Probabilistic Chain-Enhanced Parallel Genetic Algorithm for UAV Reconnaissance Task Assignment

Abstract

With the increasing diversity and complexity of tasks assigned to unmanned aerial vehicles (UAVs), the demands on task assignment and sequencing technologies have grown significantly, particularly for large UAV tasks such as multi-target reconnaissance area surveillance. While the current exhaustive methods offer thorough solutions, they encounter substantial challenges in addressing large-scale task assignments due to their extensive computational demands. Conversely, while heuristic algorithms are capable of delivering satisfactory solutions with limited computational resources, they frequently struggle with converging on locally optimal solutions and are characterized by low iteration rates. In response to these limitations, this paper presents a novel approach: the probabilistic chain-enhanced parallel genetic algorithm (PC-EPGA). The PC-EPGA combines probabilistic chains with genetic algorithms to significantly enhance the quality of solutions. In our approach, each UAV flight is considered a Dubins vehicle, incorporating kinematic constraints. In addition, it integrates parallel genetic algorithms to improve hardware performance and processing speed. In our study, we represent task points as chromosome nodes and construct probabilistic connection chains between these nodes. This structure is specifically designed to influence the genetic algorithm’s crossover and mutation processes by taking into account both the quantity of tasks assigned to UAVs and the associated costs of inter-task flights. In addition, we propose a fitness-based adaptive crossover operator to circumvent local optima more effectively. To optimize the parameters of the PC-EPGA, Bayesian networks are utilized, which improves the efficiency of the whole parameter search process. The experimental results show that compared to the traditional heuristic algorithms, the probabilistic chain algorithm significantly improves the quality of solutions and computational efficiency.

1. Introduction

With advancements in electronic computing power and communication technologies, large teams of unmanned aerial vehicles (UAVs) are now being deployed to collaborate on reconnaissance tasks. This trend is particularly prominent in military tasks [1]. The current research focuses on collaborative UAV monitoring of areas of interest [2,3]. Cooperative multi-UAV applications, such as power inspections, air rescue, and large formations, are increasingly common [4,5,6]. The key to the collaboration of multiple UAVs in combat is task assignment, which includes assigning one or more tasks to different UAVs within a UAV swarm under various environmental constraints and effectively prioritizing the tasks assigned to each UAV.

Task assignment in UAV operations is a challenging combinatorial optimization problem that has been studied extensively. The most commonly used models include the multiple traveling salesmen problem (MTSP) [7], multiple vehicle routing problem (MVRP) [8], and mixed-integer linear programming (MILP) [9]. The essence of the model is the NP-hard problem. The current models tend to only consider task assignments at the two-dimensional level due to the large amount of three-dimensional computation. This paper applies a multiple Dubins traveling salesmen problem (MDTSP) as the task assignment model. Based on this model, we designed a probabilistic chain-enhanced parallel genetic algorithm (PC-EPGA) as in Figure 1.

The structure of this paper is as follows: In Section 3, we introduce the modeling of multiple Dubins traveling salesmen problem models. In Section 4, the core of the PC-EPGA algorithm is presented and introduced. Section 5 presents the simulation results verifying the feasibility and effectiveness of our approach. Finally, Section 6 presents the conclusions.

The main contributions of this paper include the following:

1.

We enhance UAV task assignment by designing probabilistic chains, modeling corresponding chromosome coding, and refining genetic algorithm operations for crossovers and mutations based on these probabilistic chains.

2.

We design an adaptive fitness-based crossover operator to enhance the solution quality.

3.

We implement a Bayesian network to optimize the parameters of the heuristic algorithm, enhancing its effectiveness.

4.

This research establishes a parallel acceleration structure for genetic algorithms, significantly enhancing their computational speed at the hardware level.

2. Related Work

Deep learning algorithms, with their many parameters, are ill-suited for lightweight UAVs due to their inherent ‘black-box’ nature, which lacks interpretability. Also, deep learning methods are not suitable for NP-hard problem optimization [10]. The current mainstream algorithms are divided into linear optimization algorithms, heuristic algorithms, and contract net algorithms used for task assignment.

2.1. Linear Optimization Algorithms

The advantage of the linear optimization algorithm is that it can accurately find the global optimal solution, but the exhaustive method is too computationally intensive, and when the number of UAVs grows slightly, the computation will break the hardware computational limitations. Vincent T. [11] proposed a multi-objective branch-and-bound algorithm for solving mixed linear programming problems to search for optimal solutions (BNB), and Zhang [12] proposed linear dynamic programming to address the task assignment problem. These algorithms can obtain locally optimal solutions to low-dimensional problems, but as the number of UAVs and targets increases, the computational cost grows exponentially to the point where the solution cannot be obtained. In the realm of UAV task assignments, the large amount of computation constrains linear programming algorithms.

2.2. Contract Network Protocols

Chen [13] focused on auction-based task assignments in poor communication environments, using local communication to enable UAVs to bid for tasks, which enhances adaptability and resource allocation efficiency. Zhang [14] proposed a new hybrid scheduling-based UAV dynamic task scheduling method and proposed contract network protocols (CNPs), including buy and sell contracts, swap contracts, and replacement contracts. This algorithm, due to its rapid solution finding, has been widely applied in UAV task assignments.

2.3. Heuristic Algorithms

2.3.1. Particle Swarm Optimization

A heuristic algorithm is an empirically constructed algorithm that gives a feasible solution for each instance of the combinatorial optimization problem to be solved at an acceptable cost. The particle swarm optimization (PSO) algorithm [15] is widely used in heuristic algorithms because of its fast convergence, but it can easily fall into local optimal solutions. Geng [16] introduced the PSO to accomplish task assignments and introduced the concept of priority. Wang [17] enhanced the PSO, boosting its ability to handle multi-objective optimization problems in task assignment, enhancing convergence and diversity, yet it is still susceptible to local optima. Gong [18] designed the wind foraging-based PSO particle swarm algorithm based on wind foraging to further enhance the convergence speed. Geng [19] proposed an improved task allocation PSO (TAPSO) for task assignment to dynamically update the global best solution according to different phases to ensure more efficient task assignment.

2.3.2. Enhanced Heuristic and Combination Approaches

Cui [20] developed an improved chaotic self-adapting monkey algorithm, which enhances the efficiency of task distribution among multiple UAVs, dynamically assigning tasks based on real-time conditions, though it may face unpredictability in dynamic or complex environments. Wei et al. [21] proposed a collaborative attack task assignment using a competition–cooperation mechanism designed to dynamically allocate offensive and defensive roles among UAVs, which may struggle to balance the short-term individual objectives against long-term collective swarm goals. Wang [22] introduced an adjustable fully adaptive cross-entropy algorithm, offering not only improved task coupling and precedence handling but also the ability to adjust algorithm parameters in response to environmental changes at the cost of higher computational demands. J. Wu [23] explores UAV swarm allocation and take-off strategies to optimize flight distance and computation time, employing advanced scheduling algorithms to prioritize tasks based on their urgency and resource requirements. Zhang [24] developed a deadlock-free hybrid estimation of distribution algorithm, specifically addressing the challenge of task dependencies and temporal constraints to ensure smooth and efficient UAV task coordination without interruptions. Wei [25] presented a UAV cooperative reconnaissance task assignment model based on a heterogeneous target value and proposed an improved multi-verse optimizer (MVO) algorithm. To some extent, this algorithm solves the problem of easily falling into the local optimal solution. Gao [26] uses a grouped ant colony optimization algorithm to assign multiple UAVs to complete small-scale heterogeneous targets.

2.3.3. Genetic Algorithms

Currently, the genetic algorithm in heuristic algorithms has become a hotspot in task assignment research with its good global search ability. Edison [27] applied genetic algorithms (GA) to UAV task assignment. Deng QB [28] improved on the limitations of existing algorithms in terms of task assignment efficiency and multiple task type support and proposed an improved genetic algorithm. Wang [29] considered the task assignment problem as an optimization problem, used an opposition-based genetic algorithm (OGA) to search for the optimal solution, and designed the fitness function to evaluate the fitness of each individual.

2.4. Proposed Solution: PC-EPGA Algorithm

While PSOs converge quickly, the quality of the solution is often unsatisfactory. The MVO algorithm partially addresses this shortcoming, enhancing the quality of the solution. While contract net algorithms can be solved quickly, they tend to produce poor results in terms of solution quality. Genetic algorithms possess strong global optimization-seeking ability, but inefficient evolution and a large number of invalid evolutions limit the quality of the solution and the speed of solving.

In this context, we propose a probabilistic chain-enhanced parallel genetic algorithm. We achieve an efficient evolutionary process by introducing probabilistic chains, thus effectively addressing the limitations of existing methods. The introduction of probabilistic chains enables the algorithm to explore the solution space more efficiently and find high-quality solutions faster, solving the problem that genetic algorithms converge slowly despite their strong global contraction capability. PC-EPGA demonstrates superior performance in generating high-quality solutions within a finite number of heuristic iterations compared to other heuristic algorithms. This indicates that its task allocation scheme is more cost-effective and efficient in real-time applications.

3. MDTSP: UAV Reconnaissance Task Assignment Model

In this section, we present a comprehensive multi-UAV motion model, which includes a set of motion constraints specific to task requirements. The MDTSP model is used to model the tasking of UAVs. Based on the MTSP model, the UAV trajectory is planned using the Dubins curve.

3.1. Assignment Constraints

First and foremost, to ensure that UAV swarms neither miss nor duplicate tasks, Equation (1) is employed to constrain the task allocation scheme. As shown in Equation (1), the total number of tasks set after assignment is equal to the sum of the task set of n UAVs. ${U^i}_{tlist}$ is the set of tasks to be performed by the $i$th UAV, and $T_{\mathrm{list}}$ is an aggregation of the total number of tasks. This formula constrains the task allocation to not omit or duplicate tasks.

$$T_{\mathrm{list}}={\displaystyle \sum _{i=1}^N{U^i}_{tlist}}$$

(1)

3.2. Motion Trajectory Model

Selecting an appropriate UAV motion model for task assignment is crucial. The overall motion of an individual UAV is determined by the motion model. Flight trajectories are used to evaluate the cost of UAV tasking, and PC-EPGA generates the corresponding probabilistic chain based on the flight trajectories between reconnaissance targets. The Dubins curve, a spatial curve proposed by the famous mathematician L. E. Dubins in 1957 [30], is used as a simple but effective tool for solving problems related to robot motion planning and flight path planning. The Dubins curve is the shortest path curve that connects two points and allows the robot to travel from the starting point to the endpoint for a given turn radius. It has three basic forms, L (left turn), R (right turn), and S (straight line), and can be formed into six different sequences, namely, LSL, LSR, RSL, RSR, LRL, and RLR. These basic forms and sequences can generate the shortest path at a given initial position and direction connected to a given target position and direction. As shown in Equation (2), the motion curves of the UAV in the model designed in this study belong to the set of Dubins curves.

$$path\in \{RSL, LSR, RSR, LSL, RLR, LRL\}$$

(2)

In the context of constant altitude flight, the Dubins model provides a two-dimensional analytical framework for UAV trajectory planning, as shown in Equation (3). Equation (3) provides a mathematical description of the UAV’s position and heading angle changes on a plane under constant speed and variable heading conditions. This model assumes that a UAV maintains a steady speed V, and its instantaneous position in the plane is represented by coordinates x and y. The UAV’s trajectory is determined by the heading angle γ, with x and y evolving as a function of $V$ and γ, according to $x=V·\cos \gamma$ and $y=V·\sin \gamma$.

The rotation rate $\dot{z}$, a product of the UAV’s turning capability a_N and velocity V, is bound by the maximum turning rates ±a_N max, which reflect the UAV’s maneuvering constraints. This model thus characterizes the UAV’s path as a balance among its velocity, heading direction, and turning ability, which is essential for optimizing route efficiency within its operational parameters.

$$\left\{\begin{array}{l}\mathrm{x}=V·\cos \gamma \hfill \\ \mathrm{y}=V·\sin \gamma \hfill \\ \dot{z}=a_N/V,a_N\in \left[-a_{N\max },a_{N\max }\right]\hfill \end{array}\right.$$

(3)

The given scenario presents a Dubins curve originating from the origin O and terminating on the positive half-axis of the x-axis at point T. The path comprises two circular arcs, namely the right-turn arc with the center OR and the left-turn arc with the center OL, with the arrows indicating the direction of traversal. The six flight modes are shown in Figure 2 below.

3.3. Target Model

In recent years, scholars have categorized reconnaissance tasks into different types of targets: point targets, line targets, and surface targets [29,31]. UAV reconnaissance of these three areas is depicted in Figure 3. This paper models reconnaissance tasks based on Dubins curves, which are categorized into the following three types.

Point Target Reconnaissance Tasks: These tasks involve gathering reconnaissance information by flying over points within the UAV’s field of view.

Line Target Reconnaissance Tasks: To collect all reconnaissance information, UAVs need to fly along curves, covering the entire area of interest.

Surface Target Reconnaissance Tasks: These tasks require back-and-forth flight to ensure coverage of every area within the reconnaissance field of view.

According to Dubins curves’ motion trajectory, line targets have two entry points, plot targets have four entry points, and the angle requirement needs to be satisfied when entering, as shown in Figure 4. Depending on which port the UAV enters from, in this study, we used Euclidean distance measurements to select the port with the shortest Euclidean distance to enter based on the current position of the UAV.

3.4. Evaluation Model

In this study, the minimum cost model is defined by the highest individual flight cost of a UAV and the total flight cost of all UAVs. This is calculated using a weighted sum that incorporates both the total cost of flight paths upon the completion of all tasks and the cost associated with the UAV that undertakes the longest single flight distance, as specified in Equation (4). Such an approach ensures a fair distribution of tasks among the UAVs and aims to minimize the overall cost of task allocation.

The current popular method uses a generalized weighting of 0.3 for w₁ and 0.7 for w₂ for it to balance the two schemes [29,31]. As shown in Equation (4) in this paper, $P_i$ is the price of the ith UAV, and $J$ is an evaluation indicator for task assignment. In Equation (4), we employ the weights w₁ and w₂ to balance the maximum individual flight cost and the total sum of all flight costs, aiming to ensure each task allocation is as equitable as possible while also minimizing the overall cost to achieve optimal task distribution efficiency. The objective function $J$ is taken as the fitness of the chromosome.

$$\underset{A_i,i=1,2,\dots ,N}{\min }J=w_1·\underset{i=1,2,\dots ,N}{\max }{L}_i+w_2·\sum _{i=1}^N{L}_i$$

(4)

4. PC-EPGA: Probabilistic Chain-Enhanced Parallel Genetic Algorithm

4.1. PC-EPGA Model

The genetic algorithm (GA) is an optimization algorithm that mimics the evolutionary process observed in natural organisms. It aims to find the optimal solution to a problem by simulating mechanisms such as genetic variation, crossover, and selection. As shown in Figure 5, after encoding the task assignment scheme as a chromosome code, the optimal allocation scheme is then generated by means of crossover, mutation, and other operations. For task assignment, chromosome coding is performed to extract the corresponding probabilistic chain, and then the chromosomes that need to undergo crossover genetic operation are grouped for parallel computation after the selection operation. In crossover operations, crossover points are determined using probabilistic chains, and then different crossover strategies are performed based on fitness. In mutation operations, probabilistic chain-based mutation is performed on master and slave chromosomes. $J$ is the fitness of the chromosome.

Table 1. Probabilistic chain parallel accelerated genetic algorithm.

Probabilistic Chain Parallel Accelerated Genetic Algorithm

Create chromosome populations For I in generation:   Populations sorted by fitness   cluster compute    For chromo in population:     Obtain probabilistic chains     If crossover:       Determine intersections based on probabilistic chain       adaptive crossover     If mutation:       The master chromosome and slave chromosome are mutated       paired with each other to select the best solution to retain Obtain the optimal solution

presents a summary of the PC-EPGA’s framework, encapsulating the essential steps and the procedural workflow.

4.2. Encode Chromosome

In this study, we propose an innovative multi-chromosome coding strategy designed to precisely map the tasking scheme for UAV swarms. This method is based on a framework of a set of $(\mathrm{N}+1)$ chromosomes, where N represents the number of UAVs. In the proposed multi-chromosome coding strategy for task allocation to a fleet of UAVs, we define a master chromosome $C_m$ with $N-1$ nodes denoted by $c_{m1},c_{m2},\dots ,c_{m(N-1)}$. Additionally, we have $N$ slave chromosomes, $C_{s1},C_{s2},\dots ,C_{sN}$, each corresponding to a UAV, where the nodes represent assigned task numbers. The master chromosome dictates the task distribution, where $c_{m1}$ indicates the number of tasks for the first UAV. For each subsequent UAV $i$, where $1<i<N$, the number of tasks $T_i$ is given by the difference $c_{mi}-c_{m(i-1)}$. The tasks for the last UAV, $T_N$, are determined by subtracting the last node of the master chromosome from the total number of tasks, $M-c_{m(N-1)}$. Thus, the master chromosome organizes the division of tasks while the slave chromosomes specify the exact tasks per UAV. This coding model satisfies the tasking constraints of UAVs. This system ensures that all tasks are distributed appropriately and in accordance with the predetermined algorithmic constraints in Equation (1).

For example, a coded presentation diagram of the task assignment and sequencing for four UAVs and nine tasks is shown in Figure 6. The three nodes of the master chromosome correspond to the three cutting positions, and the nine tasks are divided into four parts to be distributed to the four slave chromosomes. The sequence of nodes of each slave chromosome corresponds to the sequence of tasks performed by the UAV.

The final task sequences assigned to the UAVs are as follows: UAV 0 performs tasks [2, 4], UAV 1 performs tasks [0, 8, 5], UAV 2 performs tasks [6, 3, 7], and UAV 3 performs task [1]. Each UAV will execute its assigned tasks in the order specified. The flight trajectories of the four UAVs are summed by adding up their Dubins curves to obtain the total flight distance of the UAV swarm. Subsequently, by comparison, UAV3 has the longest flight distance among the four UAVs. The flight trajectory lengths of UAV3 executing tasks 7, 3, and 6 are then extracted. The length is weighted and summed with the total flight distance of the UAV swarm according to Equation (4) to obtain the fitness score of this chromosome group.

We incorporated the concept of a “chain” into the chromosome structure. We encoded each task point as a node from the slave chromosome. The task number is the value of the node from the slave chromosome, and the probabilistic chain is the chain that connects the two slave chromosome nodes.

The probability of chromosome breakage is abstracted as the chain length such that longer chromosomes incur a higher cost between two nodes and are more likely to break. The length of the probabilistic chain (representing the probability of breaking) is defined by Equation (5), where L signifies the length of this probabilistic chain. This function accounts for two pivotal factors: the flight cost between tasks $P$, and the ratio of the number of tasks executed by the individual UAV $l$ to the average number of tasks across the swarm’s average task count $l_{average}$. This design not only simplifies the computation process but also directly correlates the workload of a UAV to its incurred cost, thereby reflecting task distribution equilibrium. To avoid unnecessary complexity and the risk of overfitting, we did not introduce additional coefficients into the model, ensuring that the model remains generalizable and robust.

$$L=P\times (l/l_{average})$$

(5)

4.3. PC-EPGA Algorithm Cross-Operator

A crossover operation is when two sets of chromosomes exchange their genes to generate a more optimal solution. First, we randomly merged all slave chromosomes, $C_{s1},C_{s2},\dots ,C_{sN}$, yielding a long parent strand $P$, which repleted all requisite task nodes. In the domain of GA, the crossover operation serves as a mechanism to recombine the genetic information from two parent chromosomes, $P_1=\left[p_{11},p_{12},\dots ,p_{1n}\right]$ and $P_2=\left[p_{21},p_{22},\dots ,p_{2n}\right]$, each comprising a sequence of nodes of n slave chromosomes. The process starts by selecting two crossover breakpoints, $c_1$ and $c_2$, to determine the segments $S_1=P_1[c_1:c_2]$ and $S_2=P_2[c_1:c_2]$ that will undergo exchange. Each parent strand corresponds to a task assignment scheme. The parent strand’s chains are classified into the following two types:

• Intra-chromosomal (IC): This category involves probabilistic chains within the same slave chromosome.
• Inter-chromosomal (XC): This category involves chains between different slave chromosomes.

Intra-chromosomal chains are the probabilistic chains obtained from within each slave chromosome, according to Equation (5). Inter-chromosomal chains are ordinary chains with no special significance between the slave chromosomes.

Ultimately, two breakpoints $c$ are selected from all the chains in both XC and IC, and the slave chromosome nodes of the parent strand within the two breakpoints are the segments to be crossed over. Upon the generation of a random number i, an assessment is made according to the proportions of the intra-chromosomal p_i to the inter-chromosomal p_t chain counts. If $i$ is greater than or equal to the threshold defined by $p_t/(p_t+p_i)$, the roulette wheel selection method is used based on the chain length $L$ in the Equation (5) mechanism to obtain the locations of the breaks from the $IC$. Should i fall short of the threshold, the breakpoint is arbitrarily selected from the XC set. This selection strategy is encapsulated by Equation (6):

$$c=\left\{\begin{array}{l}rws(p_i),i>=p_t/(p_t+p_i)\\ random(p_t),i<p_t/(p_t+p_i)\end{array}\right.$$

(6)

where c denotes the chosen breakpoint, the $rws(p_i)$ represents the roulette wheel selection method applied to preferentially choose breakpoints based on the intra-chromosomal weights (chain’s probabilisty). Conversely, $random(p_t)$ denotes a method where breakpoints are selected randomly from the inter-chromosomal set. This decision-making process ensures that the crossover operation is strategically optimized to enhance the genetic diversity and potential solutions of the algorithm.

To address the shortcomings of traditional partial mapping crossover (PMX), particularly its aggressive tendency to quickly converge on local optima in complex UAV task assignments [32], this study proposes a novel integration of edge reorganization (ER) [33] and order crossover (OX) operators [34]. The design idea of this study started with moderating the fast convergence property of PMX and retaining the potential high-quality solutions. The authors of this study relied on the properties of individual crossover operators to provide a comprehensive exploration of the solution space. While PMX is effective for quickly finding high-quality solutions, ER is essential for maintaining unique genetic information and expanding the search scope in complex, large-scale environments. The OX operator, meanwhile, provides a balance between aggressive and conservative crossover strategies. In this study, we designed adaptive crossover operators based on fitness by integrating the three crossover methods.

The selection of the three crossover operators is fitness-dependent: PMX for high-fitness chromosomes, OX for medium fitness, and ER for lower fitness. The adaptive crossover adjusts the selection probability of each operator during the crossover process using Bayesian guiding. The following section of the paper presents these three crossover operators.

In the realm of genetic algorithms, PMX is employed as a sophisticated strategy for the recombination of highly fit chromosomes [32]. This operator orchestrates the exchange of gene segments between two parent chromosomes while preserving the integrity of the remaining genetic material, thereby maintaining genetic diversity and robustness. During the crossover process, the paramount concern is the prevention of gene node duplication. If a selected gene node from one parent is absent in the corresponding segment of the other, a search is initiated to locate the identical gene node in the other parent and its counterpart in the first. This recursive search and exchange continues until a node is found that allows for a direct exchange without introducing duplicates into the progeny chromosomes, as shown in Figure 7. A mapping function M is defined to establish a gene correspondence between these segments. The offspring chromosomes, O1 and O2, are initially inherited from $P_1$ and $P_2$, respectively, with the mapping function M resolving any conflicts. Specifically, for any gene $P_{1i}$ in $P_1$ not present in S2 that would duplicate in O2, the process substitutes $P_{1i}$ with M−1($P_{1i}$), continuing iteratively until a non-duplicating gene is found. The same procedure is applied to $P_2$ to construct O1.

The ER operator is a nuanced crossover mechanism in genetic algorithms that primarily seeks to retain the adjacency relationships inherent in the parent chromosomes. Representing two parent chromosomes as $P_1$ and $P_2$, where each $p_{ij}$ is a node, the ER operator constructs an adjacency list for each node, reflecting its immediate neighbors in both parents. A starting node is selected for the offspring, and subsequent nodes are chosen to minimize the overlap of common neighbors, as guided by a combined adjacency list from both parents. This iterative selection is mathematically represented as choosing node $n$, such that $N(c)$ is $n=\underset{x\in N(c)}{argmin}|N(x)|$, where the set of nodes is adjacent to the current node $c$ of the offspring, and $|N(x)|$ is the count of neighbors for node $x$. By favoring nodes with fewer common neighbors, the ER operator effectively preserves distinctive features of the parent chromosomes in the progeny, analogous to biological genetic recombination, which mixes and preserves beneficial traits. In this example, we have two paternal chromosomes, as shown in Figure 8. We insert spacer nodes between different follower chromosomes. An adjacency table can be created for the chromosomes: (1, 2), (2, 3), (4, 5), (5, 6), (4, 6), (6, 1), (5, 3), and (3, 2). Then, we can choose a starting node from the adjacency table based on the probability chain break point. For example, we chose node 3 as the starting point. Then, we needed to select the next node of the progeny solution from the set of nodes adjacent to node 3. We selected the next node based on the adjacency information of the two parent nodes to minimize the number of common neighbors between the two nodes. Node 3 is adjacent to nodes 2 and 5 in the link table. Subsequently, the numbers of the nodes adjacent to node 2 are 3 and 1, while the numbers of the nodes adjacent to node 5 are 3, 4, and 6. The number of neighbors of node 5 is more than that of node 2, so node 2 is selected as the next node of node 3 according to the principle of least number of neighbors, and so on.

OX is another popular crossover operator used for permutation-based problems. The OX operator maintains the relative order of nodes in the parent chromosome, crossing over in the subsequent form. As shown in Figure 9, the OX operator can be performed as follows: Select a sub-sequence from one parent chromosome. Copy this subsequence into the offspring solution, maintaining the original order of the elements. Fill the remaining positions in the offspring solution with the nodes from the second parent, preserving their original order and excluding the nodes already present in the copied subsequence.

4.4. PC-EPGA Variation Operator

This study applied a dynamic mutation rate that responds to the population’s rate of fitness improvement in a genetic algorithm. The formula $mot=e^{-grad}$ was chosen to ensure that the mutation probability increases when the rate of fitness improvement decreases. This is because a negative gradient $grad$ signifies a slower improvement, and when exponentiated as $e^{-grad}$, results in a higher mutation probability. This design mimics natural evolutionary processes, where environmental stresses lead to higher genetic variability, providing a robust mechanism for exploring the solution space more thoroughly when progress stalls. The exponential function has a smoothing property that avoids abrupt changes in the rate of variation.

The strategy for variation outlined in this study splits into two key parts: one focuses on variations associated with the slave chromosomes of UAVs, and the other concerns the master chromosomes. For variations in the master chromosomes, we have designed a variation technique for enhancing the equitable distribution of tasks among UAVs. For slave chromosome variations, we devised two distinct methodologies. The first is a generic flip, where the slave chromosome chain determines the flip point based on the probabilistic chain. The second is based on a probabilistic chain design of continuation variants from chromosome breaks.

In master chromosome variants, nodes on the master chromosome primarily function to allocate the total number of tasks to each individual UAV. Mutations targeting the master chromosome occur at the node level rather than affecting the entire master chromosome sequence, corresponding to the intended mutation. In this study, the two slave chromosomes $C_{si}$ and $C_{s(i+1)}$, split by the master chromosome node $c_{mi}$ of the desired mutation, are first merged into a single sequence. Subsequently, all probability chains from between slave chromosomes within that sequence are normalized. Next, the selection of a specific segment from the resultant unified chain is efficiently executed using the roulette wheel method. The selected probabilistic chain segment serves as a demarcation point, enabling the reclassification of preceding and succeeding nodes into two novel slave chromosomes, $C_{si}$ and $C_{s(i+1)}$. Correspondingly, this reorganization is mapped in the master chromosome nodes $c_{mi}$, which are updated to reflect the node count of the newly formed slave chromosomes, $C_{si}$ and $C_{s(i+1)}$.

The flip mutation process for slave chromosomes is elaborated as follows: All the slave chromosomes are connected head to tail to form a long sequence containing all the task nodes. Subsequently, two slave chromosomes’ probabilistic chains are obtained by means of roulette, and all slave chromosome nodes between the two points are inverted. After the long sequence is reconstructed, it is divided into multiple slave chromosomes according to the master chromosome, where the slave chromosomes have changed. A schematic diagram of the flipping mutation is shown in Figure 10.

The continuation mutation process for slave chromosomes is elaborated as follows. All probabilistic chains from inside the chromosome are uniformly normalized, and the mutation points $pos$ are selected by means of roulette based on the length of the probabilistic chains $L$. This mutation site splits the slave chromosome $C_{si}$ into two where mutation site $pos$ is located, creating two distinct segments: the head $C_{si}[0:pos]$ and the tail $C_{si}[pos:]$. We evaluated the potential for forming a probabilistic chain by connecting the first and last nodes of a specific slave chromosome’s tail with the final nodes of all other slave chromosomes. A connection is then made with the slave chromosome that yields the shortest probabilistic chain. This intricate process is elucidated in Figure 11. The mutation point occurs on the slave chromosome corresponding to UAV2, which is divided into a head (3) and a tail (4, 5). The nodes in the tail are connected to the tails of the other slave chromosomes (2, 8, 9). An attempt is made to construct the probabilistic chain, and the shortest probabilistic chain is selected, which will be the continuation of the tail to the back of that slave chromosome.

4.5. Bayesian Network Optimization Operator

In this section, we apply Bayesian optimization to optimize the parameters of the genetic algorithm, in particular, the individual fitness thresholds of the adaptive crossover operator [35]. We utilize a Gaussian process (GP) based on the squared exponential kernel [36] function to describe the posterior distribution of the objective function, as shown in Equation (7):

$$\widehat{k}\left(x,x^{\prime }\right)={\sigma }^2\exp \left(-\frac{{\left(x-x^{\prime }\right)}^2}{2l^2}\right)$$

(7)

where σ² represents the output variance, which quantifies the dispersion of the function’s outputs around the mean, while $l$ denotes the length scale, a measure of the correlation between points in the input space, reflecting the smoothness of the function. Further, we employed an expected improvement (EI) strategy to sample our parameters. The expected improvement quantifies the anticipated benefit of sampling at a new point x, relative to the current optimum, and is articulated in Equation (8):

$$EI(x)=\left(\mu (x)-f\left(x_{\mathrm{best}}\right)-\xi \right)\mathsf{\Phi }(Z)+\sigma (x)\varphi (Z)$$

(8)

In this equation, μ(x) and σ(x) are the GP’s predicted mean and standard deviation for the objective function at point $x$. The term f(x_best) refers to the best objective function value observed thus far. The exploration parameter $\xi$ is usually set around 0.01. Φ(Z) and ϕ(Z) correspond to the cumulative and probability density functions of the standard normal distribution, respectively. The variable Z is a normalized measure of improvement over x_best, computed as $Z=\frac{\mu (x)-f\left(x_{\mathrm{best}}\right)-\xi }{\sigma (x)}$.

4.6. Parallel Acceleration of Chained Genetic Algorithms

In this section, we address the practical need for acceleration on the hardware level. Hardware-distributed acceleration techniques are now widely recognized to significantly help improve acceleration in real-world tasks. GA has inherent parallelism because individuals in the population can be evaluated and mutated independently. We assign chromosome sets to different nuclei to perform crossover mutations and thus enhance the acceleration effect. The main advantage of PC-EPGA is the ability to search the solution space of a problem in parallel using a large number of computational resources, thus speeding up the search process. For the multi-core implementation, the entire population is uniformly selected and assigned by the main core, and the corresponding chromosomes to be operated are uniformly distributed among the available cores according to the crossover probability, reducing the traffic and increasing the speed. The architecture is shown in Figure 12.

5. Simulation Experiment

5.1. Simulation Environment

Simulation experiments were conducted using Python 3.8 on a computer with an Intel^® Core^TM i9 processor. The tasks involved setting coordinates for point, line, and surface targets within a designated area and enabling a UAV swarm to navigate using Dubins motion curves, as outlined in Equation (3). These curves consider the UAV’s minimum turning radius and flight speed, which were set at 5 m and 3 m/s in assumption, respectively.

The simulation environment was a two-dimensional area of 700 by 700 m, mapping the UAVs’ flight trajectories. PC-EPGA’s optimal parameters were determined using Bayesian methods, resulting in a crossover probability of 0.89 and a selection ratio of 0.21.

To confirm the validity of our approach, the study deployed four UAVs to reconnoiter five point targets, one line target, and one surface target. The optimal solution derived from the PC-EPGA is visualized using the MDTSP model in Figure 13a. In this paper, the sequence of allocation schemes obtained from PC-EPGA was deployed on the ROS (Robot Operating System) platform for testing, and the motion trajectory of the UAV was visualized using the rviz (Robot Visualisation) tool, as shown in Figure 13b. It is evident that the motion planning trajectory aligns with the MDTSP model. This alignment clearly demonstrates the effectiveness of the PC-EPGA approach.

5.2. Comparative Experiment

In the experimental section of our study, we conducted a comprehensive comparative analysis to evaluate the performance of our proposed algorithm, PC-EPGA. The comparison involved several well-established algorithms in the field. These included Wang’s opposition-based genetic algorithm (OGA) [29], Geng’s task allocation PSO (TAPSO) [17], WEI’s refined Multi-Verse Optimizer (MVO) [25], Zhang’s contract network protocol (CNP) [19], and the traditional GA. To assess the effectiveness of various task allocation algorithms at different task volumes, the number of UAVs was consistently held at four, with task sizes of 20, 30, 40, and 50. The number of tasks is the sum of the number of the three target types performed by the UAV swarm. The motion trajectory of UAV is all composed of Dubins space curves. The task coordinates and the initial position coordinates of the UAV were randomly generated, while ensuring that each algorithm was tested under the same conditions. For reliability, each testing condition was repeated 40 times, documenting the optimal solution for each iteration. The heuristic algorithm’s performance was evaluated by averaging these optimal solutions. All heuristics employ a fixed population size of 30 and undergo 30 iterations of heuristic processing. The most important evaluation metric for heuristic algorithms is the inspired solution quality, i.e., the quality of the optimal solution that the algorithm can obtain. Additionally, the heuristic iteration speed and heuristic robustness are the secondary metrics of heuristic algorithms.

5.2.1. Comparative Analysis of Solution Quality

In the comparative analysis of heuristic algorithms, the principal metric under consideration is the quality of solutions rendered. By employing the objective function delineated in Equation (6) as a benchmark for fitness, we systematically juxtaposed the outcomes of diverse heuristic approaches. An inverse relationship was posited between the fitness scores and solution quality, with lower fitness indicative of a closer approximation to the global optimum. The aggregate mean of the optimum solutions obtained across these trials was then employed as the definitive standard for comparison. A lower value of fitness indicates a better-optimized solution.

The data in

Table 2. Comparative analysis of average fitness across different algorithms.

	Algorithm	TAPSO [17]	OGA [29]	GA	CNP [19]	MVO [25]	PC-EPGA
Targetnum
20		2848.28	2865.78	4084.93	6852.75	2994.24	2324.02
30		4376.48	4466.49	6163.37	7616.26	4485.42	3356.73
40		6445.21	6125.61	8445.39	8217.35	6264.58	4673.29
50		7596.97	7433.87	10,173.13	12,611.20	7448.93	5937.03

elucidate a pattern of superior performance of PC-EPGA across different task sizes. In instances with a target number of 20, PC-EPGA’s fitness was substantially lower than its nearest competitor, MVO. PC-EPGA consistently maintained a solution quality more than 20% ahead of that in second place as the number of tasks increased. Its performance, due to the efficient exploration of the solution space enabled by probabilistic chains, underscores its superiority in finding high-quality solutions rapidly. As the UAVs and tasks grew in number, PC-EPGA exhibited greater global optimization capabilities. The visualization is shown in Figure 14.

5.2.2. Comparative Analysis of Iterative Convergence Speed

Iterative convergence speed is a metric in evaluating iterative algorithms. Heuristic algorithms usually involve different programming methods. Consequently, assessing the iterative efficiency of heuristic algorithms offers a more objective perspective. As shown in Figure 15, it can be seen that PC-EPGA’s convergence speed and ability to break through the local optimal solution were better than those of the other algorithms at each task size.

5.2.3. Comparative Assessment of Optimization Robustness

We employed an interquartile range (IQR)-centric methodology to assess robustness. The IQR, the range between the third (Q3) and first quartiles (Q1), IQR = Q3 − Q1, serves as a robustness indicator. A constricted IQR symbolizes the algorithm’s consistent and stable performance through exhaustive testing paradigms. A lower value is indicative of a more robust optimal solution.

As shown in

Table 3. Robustness comparison of algorithms via interquartile range (IQR) metrics.

	Algorithm	TAPSO	OGA	GA	CNP	MVO	PC-EPGA
Targetnum
20		182.02	232.63	499.81	-	159.84	164.97
30		364.97	295.64	775.40	-	279.96	302.42
40		335.08	465.31	1125.50	-	242.16	469.81
50		281.33	520.12	1861.62	-	242.60	343.73

, PC-EPGA outperformed the traditional GA-based algorithms, especially in the large scope of NP-hard problems, showing a consistent ability to find better average solutions. As shown in Figure 16, PC-EPGA maintained high robustness, and all the optimal solutions achieved lower fitness than the other algorithms.

5.3. Ablation Study

In our study, we focused on the impact of probabilistic chain and adaptive crossover operators on PC-EPGA. As a baseline, a traditional genetic algorithm was compared with algorithms that use probabilistic chains but with traditional PMX, ER, and OX crossover operators, and PC-EPGA, which uses probabilistic chains and adaptive crossover operators. This comparative framework was instrumental in isolating and understanding the distinct impacts of probabilistic chains and adaptive crossover operators on the algorithmic efficacy of PC-EPGA. The aim was to determine the degree of performance improvement in PC-EPGA, ensuring that the core features are intact, and to understand their superiority over traditional GA. The effects on the quality of the average fitness of the solution are presented in

Table 4. Ablation study: average fitness of the solution.

	Probabilistic Chain	PMX Cross-Operator	ER Cross-Operator	OX Cross-Operator	ADAPTIVE Cross-Operator	20 Tasks	30 Tasks	40 Tasks	50 Tasks
A						4084.93	6163.37	8445.39	10,173.13
B	√	√				2412.51	3454.84	4860.81	6277.29
C	√		√			2516.58	3643.51	5200.57	6418.65
D	√			√		2430.25	3546.51	5105.67	6288.34
E	√				√	2324.02	3356.73	4673.29	5937.03

, and

Table 5. Interquartile range (IQR)-based robustness analysis in probabilistic chain-enhanced parallel genetic algorithm (PC-EPGA) with various crossover operators.

	Probabilistic Chain	PMX Cross-Operator	ER Cross-Operator	OX Cross-Operator	ADAPTIVE Cross-Operator	20 Tasks	30 Tasks	40 Tasks	50 Tasks
A						499.84	775.40	1125.50	1861.62
B	√	√				144.09	256.30	419.34	471.30
C	√		√			131.06	316.42	492.30	340.54
D	√			√		182.67	407.18	276.85	568.04
E	√				√	164.97	302.42	469.81	343.73

demonstrates the impact on heuristic robustness. A is the GA algorithm without using any tricks; B, C, and D are the GA algorithms using probabilistic chains with PMX, OX, and ER crossover operators, respectively; E is the GA algorithm using probabilistic chains with adaptive crossover operators.

Based on the conventional GA algorithm as a benchmark, the average fitness of the solution with probabilistic chains using a conventional crossover operator presented a 33% reduction, and the average fitness of the solution with PC-EPGA using probabilistic chains and an adaptive crossover operator presented a 40% reduction. In terms of robustness, it can be seen that the probabilistic chains still maintained stronger robustness with an increasing number of tasks relative to the traditional GA algorithm. These results validate the use of probabilistic chains for heuristic solution quality enhancement.

5.4. Parallel Acceleration Speed Improvement Comparison

In this study, the PC-EPGA algorithm was successfully deployed on a multi-core acceleration platform to achieve parallel accelerated computation. The effect of the parallel genetic algorithm was greater acceleration than that of the traditional genetic algorithm. The experimental settings were as follows: the number of populations was 20, the number of iterative rounds was 1, and the number of UAVs was 5.

Table 6. Probabilistic chain-enhanced parallel genetic algorithm (PC-EPGA) and probabilistic chain genetic algorithm (PC-GA) running speed comparison.

Target Size	10	20	30	40	50	60
PC-GA	4.60 s	5.35 s	9.02 s	8.72 s	13.08 s	12.39 s
PC-EPGA	0.83 s	1.62 s	2.34 s	3.03 s	3.98 s	4.62 s

and Figure 17 in this study compare the probabilistic chain genetic algorithm (PC-GA) with its multicore-accelerated counterpart, PC-EPGA, across various task sizes. The multicore-accelerated version shows up to 82% improvement in performance for smaller tasks and demonstrates better scalability with larger tasks due to a slower increase in runtime, indicating substantial benefits of multi-core parallel acceleration for computational genetic algorithms.

6. Conclusions

This paper focuses on the assignment and sequencing of reconnaissance tasks for large-scale UAV operations. In this paper, a probabilistic chain is designed by incorporating the Euclidean distance information between tasks and the uniformity of task allocation to help the genetic algorithm quickly obtain suboptimal solutions with higher quality. We design an adaptive crossover operator to expand the solution space better. The design of probabilistic chains and adaptive crossover operators successfully improved the quality of suboptimal solutions generated by existing algorithms. Ablation experiments showed that the probability chain and crossover operator reduced the fitness of GA solutions by about 40%. Compared to the current algorithms, such as MVO, OGA, CNP, GA, and TAPSO, our model demonstrated a 15% enhancement in performance, maintaining its efficacy despite increasing the UAV complexities and task quantities. Additionally, we implemented parallel acceleration at the hardware level, resulting in a significant 70% increase in operational speed. These achievements validate the effectiveness of our proposed algorithm. This study primarily utilized the probabilistic chain in the crossover operation to select crossover points in the mutation operation. The PC-EPGA algorithm exhibits notable efficacy and applicability in managing the allocation of large-scale reconnaissance tasks, potentially revolutionizing UAV swarm monitoring and distribution strategies as well as enhancing efficiency in low-altitude logistics delivery systems. However, heuristic algorithms are limited by mechanisms that make it difficult to obtain an optimal solution and can only continuously improve the quality of sub-optimal solutions. The next step of the research should be directed towards making improvements to the probabilistic chain itself, and more UAV and task-related elements should be incorporated into the probabilistic chain.