{i+1},P_{i+2}).

• Excessive Vertical Inclination Angle of the Path

If an inclination angle is larger than a predefined threshold \widehat{\theta}, the drones need to slow down to facilitate climbing and descending and then re-accelerate to the normal speed, which would reduce the flight efficiency. We calculate the excessive inclination cost as follows:\begin{equation*} S(\boldsymbol{P})=\frac{1}{n_{P}\widehat{\theta}}\sum_{i=0}^{n_{P}-2}\max(\theta(P_{i},P_{i+1},P_{i+2})-\widehat{\theta,}0) \tag{4} \end{equation*}View SourceRight-click on figure for MathML and additional features. where \theta(P_{i},P_{i+1},P_{i+2}) denotes the vertical angle between segments (P_{i},P_{i+1}) and (P_{i+1},P_{i+2}).

The objective function of drone path planning is defined as the weighted sum of the above costs,\begin{equation*} \min C(\boldsymbol{P})=L(\boldsymbol{P})+w_{H}H(\boldsymbol{P})+w_{R}R(\boldsymbol{P})+w_{S}S(\boldsymbol{P}) \tag{5} \end{equation*}View SourceRight-click on figure for MathML and additional features. where w_{H}, w_{R}, and w_{S} are the importance weights of the height cost, excessive horizontal rotation cost, and excessive inclination cost, respectively. As the weight of the length cost is explicitly one, we suggest that w_{H} can be set to approximately 0.5, w_{R}+w_{S}=1-w_{H}, and w_{R} can be smaller than w_{S} because horizontal rotation is typically simpler than vertical inclination.

3.2 Air Defense System Deployment Against Reconnaissance Drone Swarms

First, we assume that the holder of reconnaissance drones can always obtain the optimal path \boldsymbol{P}^{\ast} that minimizes the objective of Eq. (5). Given a set of m available air defense systems, the air defense system deployment optimization problem is to determine the location of each air defense system in the predetermined region, such that the total cost for the enemy drone swarm to pass through the region under the anti-drone threat is maximized. Each air defense system can automatically detect any drone entering its defense range (within a predefined radius r_{d}). If two or more drones enter simultaneously, we assume that the system can shoot down at least one drone. It is also assumed that the maximum height of the drones is in the attack range of air defense systems, while the drones cannot attack the defense systems. Under the above assumptions, if there is only one enemy drone, we can simply place an air defense system at any suitable position on \boldsymbol{P}^{\ast}. However, for a drone swarm, whenever the first drone is intercepted by the air defense system, the remaining drones have abilities to bypass the system.

Let A_{1} be the position of the first air defense system for intercepting the drones along the path \boldsymbol{P}^{\ast}. As illustrated in Fig. 2a, the first drone is intercepted at the intersection of \boldsymbol{P}^{\ast} and the circumference of the defense range of A_{1}. Afterward, the swarm replans its path from the new starting point S_{1} to the entry point T, where the distance between S_{1} and the intersection is \Delta d (which is typically set as the average interval among the drones in the swarm). However, the new path from S_{1} to T should try to avoid passing through the defense range of A_{1}. To reflect this, we add a new fire threat cost to the drone path as follows:\begin{equation*} E_{1}(\boldsymbol{P})=\frac{1}{\vert S,T\vert }\sum_{i=0}^{n_{P}-1}\Lambda_{1}(P_{i},P_{i+1}) \tag{6} \end{equation*}View SourceRight-click on figure for MathML and additional features. where \Lambda_{1}(P_{i},P_{i+1}) denotes the length of the part of the segment (P_{i},P_{i+1}) that is in the defense range of A_{1}.

Fig. 2 - Illustration of the path replanning of a drone swarm after being intercepted by air defense systems. Solid (black) line: original path; dash (green) line: first replanned path; dotted (blue) line: second replanned path; red circle: defense range.
Fig. 2

Illustration of the path replanning of a drone swarm after being intercepted by air defense systems. Solid (black) line: original path; dash (green) line: first replanned path; dotted (blue) line: second replanned path; red circle: defense range.

Consequently, the objective function of drone path replanning after being intercepted by the first air defense system should incur the fire threat cost in the following:\begin{equation*} \min C_{1}(\boldsymbol{P})=C(\boldsymbol{P})+w_{E}E_{1}(\boldsymbol{P}) \tag{7} \end{equation*}View SourceRight-click on figure for MathML and additional features. where w_{E} is the importance weight of the fire threat cost. Note that when we set w_{E} to a sufficiently large value, the path will not be allowed to enter the defense range.

Let \boldsymbol{P}^{(1)} be the new path that minimizes the updated objective of Eq. (7), and A_{2} be the position of the second air defense system, which should be on path \boldsymbol{P}^{(1)}. Similarly, after the swarm enters the defense range of A_{2}, it should replan the path \boldsymbol{P}^{(2)} from the new starting point S_{2} to the entry point T, and the new path should try to avoid passing through the defense ranges of A_{1} and A_{2}, as illustrated in Fig. 2b.

By analog, after placing the j-th air defense system on the path \boldsymbol{P}^{(j-1)}, the drone swarm replans the path from the new starting point S_{j} to the entry point T, and the path should consider the fire threat cost caused by the previous j air defense systems (1\leqslant j\leqslant m), \begin{equation*} E_{j}(\boldsymbol{P})=\frac{1}{\vert S,T\vert }\sum_{i=0}^{n_{P}-1}\Lambda_{j}(P_{i},P_{i+1}) \tag{8} \end{equation*}View SourceRight-click on figure for MathML and additional features. where \Lambda_{j}(P_{i},P_{i+1}) denotes the length of the part of the segment (P_{i},P_{i+1}) that is in the defense range of any one of A_{1},A_{2},\ldots,A_{j}. The objective function of the j-th path replanning is\begin{equation*} \min C_{j}(\boldsymbol{P})=C(\boldsymbol{P})+w_{E}E_{j}(\boldsymbol{P}) \tag{9} \end{equation*}View SourceRight-click on figure for MathML and additional features.

From the viewpoint of the opponent of the drone swarm, the air defense system deployment optimization problem is to determine m positions \{A_{1},A_{2}, \ldots,A_{m}\} in the region to place the m air defense systems, such that the total cost of the paths planned and replanned by the drone swarm is maximized,\begin{align*} \max f(A)= & C(\boldsymbol{P}^*[S, S_1])+ \\ & \sum_{j=1}^{m-1} C_j(\boldsymbol{P}^{(j)}[S_j, S_{j+1}])+C(\boldsymbol{P}^{(m)}[S_m, T]) \tag{10} \end{align*}View SourceRight-click on figure for MathML and additional features. where \boldsymbol{P}[S_{j},S_{j+1}] represents the partial path from S_{j} to S_{j+1} in the path \boldsymbol{P}, \boldsymbol{P}^{(j)}=\arg\min_{\boldsymbol{P}\in PATH(S_{j},T)}C_{j}(\boldsymbol{P}), and PATH(S_{j},T) is the set of all paths from S_{j} to T.

In practice, some areas are not suitable for deploying air defense systems due to terrain constraints. We use \boldsymbol{\Omega} to denote the union of all positions that are suitable for placing air defense systems, \boldsymbol{\Omega}_{1} to denote the remaining suitable positions after placing the first air defense system (by removing the defense range of A_{1} from \boldsymbol{\Omega}), \boldsymbol{\Omega}_{2} to denote the remaining suitable positions after placing the first two air defense systems (by removing the defense range of A_{2} from \boldsymbol{\Omega}_{1}), and so on. Consequently, the feasible region for placing air defense systems is \boldsymbol{\Omega} at first and updated to \boldsymbol{\Omega}_{j} after placing the first j systems (1\leqslant j\leqslant m-1).

SECTION 4

Exact Iterative Search Algorithm

To solve the above air defense system deployment problem, we first propose an iterative search algorithm for finding the exact optimal solution to the problem. Suppose that the previous (m-1) air defense systems have been placed, the position A_{m} of the m-th air defense system should be determined according to the second last replanned path \boldsymbol{P}^{(m-1)} of the drone swarm, aiming at maximizing the cost of \boldsymbol{P}^{(m-1)} and the last replanned path \boldsymbol{P}^{(m)}, \begin{align*} \max f(A_m)= & C_{m-1}(\boldsymbol{P}^{(m-1)}[S_{m-1}, S_m])+ \\ & C_m(\boldsymbol{P}^{(m)}[S_m, T]) \tag{11} \end{align*}View SourceRight-click on figure for MathML and additional features.

The position A_{m} belongs to set \boldsymbol{\Omega}_{m-1}, and its defense range should intersect with the path \boldsymbol{P}^{(m-1)}. Therefore, the procedure FindA (m,\boldsymbol{P}^{(m-1)},\boldsymbol{\Omega}_{m-1}) for determining the position A_{m} can be described by Algorithm 1 that iteratively tests all possible positions, where \Phi(A) denotes the defense range of the air defense system placed at the position A.

SECTION Algorithm 1:

FindA (m, \boldsymbol{P}^{(m-1)}, \boldsymbol{\Omega}_{m-1})

Algorithm

Next, suppose that the previous (m-2) air defense systems have been placed. The position A_{m-1} of the (m-1) -th air defense system should be determined to maximize the value of the following function:\begin{align*} & \max f(A_{m-1})=C_{m-2}(\boldsymbol{P}^{(m-2)}[S_{m-2}, S_{m-1}])+ \\ & C_{m-1}(\boldsymbol{P}^{(m-1)}[S_{m-1}, S_m])+C_m(\boldsymbol{P}^{(m)}[S_m, T]) \tag{12} \end{align*}View SourceRight-click on figure for MathML and additional features.

The procedure FindA (m-1,\boldsymbol{P}^{(m-2)},\boldsymbol{\Omega}_{m-2}) for determining the position A_{m-1} is described in Algorithm 2, which has a framework similar to Algorithm 1 in that each possible position is tested. The difference is that, when evaluating the fitness function value for each candidate position, we not only replan the drone path \boldsymbol{P}^{(m-1)} from A_{m-1}, but also call the procedure FindA (m,\boldsymbol{P}^{(m-1)},\boldsymbol{\Omega}_{m-1}) to find the optimal position A_{m} after A_{m-1} and therefore replan \boldsymbol{P}^{(m)}.

By analogy, we can derive the procedures for determining the positions A_{m-2},A_{m-3}, \ldots in a reverse manner, and finally obtain the procedure FindA (1, \boldsymbol{P}^{\ast},\boldsymbol{\Omega}) for determining the position of A_{1} for minimizing the original objective function of Eq. (10), as shown in Algorithm 3, which is an exact optimization algorithm for solving this air defense system deployment problem.

As illustrated in Fig. 3, FindA (1, \boldsymbol{P}^{\ast},\boldsymbol{\Omega}) will iteratively call the sub-procedures FindA (2, \boldsymbol{P}^{(1)},\boldsymbol{\Omega}_{1}), FindA (3, \boldsymbol{P}^{(2)}, \boldsymbol{\Omega}_{2}), …, FindA (m,\boldsymbol{P}^{(m-1)}, \boldsymbol{\Omega}_{m-1}), and finally obtain all m positions \{A_{1},A_{2}, \ldots,A_{m}\} for placing the air defense systems. Let N be the cardinality of \boldsymbol{\Omega}. The worst time complexity of the exact optimization algorithm is\begin{align*} & O(C(N, m) O(A^{\star}))= \\ & O(N(N-1)(N-2) \cdots(N-m+1) O(A^{\star})), \end{align*}View SourceRight-click on figure for MathML and additional features. where O(A^{\star}) denotes the time complexity of the A^{\star} algorithm (typically does not exceed O(N^{2})). Nevertheless, in practice, the number of points that intersect the designated paths with the circumference of defense ranges is usually much smaller than n. Moreover, with the progress of drone path replanning, the new starting point continually becomes closer to the exit point T, and therefore the length of new replanned paths continuously becomes shorter. Consequently, the average time complexity of the algorithm is not so high.

SECTION Algorithm 2

FindA (m-1, \boldsymbol{P}^{(m2)}, \boldsymbol{\Omega}_{m-2})

Algorithm
SECTION Algorithm 3

FindA (1, \boldsymbol{P}^{\ast},\boldsymbol{\Omega})

Algorithm
Fig. 3 - Recursive invocation of the sub-procedure FindA().
Fig. 3

Recursive invocation of the sub-procedure FindA().

SECTION 5

Evolutionary Algorithm

As the running time of the above exact optimization algorithm increases exponentially with N and m, it is only suitable for situations where the region is relatively small and the number of air defense systems is also not large. Otherwise, the running time would be unacceptable. To solve large-size instances of the problem, we propose an evolutionary framework that evolves a population of candidate solutions to search for optimal or near-optimal solutions. Note that although this problem determining m positions among N candidates can be regarded as a multi-knapsack problem, existing evolutionary algorithms for the multi-knapsack problem are not efficient because they often test many candidate positions that are unable or inefficient to intercept drones.

To efficiently solve the air defense system deployment problem, we design a specific encoding and decoding scheme. Each solution to the problem is encoded as an m-dimensional real-valued vector \boldsymbol{x}=\{x_{1},x_{2}, \ldots,x_{m}\}, where the value of each component x_{i} is in the range of [0, 1]. The solution decoding procedure consists of the following steps:

  • Step 1:

    Find out the set of all candidate positions whose defense range intersects with the path \boldsymbol{P}^{\ast}, and sort these positions in order of their intersections with the path from front to back (i.e., the closer the intersection to the starting point S, the higher the rank;

  • Step 2:

    Let N_{1} be the number of candidate positions and k be the integer closest to N_{1}x_{1}; take the k-th candidate position to place the first air defense system;

  • Step 3:

    Let j=1;

  • Step 4:

    Replan the drone path \boldsymbol{P}^{(j)} based on A_{j};

  • Step 5:

    Find out the set of all candidate positions whose defense range intersects with the path \boldsymbol{P}^{(j)}, and sort these positions in order of their intersections with the path from front to back;

  • Step 6:

    Let N_{j+1} be the number of candidate positions and k be the integer closest to N_{j+1}x_{j+1}; take the k-th candidate position to place the j-th air defense system;

  • Step 7:

    Set j=j+;1 if j > m, then stop; otherwise, go to Step 4.

The above scheme encodes the position of each air defense system as the rank of its intersection with the path of the drone swarm to relate the position to the anti-drone task. The use of real-valued encoding makes it easy to adapt existing evolutionary algorithms for continuous optimization to our problem.

In the following subsections, we use GA[6], PSO[7], EBO[8], QTS[9], DE[10], and WWO[11] to implement the framework.

5.1 Ga With Variable Mutation Rate

GA performs a stochastic search based on the principle of genetic crossover and mutation. On the basis of the above encoding/decoding scheme, we use the breeder crossover operation[43] that generates offsprings \boldsymbol{x}^{c} and \boldsymbol{x}^{b} from two parents \boldsymbol{x}^{a} and \boldsymbol{x}^{b} as follows (where r is a random value uniformly distributed in [0,1],1\leqslant j\leqslant m): \begin{align*} & x_j^c=r x_j^a+(1-r) x_j^b, \\ &\ x_j^d=(1-r) x_j^a+r x_j^b \tag{13} \end{align*}View SourceRight-click on figure for MathML and additional features.

Another key difference of our algorithm from classical GA is that it uses a variable mutation rate r_{m}(\boldsymbol{x}) calculated as inversely proportional to the fitness of solution \boldsymbol{x} as follows (such that low-fitness solutions are changed greatly while high-fitness solutions are more stable):\begin{equation*} r_{\boldsymbol{m}}(\boldsymbol{x})=\frac{f(\boldsymbol{x})}{f_{\max}}r_{m}^{\max} \tag{14} \end{equation*}View SourceRight-click on figure for MathML and additional features. where f_{\max} denotes the maximum objective function value in the population, and r_{m}^{\max} is a parameter controlling the maximum mutation rate. The mutation is conducted by randomly setting each dimension in the value range.

Algorithm 4 presents the pseudo-code of the GA, where \boldsymbol{r}_{c} is the crossover rate, and rand (0, 1) produces a random value uniformly distributed in [0, 1]. The time complexity of the fitness evaluation is O(mN^{3}). The average mutation rate is expected to be 0.5. Let G be the maximum number of generations and N_{P} be the population size; the average time complexity of the GA is O(1.5GN_{P}mN^{3}).

5.2 Adaptive Comprehensive Learning PSO

PSO[44] associates each solution (particle) \boldsymbol{x}_{i} with a velocity vector \boldsymbol{v}_{i} of the same dimensionality, which is used to move the solution at each iteration,\begin{equation*} x_{i,j}=x_{i,j}+v_{i,j} \tag{15} \end{equation*}View SourceRight-click on figure for MathML and additional features.

Algorithm 4 GA for the air defense system deployment optimization problem

Algorithm

In the classical PSO, each velocity vector is adjusted by learning from both the personal historical best pbest_{i} of the current solution and the global best gbest of the whole population. Comprehensive learning PSO[45] improves the classical PSO by utilizing all other particles' historical best to update the velocity of each particle, such that the information of different particles is exchanged more thoroughly,\begin{equation*} v_{i,j}=wv_{i,j}+c\times rand(0,1)(pbest_{i^{\prime},j}-x_{i,j}) \tag{16} \end{equation*}View SourceRight-click on figure for MathML and additional features.

The exemplar solution pbest_{i^{\prime}} is randomly selected based on a learning probability at each dimension j, and pbest_{i^{\prime},j} is the corresponding component of pbest_{i^{\prime}}. w and c are two control parameters. We also adaptively adjust the values of the two parameters according to the evolutionary states, i.e., the improvement of each individual and the whole population over iterations[7].

Algorithm 5 presents the pseudo-code of the adaptive comprehensive learning PSO algorithm for the problem. The average time complexity of the algorithm is O(GN_{P}mN^{3}).

5.3 Ebo

EBO is an extended version of Biogeography-Based Optimization (BBO)[46], which associates each solution \boldsymbol{x} with an immigration rate r_{\mu}(\boldsymbol{x}) and an emigration rate r_{v}(\boldsymbol{x}) as follows:\begin{align*} & r_\mu(\boldsymbol{x})=\frac{f_{\max }-f(\boldsymbol{x})+\epsilon}{f_{\max }-f_{\min }+\epsilon}, \\ &\ \ r_v(\boldsymbol{x})=\frac{f(\boldsymbol{x})-f_{\min }+\epsilon}{f_{\max }-f_{\min }+\epsilon} \tag{17} \end{align*}View SourceRight-click on figure for MathML and additional features. where f_{\min} is the minimum objective function values in the population, and \epsilon is a small positive number to avoid division by zero.

Algorithm 5 PSO for the air defense system deployment optimization problem

Algorithm

Compared to the basic BBO, EBO employs a local neighborhood structure[47] and differentiates local migration and global migration. Local migration sets the current dimension of the immigrating solution \boldsymbol{x} by blending with the corresponding dimension of the emigrating solution \boldsymbol{x}^{\prime}, where \boldsymbol{x}^{\prime} is a neighbor of \boldsymbol{x} selected with a probability proportional to the emigrating rate. The corresponding dimention x_{j}^{\prime} of the emigrating solution \boldsymbol{x}^{\prime} is as follows:\begin{equation*} x_{j}=x_{j}+rand(0,1)(x_{j}^{\prime}-x_{j}) \tag{18} \end{equation*}View SourceRight-click on figure for MathML and additional features.

Global migration selects two emigrating solutions \boldsymbol{x}^{\prime} and \boldsymbol{x}^{\prime \prime}, where \boldsymbol{x}^{\prime} is from the neighboring solutions of \boldsymbol{x}, and \boldsymbol{x}^{\prime \prime} is from the non-neighboring solutions. A global migration operation is performed according to the fitness comparison of the two emigrating solutions as follows:\begin{equation*} x_j=\begin{cases} x_j^{\prime}+rand(0,1)(x_j^{\prime \prime}-x_j), & f(\boldsymbol{x}^{\prime}) > f(\boldsymbol{x}^{\prime \prime}); \\ x_j^{\prime \prime}+rand(0,1)(x_j^{\prime}-x_j), & f(\boldsymbol{x}^{\prime}) < f(\boldsymbol{x}^{\prime \prime}) \end{cases} \tag{19} \end{equation*}View SourceRight-click on figure for MathML and additional features.

Whether a migration operation is a local or global migration is determined by a parameter \eta that linearly decreases with iteration, such that global migration has a high probability in early stages for diversifying the search, whereas local migration is more probable in later stages for improving solution accuracy.

Algorithm 6 presents the pseudo-code of the EBO algorithm, the time complexity of which is O(GN_{P}mN^{3}).

Algorithm 6 EBO for the air defense system deployment optimization problem

Algorithm

5.4 Qts

Tabu search[48] is an extension of basic local search using short-term memory to save recently obtained local optimal solutions to avoid repeated searches. QTS[9] introduces quantum-inspired bits and gates into the algorithm to suppress premature convergence and better balance exploration and exploitation. The QTS algorithm first initializes a quantum matrix Q to represent the solution, and then continually performs neighborhood search by measuring the quantum matrix several times. The neighboring solutions are further improved by entanglement and local search. The search procedure continues until the stop condition is satisfied. The quantum matrix is also iteratively updated to push new solutions toward the current best solution while keeping away the current worst.

Algorithm 7 presents the pseudo-code of the QTS algorithm, the time complexity of which is O(t_{\max}N_{b}mN^{3}), where N_{b} is the neighborhood size.

5.5 Dual-Strategy Differential Evolution

DE is a simple but fast evolutionary algorithm that evolves solutions according to the difference between them[49]. At each iteration, a mutation vector is produced for each solution by adding the difference between two randomly selected solutions to a third one; a trial vector is then produced by the crossover of the solution and its mutation vector; finally, the better one between the solution and the trial vector is chosen for the next generation.

Dual-strategy DE[10] uses multiple sub-populations of solution. In each sub-population, the solutions are sorted in decreasing order of fitness, and the following DE/rand/1 mutation scheme is applied to the first half of the solutions:\begin{equation*} \boldsymbol{v}_{j}=\boldsymbol{x}_{r_{1}}+F(\boldsymbol{x}_{r_{2}}-\boldsymbol{x}_{r_{3}}) \tag{20} \end{equation*}View SourceRight-click on figure for MathML and additional features. where r_{1}, r_{2}, and r_{3} are three random indices in the population, and F is a control parameter called the scale factor.

Algorithm 7 QTS for the air defense system deployment optimization problem

Algorithm

The following DE/lbest/1 mutation scheme is applied to the second half of the solutions:\begin{equation*} \boldsymbol{v}_{i}=\boldsymbol{x}_{lbest}+F(\boldsymbol{x}_{r_{1}}-\boldsymbol{x}_{r_{2}}) \tag{21} \end{equation*}View SourceRight-click on figure for MathML and additional features. where \boldsymbol{x}_{lbest} denotes the current best solution in the sub-population.

Algorithm 8 presents the pseudo-code of the dual-strategy DE algorithm, the time complexity of which is O(GMN_{S}mN^{3}), where N_{S} denotes the size of sub-populations.

5.6 Wwo

Inspired by the shallow water wave theory, WWO assigns each solution \boldsymbol{x} with a “wavelength” \lambda(\boldsymbol{x}), whose value is initialized as 0.5 and updated as inversely proportional to the solution fitness as follows:\begin{equation*} \lambda(\boldsymbol{x})=\lambda(\boldsymbol{x}) \alpha^{-(f(x)-f_{\min }+\epsilon) /(f_{\max }-f_{\min }+\epsilon)} \tag{22} \end{equation*}View SourceRight-click on figure for MathML and additional features. where \alpha is a control parameter set to 1.0026.

At each iteration, WWO uses a propagation operation to produce an offspring \boldsymbol{x}^{\prime} for each solution \boldsymbol{x} within a search range proportional to \lambda(\boldsymbol{x}) as follows: \begin{equation*} x_{j}^{\prime}=x_{j}+rand(-1,1)\lambda(\boldsymbol{x})L_{j} \tag{23} \end{equation*}View SourceRight-click on figure for MathML and additional features. where L_{j} denotes the search range of the j-th dimension (1 for all dimensions of our problem). In this way, high (low) fitness solutions have small (large) search ranges to balance global and local search. The better one between the original solution and its offspring will be selected into the population.

Algorithm 8 Dual-strategy DE for the air defense system deployment optimization problem

Algorithm

WWO also performs a breaking operation on each newly found best solution \boldsymbol{x}^{\ast} by generating k_{N} neighboring solutions, each at a distinct randomly selected dimension j as follows:\begin{equation*} x_{j}^{\prime}=x_{j}^{\ast}+N(0,1)\beta L_{j} \tag{24} \end{equation*}View SourceRight-click on figure for MathML and additional features. where N(0,1) generates a Gaussian random number with mean 0 and standard deviation 1, and \beta is a control parameter set to 0.002. The best neighbor, if better than \boldsymbol{x}^{\ast}, will replace \boldsymbol{x}^{\ast} in the population.

We use a variable population size[50], which linearly decreases from N_{P}^{\max} to N_{P}^{\min} (two control parameters). The size reduction is performed by removing the worst solution from the population.

Algorithm 9 presents the pseudo-code of the WWO algorithm. The time complexity of the algorithm is O(G(N_{P}+K_{N})mN^{3}), where N_{P}=(N_{P}^{\max}+N_{P}^{\min})/2.

SECTION 6

Computational Experiment

The experiments were conducted on eight test instances, which were constructed based on four selected maps with different topographic features in South East China. Table 1 presents the basic features of the instances, including the regional area (2nd column), number of waypoints (3rd column), number of available air defense systems (4th column), and number of drones in the swarm (5th column). The important weights w_{H}, w_{R}, and w_{S} in the objective function of Eq. (5) and w_{E} in Eq. (9) were set to 0.5, 0.2, 0.3, and 1.0, respectively.

SECTION Algorithm 9

WWO for the air defense system deployment optimization problem

Algorithm

For each test instance, we first use the A^{\star} algorithm to plan the path of the drone swarm and calculate the cost of the path \boldsymbol{P}^{\ast} without interception of air defense systems. Next, we try to use the exact Iterative Search algorithm (IS) to solve the instance (until its running time exceeds 12 hours). Afterward, we use the six evolutionary algorithms, i.e., GA, PSO, EBO, QTS, DE, and WWO, to solve the instance, each being repeated 30 times. The control parameters of each evolutionary algorithm were tuned on the whole set of instances. The stop condition of the evolutionary algorithms was set in such a way that the running time reached 30 min.

Figures 4–​11 present the experimental results on the eight test instances, respectively. In each figure, the dashed line denotes the path cost C(\boldsymbol{P}^{\ast}) of the drone swarm without considering air defense systems, and each box plot gives the median, minimum, maximum, first quartile (25%, denoted by Q1), and third quartile (75%, denoted by Q3) of the objective values obtained by the corresponding algorithm over the 30 runs. Within the predefined time limit of 12 hours, the exact IS algorithm only stops on the first two instances, and therefore its results are only given in Figs. 4 and 5. The results showed that using solutions obtained by the proposed algorithms to deploy air defense systems can increase the path cost of drone swarms to 500%-1100% of the original cost without air defense

Fig. 4 - Computational results on instance 1.
Fig. 4

Computational results on instance 1.

Fig. 5 - Computational results on instance 2.
Fig. 5

Computational results on instance 2.

Fig. 6 - Computational results on instance 3.
Fig. 6

Computational results on instance 3.

Fig. 7 - Computational results on instance 4.
Fig. 7

Computational results on instance 4.

Table 1 Basic features of the test instances.
Table 1- Basic features of the test instances.
systems, and the ratio increases with the instance size. This finding demonstrates that the proposed method can effectively deploy air defense systems to defend against reconnaissance drone swarms especially in large areas.
Fig. 8 - Computational results on instance 5.
Fig. 8

Computational results on instance 5.

Fig. 9 - Computational results on instance 6.
Fig. 9

Computational results on instance 6.

Fig. 10 - Computational results on instance 7.
Fig. 10

Computational results on instance 7.

Fig. 11 - Computational results on instance 8.
Fig. 11

Computational results on instance 8.

On smallest-size Instance 1, five evolutionary algorithms (except GA) obtain the same median value, which is the exact optimal objective value obtained by IS (but the computation time of the evolutionary algorithms is much shorter than that of IS). EBO, DE, and WWO always obtain the optimal solution, whereas the minimum values of PSO and QTS are smaller than the optimal value, i.e., the two algorithms occasionally fail to obtain the optimal solution. The median value of GA is smaller than the optimal value, although its maximum value is the optimal, i.e., GA occasionally obtains the optimal solution, but fails to do so in most cases. On Instance 2 where the region is the same as that of Instance 1 but the numbers of waypoints, air defense systems, and drones are all larger than those of Instance 1, WWO obtains the best median value, EBO, DE, and WWO occasionally obtain the optimal solution as IS, but GA, PSO, and QTS never do so.

On the remaining larger-size instances, IS cannot obtain the optimal solution within 12 hours, indicating that the exact algorithm can only solve small-size instances. DE obtains the best median value on Instance 4, and EBO obtains the best median values on the other five instances.

We conducted a nonparametric Wilcoxon rank sum test on the results of the six evolutionary algorithms to evaluate their differences on each instance. The results (all at a confidence level of 95%) show that, on either Instance 1 or Instance 2, there is no significant difference among the results of PSO, EBO, QTS, DE, and WWO, and they are all significantly better than the result of GA. On Instances 3 and 6, the result of EBO is significantly better than those of GA, PSO, and QTS, but it is not significantly different than those of DE and WWO. On Instance 4, the result of DE is significantly better than those of GA, EBO, and QTS, but it is not significantly different than those of PSO and WWO. On Instances 5, 7, and 8, the result of EBO is significantly better than those of the other five algorithms. Tables 2–​7 present the statistical test results on Instances 3–8, respectively, where “>” denotes that the result of the algorithm in the current row is significantly better than that in the current column.

Table 8 presents the rank of the median value of each algorithm on each test instance, and Table 9 presents the total rank number (row 2), the number of times of obtaining the best median value (row 3, abbreviated to “best”), and the number of times of being significantly better than other algorithms (row 4, abbreviated to “significantly better”) over all eight instances. EBO exhibits the best overall performance, WWO performs the second best, and GA performs the worst because it is easily trapped in local optima, given that its crossover operator is relatively weak in global search. QTS only performs better than GA, as its neighborhood search mechanism is not also inefficient in searching a large-size solution space. PSO's particle motion mechanism makes it converge fast to some good solutions on small-size instances, but often leads to premature convergence on large-size instances. The differential crossover operator and multi-population strategy make DE more suitable for searching large-size solution spaces than small-size ones. The integration of global and local migrations in EBO and the wavelength-based propagation in WWO can achieve a quite good balance between global search and local search, which is the main reason why the two algorithms exhibit good performance on the test instances. Comparatively, the migration mechanism endows EBO with a stronger global search ability to solve large-size instances.

Table 2 Statistical test results on instance 3 (“>” indicates “significantly better”).
Table 2- Statistical test results on instance 3 (“>” indicates “significantly better”).
Table 3 Statistical test results on instance 4 (“>” indicates “significantly better”).
Table 3- Statistical test results on instance 4 (“>” indicates “significantly better”).
Table 4 Statistical test results on instance 5 (“>” indicates “significantly better”).
Table 4- Statistical test results on instance 5 (“>” indicates “significantly better”).
Table 5 Statistical test results on instance 6 (“>” indicates “significantly better”).
Table 5- Statistical test results on instance 6 (“>” indicates “significantly better”).
Table 6 Statistical test results on instance 7 (“>” indicates “significantly better”).
Table 6- Statistical test results on instance 7 (“>” indicates “significantly better”).
Table 7 Statistical test results on instance 8 (“>” indicates “significantly better”).
Table 7- Statistical test results on instance 8 (“>” indicates “significantly better”).
Table 8 Ranks of the six evolutionary algorithms on the test instances.
Table 8- Ranks of the six evolutionary algorithms on the test instances.
Table 9 Summary of the comparative performance of the six evolutionary algorithms on the test instances.
Table 9- Summary of the comparative performance of the six evolutionary algorithms on the test instances.

The computational results also validate that increasing the number of air defense systems can effectively increase the cost of enemy drones. For example, the ratio of the path cost after the deployment of the air defense systems (in terms of the maximum cost achieved by the algorithms) to the path cost before the deployment is around 500% on Instance 1 with five air defense systems, but the ratio increases to over 580% on Instance 2 with six systems. From the viewpoint of the drone holder, under the threat of air defense systems, increasing the number of drones can effectively decrease its total path cost. Facing the invasion of a large swarm of drones, it is crucial to obtain high-quality air defense system deployment solutions to defend against reconnaissance drones.

SECTION 7

Conclusion

This paper presents an optimization problem of air defense system deployment against enemy reconnaissance drone swarms, which determines the locations of air defense systems to intercept drones and force them to continually change their paths, such that the total cost of replanned paths of the drone swarm is maximized. We propose an exact iterative search algorithm and an evolutionary framework implemented using six concrete algorithms to solve the problem. The computational experimental results validate the performance of air defense system deployment solutions obtained by the algorithms for defending against reconnaissance drones.

This study only considers defense against reconnaissance drones that cannot attack our defense systems. In our ongoing study, we consider that the target drone swarm can have ground attack drones[51], and the problem should take the possible damage caused by the drones to the air defense systems into consideration, aiming at maximizing the cost of drones and minimizing the loss of air defense systems simultaneously. In this study, we only implemented the evolutionary framework with six popular individual algorithms, and the ongoing study also includes implementing more hybrid algorithms for further possible performance improvement.

Although the countermeasure of air defense is more active than spoofing and interference, it is not sufficiently active because air defense systems are relatively static after deployment. Currently, we are also studying more comprehensive air defense systems that incorporate the cooperation of ground weapons, attacking drones, as well as human soldiers to fight against enemy droncs[52]. Future studies will consider air-ground cooperative defense against enemy drones, which could be significantly more complex and require algorithms to be more efficient and intelligent in predicting drone behaviors and utilizing problem-solving knowledge [53].

ACKNOWLEDGMENT

This work was supported by the National Natural Science Foundation of China (No. 61872123) and the Natural Science Foundation of Zhejiang Province (No. LR20F030002).

References

References is not available for this document.