Cooperative Task Allocation for Unmanned Aerial Vehicle Swarm Using Multi-Objective Multi-Population Self-Adaptive Ant Lion Optimizer

Highlights

What are the main findings?

• A double-layer encoding mechanism and adaptive penalty function were designed for constraint handling.
• A multi-objective multi-population self-adaptive ant lion optimizer is proposed.

What are the implications of the main findings?

• The coupling constraints imposed by heterogeneous tasks, such as task sequencing and time-window requirements, are addressed.
• The diversity of task-allocation outcomes is enhanced.

Abstract

The rational allocation of tasks is a critical issue in enhancing the mission execution capability of unmanned aerial vehicle (UAV) swarms, which is difficult to solve exactly in polynomial time. Evolutionary-algorithm-based approaches are among the popular methods for addressing this problem. However, existing methods often suffer from insufficiently rigorous constraint settings and a focus on single-objective optimization. To address these limitations, this paper considers multiple types of constraints—including temporal constraints, time window constraints, and task integrity constraints—and establishes a model with optimization objectives comprising task reward, task execution cost, and task execution time. A multi-objective multi-population self-adaptive ant lion optimizer (MMSALO) is proposed to solve the problem. In MMSALO, a sparsity-based selection mechanism replaces roulette wheel selection, effectively enhancing the global search capability. A random boundary strategy is adopted to increase the randomness and diversity of ant movement around antlions, thereby improving population diversity. An adaptive position update strategy is employed to strengthen exploration in the early stages and exploitation in the later stages of the algorithm. Additionally, a preference-based elite selection mechanism is introduced to enhance optimization performance and improve the distribution of solutions. Finally, to handle complex multiple constraints, a double-layer encoding mechanism and an adaptive penalty strategy are implemented. Simulation experiments were conducted to validate the proposed algorithm. The results demonstrate that MMSALO exhibits superior performance in solving multi-task, multi-constraint task-allocation problems for UAV swarms.

1. Introduction

The rapid development of unmanned aerial vehicle (UAV) technology has garnered widespread global attention. As a technology with immense potential, UAVs have demonstrated significant application prospects in military, civilian, and commercial domains [1,2,3,4]. With advancements in technology and reductions in cost, UAVs are gradually transitioning from single-platform applications to swarm-based applications [5]. A UAV swarm system refers to a system where multiple UAVs work collaboratively, sharing information and distributing tasks to accomplish complex missions [6]. This system offers numerous advantages, such as high task-execution efficiency, strong reliability, and high flexibility. However, a key challenge facing UAV swarm systems is the rational allocation of tasks to achieve optimal system performance. The task-allocation problem for UAV swarms involves multiple factors, including task priority, UAV capabilities, and constraints, making it complex and diverse. It is a typical NP-hard problem, and the main difficulties in addressing it are problem modeling and model solving [7].

In terms of problem modeling, the research on multi-task multi-constraint task-allocation problems is relatively limited, and there has been insufficient attention paid to multi-objective optimization algorithms. One of the significant advantages of UAV swarms over individual UAVs is their ability to perform diverse and complex tasks. Therefore, multi-task types are a key focus in the study of UAV swarm task allocation, and the task sequence constraints brought by multi-task types must also be taken into account. Some studies on this subject did not consider multiple task types and temporal constraints of tasks, lacking rigor [8,9]. Due to the complexity and diversity of UAV swarm task allocation, evaluation criteria for task-allocation schemes should not be overly singular. Thus, traditional single-objective optimization strategies are not applicable, and a multi-objective optimization strategy is more appropriate. Other studies converted multi-objective optimization problems into single-objective optimization problems by weighting, which leads to the solution schemes being subjectively influenced by the weights. Therefore, it is not conducive to comprehensive decision-making [10,11].

In response to the aforementioned issues, this paper describes the collaborative multi-task-allocation model as a multiple traveling salesman problem. The model considers multiple task types, temporal constraints, and multi-objective optimization. First, UAVs can perform reconnaissance, supplies delivery, and assessment tasks, and a UAV swarm is required to execute all three types of tasks for all targets. Second, the three tasks on each target must be strictly carried out in the order from reconnaissance, supplies delivery, to assessment, and some tasks have specific requirements for the time period of execution. Finally, three optimization objectives need to be considered: task reward, task execution cost, and task execution time.

Generally, solution approaches can be categorized into two main types: exact methods and approximate methods. Exact methods yield optimal solutions at the expense of higher computational time, whereas approximate methods obtain satisfactory solutions within limited time.

Exact methods, such as dynamic programming [12] and branch-and-bound algorithms [13], can provide accurate references for the analysis of task-allocation problems. Dynamic programming is a global optimization method for solving multi-step optimization problems. Task allocation, which involves optimizing the execution sequence of multiple tasks by UAVs, is well-suited for dynamic programming. To address the multi-area reconnaissance coverage problem, Xie et al. [14] employed dynamic programming to recursively determine the optimal visiting sequence and entry–exit positions for multiple regions. In scenarios involving cooperative multi-UAV operations in adversarial environments, Alighanbari et al. [15] applied a two-step look-ahead strategy to derive action plans for UAVs, achieving optimal solutions with reduced computation time. Branching methods, a class of tree search strategies, primarily include branch and bound, branch and price, and branch and cut. The branch-and-bound method prunes suboptimal branches during the solution of mixed-integer linear programming models, thereby avoiding extensive unnecessary searches. Casbeer et al. [16] utilized a branch and price approach to solve temporally coupled problems. By leveraging the sparsity of constraint distributions, they decomposed the problem into smaller subproblems via column generation, progressively converging to the optimal solution while significantly reducing computational effort. Subramanyam et al. [17] formulated a robust linear programming model for heterogeneous vehicle routing under worst-case scenarios and employed a branch and cut algorithm to dynamically introduce constraints, thereby approaching the optimal solution and obtaining a tight lower bound for the problem.

In contrast, approximate methods do not aim to obtain the optimal solution. Instead, they utilize heuristic information to guide the search process in order to achieve satisfactory solutions within polynomial time. In the field of task allocation for UAV swarms, evolutionary algorithms have emerged as popular and efficient approximate methods. Commonly used algorithms include particle swarm optimization (PSO) [18], genetic algorithms (GA) [19], and ant colony optimization (ACO) [20]. Researchers have tailored these algorithms for specific problem environments through various improvements, such as strategy optimization [21], algorithm hybridization [22], and enhanced encoding schemes [23]. These modifications have effectively advanced the development of UAV swarm task allocation. Based on the context of multi-UAV cooperative reconnaissance, Wang et al. [21] further enhanced the population diversity and convergence capability of the algorithm by introducing a constraint-based particle initialization strategy, an experience-pool-based particle position reconstruction strategy, and a time-varying parameter adjustment strategy into the PSO. Liu et al. [7] employed a greedy selection operator and introduced elite opposition-based learning, constructing opposition solutions from feasible solutions of the current problem to increase population diversity, thereby improving the convergence speed of the GA. He et al. [24] first performed a preliminary search on the data using a simplified artificial fish swarm algorithm to generate an initial pheromone distribution, which was then transferred to an improved ACO for solution. This approach reduced the initial search blindness of the ACO and accelerated the algorithm’s convergence.

Ant lion optimizer (ALO) [25] is a swarm intelligence algorithm proposed by Seyedali Mirjalili in 2015. It simulates the foraging behavior of antlions trapping ants. The main steps include random walking of ants, setting traps by antlions, ants falling into traps, antlions preying on ants, and antlions rebuilding traps. It has been widely applied in various fields, including computer science, engineering, energy, and operations management due to its ease of implementation, strong scalability, and high flexibility [26].

This paper proposes a multi-objective multi-population self-adaptive ant lion optimizer (MMSALO) to efficiently solve the model. To enhance global exploration, MMSALO replaces the roulette-wheel operator with a sparsity selection mechanism. A random boundary strategy redefines the margins of the random walk that each ant performs around an antlion, thereby increasing the population diversity. An adaptive position update schedule then biases early iterations toward sparsity-selected antlions to favor exploration, whereas later iterations pivot around the elite antlion to intensify exploitation. Finally, a preference-based elite selection scheme drives each ant to track the best individual discovered thus far along distinct objective dimensions, which strengthens the algorithm’s convergence behavior and improves the spread of the resulting Pareto front. This study implemented a comprehensive experiment comparing MMSALO with four representative algorithms on the DTLZ test function suite, multi-objective 0-1 knapsack problem (MOKP), and two task-allocation models. To address the complex, multiple constraints inherent in the task-allocation models, a dual-layer encoding scheme, and an adaptive penalty strategy are proposed. The results demonstrate the effectiveness and superiority of the proposed approach.

The remainder of this paper is organized as follows. Section 2 formulates the task-allocation model for UAV swarms. Section 3 presents the MMSALO algorithm in detail. Section 4 describes the simulations and results. Finally, the conclusions of this study are presented in Section 5.

2. Task-Allocation Model for UAV Swarms

2.1. Scenario Description

In the scenario, multiple UAVs and multiple targets are present, with each target involving individuals in need of supplies assistance. To fulfill rescue requirements, the UAV swarm must depart from their initial positions and, based on task-allocation results, sequentially conduct reconnaissance, supplies delivery, and assessment for all targets. Effective collaboration among the UAVs is essential to executing these complex missions. Accordingly, the following assumptions are established to analyze the cooperative task-allocation problem:

1.

All targets are stationary and their locations are known. The operational area is simplified to a two-dimensional region.

2.

All UAVs have limited range and onboard resources. Tasks such as supplies delivery consume these onboard resources. Since the drones operate at different altitudes, the possibility of collision between them is not considered.

3.

During task execution, temporal sequence constraints and time-window constraints must be considered.

Current methods for solving task-allocation problems are primarily categorized into centralized, distributed, and mixed approaches [27]. Centralized methods utilize a central node to collect information and assign tasks, thereby enabling the full coordination of various resources and reflecting global characteristics, suitable for scheduling based on the integration of known information prior to task execution. Compared with distributed methods, centralized methods can effectively avoid task competition among UAVs, eliminating the need for additional resources to coordinate conflicts. In contrast to hybrid methods, centralized methods offer more concentrated resources, and the unified scheduling by the central node can achieve better global scheduling outcomes. Therefore, this study adopted a centralized allocation strategy.

2.2. Basic Model Definition

In this section, three fundamental models are defined: the target model, the task model, and the UAV model.

2.2.1. Modeling of Target and Task

This study primarily investigated static targets. Figure 1 illustrates a schematic diagram of UAV swarm task allocation, where “RE” denotes reconnaissance tasks, “AT” represents supplies-delivery tasks, and “AS” indicates assessment tasks. There are $N_T$ targets with known location information, and each target $T_j$$(j=1,2,\dots ,N_T)$ comprises three types of tasks $(N_{type}=3)$: reconnaissance, supplies delivery, and assessment. The three tasks on each target must be executed strictly in order from reconnaissance, to supplies delivery, to assessment. Let $M_k$$(k=1,2,\dots ,N_M)$ denote the set of tasks for the k-th task and $N_M=N_T{N}_{type}$ represent the total number of tasks.

Table 1. Attributes of targets and tasks.

Model	Attribute	Parameter
Target, $T_j$	Target number	$N_T$
	Task type number	$N_{type}$
	Target position	$X_j^T$
Task, $M_k$	Task number	$N_M$
	Task value	$V_k^M$
	Onboard resources requirement	$R_k$
	Task timetable	$[t_k^s,t_k^e]$
	Task execution time	$t_k^{do}$
	Task time window	$[E{T}_k,L{T}_k]$
	Probability of task failure	$S_k$

provides detailed attributes of targets and tasks. For targets, $X_j^T$ denotes the location of the target. For tasks, $V_k^M$ represents the value of the task; $R_k$ represents the demand of the supplies-delivery task for onboard resources; $[t_k^s,t_k^e]$ indicates the task schedule, where $t_k^s$ and $t_k^e$ represent the start and completion times, respectively; $t_k^{do}$ denotes the task execution time; $[E{T}_k,L{T}_k]$ represents the task time window, where $E{T}_k$ is the earliest allowable start time, and $L{T}_k$ is the latest allowable completion time; $S_k$ denotes the probability of task failure.

2.2.2. Modeling of UAVs

Table 2. Attributes of UAVs.

Model	Attribute	Parameter
UAV, $U_i$	UAV number	$N_V$
	UAV position	$X_i^U$
	UAV speed	$v_i$
	Task capability parameter	$P_{ik}$
	UAV maximum range	$D_i^U$
	UAV value	$V_i^U$
	Amount of UAV onboard resources	$L_i$

presents the detailed attributes of UAVs. Specifically, $X_i^U$ denotes the location of the UAV, $v_i$ represents the speed of the UAV, $P_{ik}$ is the task capability parameter of the UAV, $D_i^U$ represents the maximum range of the UAV, $V_i^U$ refers to the value of the UAV, and $L_i$ signifies the amount of onboard resources of the UAV.

Based on the target model, task model, and UAV model, the relationship between the UAV and the task is described using $x_{ik}$ as the decision variable, which is defined as follows:

$$x_{ik}=\left\{\begin{array}{cc}1,& \mathrm{allocate}\mathrm{task}{M}_k\mathrm{to}\mathrm{UAV}{U}_i\\ 0,& \mathrm{else}\end{array}\right.$$

(1)

2.3. Constraints

Based on the three fundamental models, this section establishes the model constraints. All constraints can be categorized into two parts: UAV constraints and task constraints. UAV constraints are derived from the limitations of UAV capabilities, including collaboration constraints, range constraints, and supplies-delivery-capability constraints. Task constraints are related to the requirements of the task, such as integrity constraints, temporal sequence constraints, and time-window constraints.

Collaborative constraints: A UAV can perform only one task for a single target at any given moment.

$$\sum _{i=1}^{N_V}{x}_{ik}=1,\forall k\in N_M$$

(2)

Range constraints: The flight distance of the UAV must not exceed its maximum range.

$$\left(max\left\{x_{ik}{t}_k^e\right\}-\sum _{k=1}^{N_M}{x}_{ik}{t}_k^{do}\right)v_i\le D_i^U,\forall i\in N_V$$

(3)

where $max\left\{x_{ik}{t}_k^e\right\}$ denotes the completion time of the last task assigned to the UAV i, ${\sum }_{k=1}^{N_M}{x}_{ik}{t}_k^{do}$ represents the total time taken by the UAV i to complete all its tasks, and $\left(max\left\{x_{ik}{t}_k^e\right\}-{\sum }_{k=1}^{N_M}{x}_{ik}{t}_k^{do}\right)$ corresponds to the actual flight duration of the UAV i. The actual flight distance of the UAV i is then calculated as $\left(max\left\{x_{ik}{t}_k^e\right\}-{\sum }_{k=1}^{N_M}{x}_{ik}{t}_k^{do}\right)v_i$.

Supplies-delivery-capability constraints: The amount of supplies delivered by each UAV shall not exceed its onboard resources.

$$\sum _{k=1}^{N_M}{x}_{ik}{R}_k\le L_i,\forall i\in N_V$$

(4)

Integrity constraints: Each task on every target must be executed and only executed once.

$$\sum _{i=1}^{N_V}\sum _{k=1}^{N_M}{x}_{ik}=N_M$$

(5)

Temporal sequence constraints: Tasks on each target must be executed in the sequence of reconnaissance, supplies delivery, and assessment.

$$\left\{\begin{array}{c}{t}_{reconnaissance}^e\le t_{suppliesdelivery}^s\\ t_{suppliesdelivery}^e\le t_{assessment}^s\end{array}\right.$$

(6)

Time-window constraints: Some tasks must be executed within specific time windows, which restrict the earliest start time and latest end time of the tasks.

$$E{T}_k\le t_k^s<t_k^e\le L{T}_k,\forall k\in N_M$$

(7)

2.4. Optimization Objectives

In the task-allocation model, three optimization objectives are defined to evaluate various allocation schemes: task rewards, task execution costs, and task execution time.

Task rewards: Task rewards refer to the value obtained upon successful execution of a task. Rewards are categorized into three types based on task types: reconnaissance rewards, supplies-delivery rewards, and assessment rewards. Reconnaissance rewards refer to the amount of information acquired by the UAV about the target. The purpose of reconnaissance is to verify the status of the target and environmental information, thereby reducing uncertainty. The supplies-delivery reward is associated with the accurate transportation of required supplies to designated locations, ensuring effective resource replenishment. The assessment reward pertains to the evaluation of task effectiveness after supplies delivery, examining both delivery accuracy and execution performance. These three types of rewards are associated with task value, UAV task capability parameters, and the probability of task failure. To minimize all optimization objectives, the reward optimization function was set as the loss value of the objective, with $MR$ representing the maximum value of the objective. A lower reward objective value indicates better task completion.

$$F_{reward}=MR-\sum _{i=1}^{N_V}\sum _{k=1}^{N_M}\left(1-\prod _{i=1}^{N_V}\left(1-x_{ik}{P}_{ik}\left(1-S_k\right)\right)\right)V_k^M$$

(8)

In the equation, $\left(1-{\prod }_{i=1}^{N_V}\left(1-x_{ik}{P}_{ik}\left(1-S_k\right)\right)\right)$ represents the probability of successfully performing task k.

Task execution costs: Costs incurred by UAVs for reconnaissance, supplies delivery, and assessment tasks are defined as task execution costs. Task execution costs are related to the UAVs’ value and the probabilities of task failure.

$$F_{cost}=\sum _{i=1}^{N_V}\sum _{k=1}^{N_M}{x}_{ik}{S}_k{V}_i^U$$

(9)

Task execution time: The task execution time is defined as the total time required for the UAV swarm to complete all tasks, that is, the latest task completion time in the UAV task timetable.

$$F_{time}=max\left\{x_{ik}{t}_k^e\right\},\forall \left(i,k\right)\in \left(N_V,N_M\right)$$

(10)

3. Method for Multi-Objective Optimization

This section first presents a concise overview of both ALO and multi-objective ant lion optimizer (MOALO) and then elaborates on the improvement strategies implemented in MMSALO.

3.1. ALO

ALO mimics the process of antlions preying on ants in nature. The main steps of the algorithm include the random walking of ants, trap setting by antlions, ants falling into the traps, antlions preying on ants, and trap reconstruction by antlions.

The random walk of ants is demonstrated as follows:

$$\mathit{X}\left(t\right)=\left[0,cs\left(r\left(t_1\right)\right),cs\left(r\left(t_2\right)\right),\dots ,cs\left(r\left(t_T\right)\right)\right]$$

(11)

where t denotes the current iteration number, T represents the maximum number of iterations, $\mathit{X}\left(t\right)$ indicates the position of the random walk, $cs$ represents the cumulative sum of the random walk step lengths, and $r\left(t\right)$ takes the value of 1 or −1 with equal probability.

During the iterative process, antlions set traps in the search space to prey on ants, and the random walk of ants is influenced by these traps.

$${\mathit{c}}_i^t={\mathit{Antlion}}_j^t+{\mathit{c}}^t$$

(12)

$${\mathit{d}}_i^t={\mathit{Antlion}}_j^t+{\mathit{d}}^t$$

(13)

where ${\mathit{c}}^t$ denotes the lower bounds of the ant colony’s individual values across all dimensions at the t-th iteration, ${\mathit{d}}^t$ represents the upper bounds of the ant colony’s individual values across all dimensions at the t-th iteration, and ${\mathit{Antlion}}_j^t$ indicates the position of the j-th antlion at the t-th iteration.

After constructing pits, antlions preys on ants that wander randomly. Once an ant enters an antlion’s pit, the antlion will eject sand towards the center of the pit to reduce the ant’s range of movement. The specific process is as follows:

$${\mathit{c}}^t={\displaystyle \frac{\mathit{c}}{I}}$$

(14)

$${\mathit{d}}^t={\displaystyle \frac{\mathit{d}}{I}}$$

(15)

$$I={10}^w\times {\displaystyle \frac{t}{T}}$$

(16)

where $\mathit{c}$ represents the lower bounds of the values of each dimension for individual ants, and $\mathit{d}$ represents the upper bounds of the values of each dimension for individual ants. The value of w is as follows:

$$w=\left\{\begin{array}{cc}0,& t\le 0.1T\\ 2,& 0.1T<t\le 0.5T\\ 3,& 0.5T<t\le 0.75T\\ 4,& 0.75T<t\le 0.9T\\ 5,& 0.9T<t\le 0.95T\\ 6,& t>0.95T\end{array}\right.$$

(17)

Antlions that are better adapted to the environment possess stronger predatory capabilities. To simulate this scenario, ALO retains the antlion with the optimal fitness obtained in each iteration as the elite antlion. Each ant then performs a random walk around the elite antlion. Additionally, each ant selects another antlion based on its fitness. This selection is done using the roulette-wheel method. Once selected, the ant performs a random walk around this chosen antlion.

After the ants update their positions through random walks, they are considered to be preyed upon by the selected antlion. The position update is described as follows:

$${\mathit{Ant}}_i^t={\displaystyle \frac{{\mathit{R}}_e^t+{\mathit{R}}_r^t}{2}}$$

(18)

where ${\mathit{R}}_e^t$ represents the vector obtained by the ant performing random walks around the elite antlion during the t-th iteration, while ${\mathit{R}}_r^t$ denotes the vector obtained by the ant performing random walks around the roulette-wheel-selected antlion during the t-th iteration.

When an ant slides into the bottom of a pit and is captured by an antlion, the antlion assesses whether this location is more suitable for capturing ants than its original position. If it is deemed more advantageous, the antlion relocates to this position and reconstructs its pit. The specific process is as follows:

$${\mathit{Antlion}}_j^t={\mathit{Ant}}_i^t,f\left({\mathit{Ant}}_i^t\right)>f\left({\mathit{Antlion}}_j^t\right)$$

(19)

where ${\mathit{Ant}}_i^t$ denotes the position of the i-th ant at the t-th iteration.

At the end of the iteration, the antlion population is updated. After which, the antlion with the optimal fitness value is designated as the elite.

3.2. MOALO

Pareto optimality is a core concept in multi-objective optimization. In multi-objective optimization, given two solutions $\mathit{x}$ and $\mathit{y}$ in the decision space, $\mathit{x}$ is said to dominate $\mathit{y}$ if $\mathit{x}$ is superior to $\mathit{y}$ in at least one objective and not inferior to $\mathit{y}$ in the remaining objectives. A solution $\mathit{x}$ is defined as a non-dominated solution or a Pareto optimal solution when no other solution in the decision space dominates it. The set of all Pareto optimal solutions in the decision space is the Pareto set (PS), and the corresponding set of objective vectors is the Pareto front (PF).

The specific algorithm for maintaining the PS is presented in Algorithm 1. In each iteration, after updating all individuals in the population, the algorithm merges the current population with the PS. Subsequently, it employs the Pareto dominance strategy to select non-dominated solutions, which are then stored in the PS.   

Algorithm 1 PS Maintenance algorithm

Input:  Original Pareto set $PS$; Population P Output:  New Pareto set $P{S}_{new}$ 1: Combine $PS$ and P into $P{S}_{tmp}$; 2: for $\mathit{x}$ in $PS$ do 3:     if $\mathit{x}$ is not dominated by any solution in $P{S}_{tmp}$ then 4:         Designate $\mathit{x}$ as a non-dominated solution; 5:     end if 6: end for 7: Preserve all non-dominated solutions in $P{S}_{new}$;

   MOALO [28] incorporates the concept of sparsity to determine the density of the location occupied by a non-dominated solution. The specific algorithm for calculating sparsity is presented in Algorithm 2, where $distance(\mathit{x},\mathit{y})$ denotes the Euclidean distance between $\mathit{x}$ and $\mathit{y}$. In brief, the sparsity of a non-dominated solution is defined as the number of other non-dominated solutions within its local neighborhood.

Algorithm 2 Sparsity calculation

Input:  Pareto set $PS$; The number of optimization objectives $Obj$ Output:  The sparsity S of all solutions in $PS$ 1: Calculate the neighborhood radius r for all objectives; 2: S is initialized to 0; 3: for $\mathit{x}$ in $PS$ do 4:     for $\mathit{y}$ in $PS$ do 5:         if $distance(\mathit{x},\mathit{y})<r$ in all objectives then 6:            $S\left[\mathit{x}\right]=S\left[\mathit{x}\right]+1$; 7:         end if 8:     end for 9: end for

After obtaining the sparsity of all solutions in PS, MOALO employs a sparsity-based roulette wheel to select solutions from the PS as roulette-wheel antlions. The specific algorithm for the sparsity-based roulette-wheel selection is presented in Algorithm 3.

Algorithm 3 Sparsity roulette algorithm

Input:  The sparsity S of all solutions in $PS$ Output:  The antlion index selected by the roulette wheel based on sparsity   1: The reciprocal of the sparsity is calculated and stored in X;   2: for x in X do   3:     The cumulative sum from $X\left[0\right]$ to x is calculated and stored in W;   4: end for   5: The sum of the reciprocals of all sparsities is calculated and then multiplied by a random number between 0 and 1 to obtain the selection probability p;   6: for w in W do   7:     if $w>p$ then   8:         The index value of w is selected as the antlion index;   9:         Terminate the loop; 10:     end if 11: end for

   When the number of non-dominated solutions in PS exceeds its maximum capacity, MOALO also employs the sparsity-based roulette-wheel selection to remove the surplus solutions. The method follows the same procedure as Algorithm 3, except that the data stored in X are replaced by the sparsity itself.

3.3. MMSALO

3.3.1. Sparsity Selection Mechanism

The conventional MOALO employs a sparsity-based roulette wheel to maintain the Pareto set and to select ant lions, thereby injecting stochasticity into the search. However, this roulette mechanism occasionally retains inferior solutions, which hinders global exploration. Consequently, the present study introduces a deterministic selection to replace the original roulette wheel. This method is guided strictly by sparsity magnitude, which significantly enhances the algorithm’s global search capability. The modified PS maintenance algorithm and antlion selection algorithm are presented in Algorithm 4 and Algorithm 5, respectively.   

Algorithm 4 Modified PS maintenance algorithm

Input:  Original Pareto set $PS$; Population P; $PS$ capacity c; The sparsity S of all solutions in $PS$ Output:  New Pareto set $P{S}_{new}$   1: Combine $PS$ and P into $P{S}_{tmp}$;   2: for $\mathit{x}$ in $PS$ do   3:     if $\mathit{x}$ is not dominated by any solution in $P{S}_{tmp}$ then   4:         Designate $\mathit{x}$ as a non-dominated solution;   5:     end if   6: end for   7: Preserve all non-dominated solutions in $P{S}_{new}$;   8: while the number of solutions in $P{S}_{new}$ > c do   9:     Delete the solution with the highest sparsity; 10:     Delete the corresponding sparsity from S; 11: end while

Algorithm 5 Sparsity selection algorithm

Input:  The sparsity S of all solutions in $PS$ Output:  The antlion index selected based on sparsity 1: Find the minimum value s in S; 2: The index value of s is selected as the antlion index;

3.3.2. Random Boundary Strategy

In the classical MOALO, the value of I decreases with increasing iteration counts, exhibiting a piecewise linear trend. Moreover, in each iteration, all ants traverse the same boundaries, which reduces the diversity of the algorithm and is unfavorable for locating the global optimum. To address this issue, this study proposes a stochastic boundary strategy that modifies the calculation of I as follows to enhance the diversity of ants’ movement around antlions:

$$I={10}^w{\displaystyle \frac{t}{T}}\left(1+sin\left({\displaystyle \frac{\pi \left(2t-T\right)}{6T}}\right)·rand\right)$$

(20)

where $rand$ denotes a random number between 0 and 1. Because $sin\left({\displaystyle \frac{\pi \left(2t-T\right)}{6T}}\right)$ exhibits a nonlinear increasing trend within the range of $[-0.5,0.5]$ as t increases, $\left(1+sin\left({\displaystyle \frac{\pi \left(2t-T\right)}{6T}}\right)·rand\right)$ displays a stochastic nonlinear increasing trend within the range of $[0.5,1.5]$. This introduces a degree of randomness into the variation process of I.

Overall, the value of I exhibits a nonlinear increasing trend with a certain degree of randomness as the number of evolutionary generations increases. Consequently, the boundary size decreases nonlinearly with a certain degree of randomness as the number of evolutionary generations progresses. This phenomenon enhances the randomness and diversity of the ants’ movement around antlions, thereby improving the diversity of the ant population to some extent.

3.3.3. Adaptive Position Update Strategy

The standard MOALO employs the method of calculating the average values of two types of random walks, ${\mathit{R}}_e^t$ and ${\mathit{R}}_r^t$, to balance exploration and exploitation capabilities. However, it overlooks the issue that the proportions of exploration and exploitation should vary at different stages. In this study, we improved the proportionality coefficients of ${\mathit{R}}_e^t$ and ${\mathit{R}}_r^t$, allowing the two types of walks to have different weights at various stages. In the early stages of the algorithm, the random walk around the sparsity-selected antlion is dominant, while in the later stages, the random walk around the elite antlion prevails. The improved position update strategy is described as follows:

$${\mathit{Ant}}_i^t={\displaystyle \frac{\left(1+\frac{t}{T}·rand\right){\mathit{R}}_e^t+\left(1-\frac{t}{T}·rand\right){\mathit{R}}_r^t}{2}}$$

(21)

Through these improvements, the ants primarily wandered around the sparsity-selected antlion in the early stages. At the same time, the algorithm gradually enhanced the directional influence of the elite antlion. In the later stages, the ants mainly wandered around the elite antlion, with a certain degree of randomness retained. This approach effectively enhanced the exploration capability of the algorithm in the early stages and the exploitation capability in the later stages.

3.3.4. Preference-Based Elite Selection Mechanism

As shown in Figure 2, MMSALO divides the original population into multiple subpopulations based on the number of optimization objectives. Each subpopulation focuses on a single optimization objective and selects the solution with the best performance on that objective as its elite antlion. During the iterative process, each ant first selects an antlion using a sparsity selection mechanism. Then, it performs random walks around two key targets: the elite antlion of its own subpopulation and the sparsity-selected antlion. This process ultimately generates new individual solutions.    

By using a preference-based elite selection mechanism, each ant iteratively updates its search around the best individual for a distinct objective. This enhances the algorithm’s optimization capability and produces a more widely distributed set of solutions.

3.3.5. Complete Algorithm Framework

The MMSALO procedure is illustrated in Figure 3. The algorithm first initializes the ant population. The fitness of each ant is then calculated, and Pareto solutions are selected from the population based on Pareto dominance relationships and preserved in the PS. The sparsity of the antlions is subsequently computed, and elite antlions are selected for each optimization objective according to a preference-based elite selection mechanism. During the iterative process, each initialized ant performs a random walk around the elite antlion of its own population and an antlion selected by sparsity, employing a random boundary strategy. New ants are generated using an adaptive position update strategy. The fitness of the new ant population is calculated, and these ants are incorporated into the archive, followed by an update of the archive and elite antlions. Finally, upon reaching the maximum number of iterations, the task-allocation scheme constructed by the antlions is output.    

An analysis of the time complexity of MMSALO is presented as follows. Let D denote the dimensionality of the decision variables, N the population size, and T the maximum number of iterations. The time complexity of the ant random walk iteration is $O(N\times D\times T)$, while the time complexity of computing the dominance relations and sparsity of solutions is $O(N^2\times T)$. In summary, the overall time complexity of MMSALO is $O(N\times T\times (N+D\left)\right)$.

4. Simulations and Results

Two experiments were designed to verify the effectiveness of the proposed MMSALO in solving the multi-task and multi-constraint task-allocation problem for UAV swarm. In experiment 1, the DTLZ test function suite and MOKP were employed to evaluate the distribution and convergence performance of MMSALO on both continuous and discrete problems, thereby validating its capability for solving multi-objective optimization problems. Experiment 2 employed a battlefield model to create a simulation environment and solved the multi-task and multi-constraint task-allocation model to verify the algorithm’s effectiveness. In the comparison, four algorithms were employed: MOALO, improved multi-objective particle swarm optimization (IMOPSO) [29], improved non-dominated sorting genetic algorithm-II (INSGA-II) [30], and improved non-dominated sorting genetic algorithm-III (INSGA-III) [31]. These algorithms are specifically devised for the task-allocation problem of UAV swarms. IMOPSO balances diversity and convergence by adaptive angular region partitioning and a dedicated reuse rule for infeasible solutions, which effectively mitigates premature convergence. INSGA-II reshapes the non-dominated sorting criterion through a “constraint-tolerance” mechanism, significantly improving both convergence and feasibility under intricate constraints. INSGA-III introduces an enhanced reference-point association strategy to promote uniform solution distribution and employs a dynamic elite-selection scheme to balance exploration and exploitation, enabling rapid and evenly distributed convergence to optimal solutions in high-dimensional objective spaces with complex constraints. The detailed parameters of these algorithms are listed in

Table 3. Basic parameter for algorithms.

Algorithm	Parameter Setting
MOALO	$S_e=0.5,S_r=0.5$
IMOPSO	$w=0.4,c_1=1,c_2=1$
INSGA-II	$P_c=0.7,P_m=0.3$
INSGA-III	$P_c=0.7,P_m=0.3$

. Among these parameters, $S_e$ and $S_r$ denote the weighting coefficients for the ant’s random walk around the elite antlion and the roulette-wheel-selected antlion, respectively; w denotes the weight vector; $P_c$ and $P_m$ represent the crossover probability and mutation probability, respectively. Both experiments were conducted on a computer equipped with an Intel i7-12650H CPU (2.30 GHz) and 16.0 GB of memory.

The performance of all algorithms was evaluated using hyper volume (HV) [32]. The formulas for calculating HV is shown below:

$$HV=volume(\bigcup _{v\in P}v)$$

(22)

where P represents the set of non-dominated solutions within the target vector set calculated by the algorithm, while volume refers to the hypervolume calculated from the vector to the reference point.

4.1. Simulation on the Test Functions

In the experiment, the test functions were employed to evaluate the performance of MMSALO in solving multi-objective optimization problems. Given that the problem model in this study was a three-objective model, the number of objective functions, M, was set to 3.

The algorithms were assessed on DTLZ1 and DTLZ3, while $|{\mathit{x}}_M|$ was set to 2 and 8. Because the dimensionality of the decision variables is $n=M+|{\mathit{x}}_M|-1$, the resulting problem dimensions are 4 and 10. The analytical expressions of the three-objective DTLZ1 and DTLZ3 are listed in

Table 4. Expressions for the three-objective DTLZ1 and DTLZ3.

Instance	Search Space	Objectives, PS and PF
DTLZ1	${[0,1]}^n$	$f_1\left(\mathit{x}\right)={\displaystyle \frac{1}{2}}{x}_1{x}_2(1+g\left({\mathit{x}}_M\right))$
		$f_2\left(\mathit{x}\right)={\displaystyle \frac{1}{2}}{x}_1(1-x_2)(1+g\left({\mathit{x}}_M\right))$
		$f_3\left(\mathit{x}\right)={\displaystyle \frac{1}{2}}(1-x_1)(1+g\left({\mathit{x}}_M\right))$
		$g\left({\mathit{x}}_M\right)=100\left(\right|{\mathit{x}}_M|+{\sum }_{x_i\in {\mathit{x}}_M}({(x_i-0.5)}^2-cos\left(20\pi (x_i-0.5)\right)))$
		PS: $0\le x_1,x_2\le 1,x_i=0.5,$for$i=3,\dots ,n$
		PF: $f_1={\displaystyle \frac{1}{2}}uv,f_2={\displaystyle \frac{1}{2}}u(1-v),f_3={\displaystyle \frac{1}{2}}(1-u),0\le u,v\le 1$
DTLZ3	${[0,1]}^n$	$f_1\left(\mathit{x}\right)=cos\left({\displaystyle \frac{\pi }{2}}{x}_1\right)cos\left({\displaystyle \frac{\pi }{2}}{x}_2\right)(1+g\left({\mathit{x}}_M\right))$
		$f_2\left(\mathit{x}\right)=cos\left({\displaystyle \frac{\pi }{2}}{x}_1\right)sin\left({\displaystyle \frac{\pi }{2}}{x}_2\right)(1+g\left({\mathit{x}}_M\right))$
		$f_3\left(\mathit{x}\right)=sin\left({\displaystyle \frac{\pi }{2}}{x}_1\right)(1+g\left({\mathit{x}}_M\right))$
		$g\left({\mathit{x}}_M\right)=100\left(\right|{\mathit{x}}_M|+{\sum }_{x_i\in {\mathit{x}}_M}({(x_i-0.5)}^2-cos\left(20\pi (x_i-0.5)\right)))$
		PS: $0\le x_1,x_2\le 1,x_i=0.5,$for$i=3,\dots ,n$
		PF: $f_1=cos\left(u\right)cos\left(v\right),f_2=cos\left(u\right)sin\left(v\right),f_3=sin\left(u\right),0\le u,v\le {\displaystyle \frac{\pi }{2}}$

.

   Given a set of n items and a set of m knapsacks, MOKP can be stated as Equation (23):

$$\left\{\begin{array}{cc}\mathrm{maximize}& f_i\left(x\right)=\sum _{j=1}^n{p}_{ij}{x}_j,i=1,\dots ,m\\ \mathrm{subject}\mathrm{to}& \sum _{j=1}^n{w}_{ij}{x}_j⩽c_i,i=1,\dots ,m\\ & x={(x_1,\dots ,x_n)}^T\in {\left\{0,1\right\}}^n\end{array}\right.$$

(23)

where $p_{ij}\ge 0$ is the profit of item j in knapsack i, $w_{ij}\ge 0$ is the weight of item j in knapsack i, and $c_i$ is the capacity of knapsack i. $x_j=1$ means that item j is selected and put in all the knapsacks. Since this study addressed a three-objective optimization problem, the parameter m was set to 3 to evaluate the performance of all algorithms when n was set to 200 and 800, respectively.

Each algorithm-MMSALO, MOALO, IMOPSO, INSGA-II, and INSGA-III-was independently executed 20 times with a population size of 100 and a PS size of 100. For DTLZ1 and DTLZ3, the maximum number of iterations was set to 1000, and the reference point was defined as (5, 5, 5). For MOKP, the maximum number of iterations was set to 100, with a reference point at (1, 1, 1).

The comparison of HV results obtained by five algorithms on DTLZ1, DTLZ3, and MOKP is presented in

Table 5. Hyper volume index of algorithms on the test functions.

Test Function	n	HV	MMSALO	MOALO	IMOPSO	INSGA-II	INSGA-III
DTLZ1	4	Avg.	$1.24\times {10}^2$	$1.23\times {10}^2$	$8.42\times {10}^1$	$1.20\times {10}^2$	$\mathbf{1}.\mathbf{24}\times {\mathbf{10}}^{\mathbf{2}}$
		Std.	$8.48\times {10}^{-1}$	$6.02\times {10}^{-1}$	$2.43\times {10}^1$	$6.19\times {10}^0$	$9.39\times {10}^{-1}$
	10	Avg.	$\mathbf{3}.\mathbf{85}\times {\mathbf{10}}^{\mathbf{1}}$	$1.92\times {10}^1$	$0.00\times {10}^0$	$3.16\times {10}^1$	$1.71\times {10}^1$
		Std.	$3.13\times {10}^1$	$3.44\times {10}^1$	$0.00\times {10}^0$	$2.78\times {10}^1$	$2.23\times {10}^1$
DTLZ3	4	Avg.	$\mathbf{1}.\mathbf{14}\times {\mathbf{10}}^{\mathbf{2}}$	$1.10\times {10}^2$	$4.62\times {10}^1$	$9.59\times {10}^1$	$9.05\times {10}^1$
		Std.	$4.81\times {10}^0$	$4.39\times {10}^0$	$3.06\times {10}^1$	$3.32\times {10}^1$	$3.64\times {10}^1$
	10	Avg.	$\mathbf{1}.\mathbf{82}\times {\mathbf{10}}^{\mathbf{0}}$	$0.00\times {10}^0$	$0.00\times {10}^0$	$1.93\times {10}^{-1}$	$6.24\times {10}^{-1}$
		Std.	$7.63\times {10}^0$	$0.00\times {10}^0$	$0.00\times {10}^0$	$8.43\times {10}^{-1}$	$2.23\times {10}^0$
MOKP	200	Avg.	$9.99502\times {10}^{-1}$	$9.99499\times {10}^{-1}$	$\mathbf{9}.\mathbf{99532}\times {\mathbf{10}}^{-\mathbf{1}}$	$9.99485\times {10}^{-1}$	$9.99481\times {10}^{-1}$
		Std.	$8.68\times {10}^{-6}$	$8.54\times {10}^{-6}$	$7.58\times {10}^{-6}$	$8.59\times {10}^{-6}$	$7.15\times {10}^{-6}$
	800	Avg.	$9.99869\times {10}^{-1}$	$9.99869\times {10}^{-1}$	$\mathbf{9}.\mathbf{99874}\times {\mathbf{10}}^{-\mathbf{1}}$	$9.99867\times {10}^{-1}$	$9.99866\times {10}^{-1}$
		Std.	$1.08\times {10}^{-6}$	$1.46\times {10}^{-6}$	$1.32\times {10}^{-6}$	$9.27\times {10}^{-7}$	$1.19\times {10}^{-6}$

Bold denotes the optimal result.

and Figure 4. In the comparison of the mean values of the HV results, it can be observed that among the six test functions, MMSALO outperforms the other four algorithms on DTLZ1 (n = 10) and DTLZ3 and achieves the second-best performance on DTLZ1 (n = 4) and MOKP. This demonstrates the overall superiority of MMSALO on these test functions. Compared to MOALO, MMSALO yields better results across all test functions, indicating that the proposed improvements enhance the algorithm’s exploitation capability and global search ability, thereby ensuring solution diversity. Furthermore, in the comparison of the standard deviations of the HV results, it can be seen that the standard deviations of MMSALO and MOALO are comparable, with each performing better on certain test functions. This confirms that the proposed modifications do not compromise the stability of the algorithm.

Table 6. Wilcoxon rank-sum test analysis of HV results on the test functions.

Test Function	n	MOALO	IMOPSO	INSGA-II	INSGA-III
DTLZ1	4	$1.85\times {10}^{-1}$	$\mathbf{6}.\mathbf{30}\times {\mathbf{10}}^{-\mathbf{8}}$	$1.17\times {10}^{-1}$	$2.04\times {10}^{-1}$
	10	$\mathbf{1}.\mathbf{49}\times {\mathbf{10}}^{-\mathbf{2}}$	$\mathbf{4}.\mathbf{26}\times {\mathbf{10}}^{-\mathbf{6}}$	$5.89\times {10}^{-1}$	$\mathbf{2}.\mathbf{56}\times {\mathbf{10}}^{-\mathbf{2}}$
DTLZ3	4	$\mathbf{3}.\mathbf{26}\times {\mathbf{10}}^{-\mathbf{2}}$	$\mathbf{6}.\mathbf{30}\times {\mathbf{10}}^{-\mathbf{8}}$	$\mathbf{3}.\mathbf{49}\times {\mathbf{10}}^{-\mathbf{2}}$	$\mathbf{2}.\mathbf{00}\times {\mathbf{10}}^{-\mathbf{2}}$
	10	$5.89\times {10}^{-1}$	$5.89\times {10}^{-1}$	$7.87\times {10}^{-1}$	$1.00\times {10}^0$
MOKP	200	$4.33\times {10}^{-1}$	$\mathbf{1}.\mathbf{33}\times {\mathbf{10}}^{-\mathbf{7}}$	$\mathbf{3}.\mathbf{74}\times {\mathbf{10}}^{-\mathbf{6}}$	$\mathbf{6}.\mathbf{01}\times {\mathbf{10}}^{-\mathbf{7}}$
	800	$6.75\times {10}^{-1}$	$\mathbf{6}.\mathbf{80}\times {\mathbf{10}}^{-\mathbf{8}}$	$\mathbf{2}.\mathbf{87}\times {\mathbf{10}}^{-\mathbf{6}}$	$\mathbf{6}.\mathbf{45}\times {\mathbf{10}}^{-\mathbf{7}}$

Bold denotes statistical significance.

reports the p-values obtained from the Wilcoxon rank-sum test on HV. At $\alpha$ = 0.05, most p-values for the two MMSALO-optimal and two MMSALO-second-best metrics were significantly lower than $\alpha$, demonstrating the statistically significant superiority of MMSALO over the comparison algorithms.

Table 7. Runtime performance comparison on the test functions.

Test Function	n	MMSALO	MOALO	IMOPSO	INSGA-II	INSGA-III
DTLZ1	4	$6.23\times {10}^1$	$2.68\times {10}^1$	$1.00\times {10}^1$	$\mathbf{1}.\mathbf{20}\times {\mathbf{10}}^{\mathbf{0}}$	$4.86\times {10}^0$
	10	$9.08\times {10}^1$	$7.06\times {10}^1$	$9.62\times {10}^0$	$\mathbf{1}.\mathbf{01}\times {\mathbf{10}}^{\mathbf{0}}$	$3.84\times {10}^0$
DTLZ3	4	$6.73\times {10}^1$	$5.18\times {10}^1$	$8.71\times {10}^0$	$\mathbf{1}.\mathbf{16}\times {\mathbf{10}}^{\mathbf{0}}$	$4.06\times {10}^0$
	10	$1.31\times {10}^2$	$9.12\times {10}^1$	$1.38\times {10}^1$	$\mathbf{1}.\mathbf{09}\times {\mathbf{10}}^{\mathbf{0}}$	$3.84\times {10}^0$
MOKP	200	$3.63\times {10}^1$	$4.12\times {10}^1$	$2.06\times {10}^0$	$\mathbf{5}.\mathbf{97}\times {\mathbf{10}}^{-\mathbf{1}}$	$8.17\times {10}^{-1}$
	800	$1.52\times {10}^2$	$1.32\times {10}^2$	$3.41\times {10}^0$	$\mathbf{1}.\mathbf{94}\times {\mathbf{10}}^{\mathbf{0}}$	$2.10\times {10}^0$

Bold denotes the shortest runtime.

presents the average time required for all algorithms to run the test functions once. It can be observed that the computation times of MMSALO and MOALO are within the same order of magnitude. Compared to MOALO, MMSALO requires a longer computation time because its PS reaches the capacity limit more rapidly, owing to the sparsity selection mechanism, thus necessitating additional PS update operations. However, on the MOKP (n = 200) instance, MMSALO exhibits a shorter runtime than does MOALO. This occurs because the test problem is relatively simple, leading both algorithms to approach the capacity limit almost simultaneously. Since MMSALO employs a direct sparsity selection mechanism while MOALO utilizes a sparsity-based roulette wheel selection, MMSALO achieves faster updates once the PS capacity is reached, resulting in a shorter overall computation time for MMSALO in this scenario.

Table 8. Results from ablation studies on the test functions.

Test Function	n	MMSALO	SS	RB	APU	PBES	MOALO
DTLZ1	4	$1.2359\times {10}^2$	$1.2331\times {10}^2$	$1.2358\times {10}^2$	$1.2329\times {10}^2$	$1.2342\times {10}^2$	$1.2326\times {10}^2$
Improvement		1.0027	1.0004	1.0025	1.0002	1.0013	1.0000
Runtime		$6.2275\times {10}^1$	$6.0715\times {10}^1$	$2.6168\times {10}^1$	$2.5005\times {10}^1$	$2.5745\times {10}^1$	$2.6802\times {10}^1$
DTLZ3	4	$1.1412\times {10}^2$	$1.1145\times {10}^2$	$1.1084\times {10}^2$	$1.1133\times {10}^2$	$1.1072\times {10}^2$	$1.1048\times {10}^2$
Improvement		1.0329	1.0088	1.0032	1.0077	1.0022	1.0000
Runtime		$6.7328\times {10}^1$	$6.9311\times {10}^1$	$2.4247\times {10}^1$	$2.2199\times {10}^1$	$2.3500\times {10}^1$	$5.1819\times {10}^1$

presents the HV and runtime results of the ablation studies conducted for MMSALO on the DTLZ test suite. In the table, “SS” denotes the sparsity selection mechanism component, “RB” represents the random boundary strategy component, “APU” indicates the adaptive position update strategy component, and “PBES” refers to the preference-based elite selection mechanism component. A comparative analysis between MMSALO and MOALO on MOKP is omitted here due to the negligible differences observed in their HV results. As can be observed, the performance contributions of the different components are relatively balanced, with each component demonstrating distinct advantages across different problems. Collectively, these components contribute to the enhanced overall performance of MMSALO.

4.2. Simulation on the Task Allocation Model

To address the intricate and multiple constraints introduced by the task-allocation model, we introduce a double-layer encoding scheme and an adaptive penalty strategy.

4.2.1. Double-Layer Encoding

The present study employs a real-number encoding method, representing individual positions as real-valued vectors. These vectors are then mapped to the discrete solution space of task allocation through a specific approach. This method allows individuals to search in the continuous domain space and converts the continuous encoding into practical task-allocation schemes through a particular decoding process.

The task-allocation scheme is represented by a task-allocation vector, which comprises two rows. The length of each row corresponds to the number of targets multiplied by the number of task types. Each element in the first row corresponds to a task assigned to a target and is encoded as a real number. The integer part of this value indicates the UAV assigned to execute the task, while the fractional part denotes the task’s priority, with a smaller value representing higher priority. The elements in the second row are also real numbers and are used to determine whether the corresponding task is reconnaissance, supplies delivery, or assessment. For tasks associated with the same target, they are classified as reconnaissance, supplies delivery, and assessment in ascending order of the values in the second row. Figure 5 illustrates the mapping between the algorithm encoding and the task-allocation result in a scenario where two UAVs are assigned to two targets. Specifically, given the task-allocation result [[1.2837, 2.8449, 2.5364, 1.0482, 2.4619, 1.2984], [1.3283, 2.2581, 1.9564, 1.1012]], the following interpretation applies: tasks 1, 4, and 6 on target 2 are assigned to UAV 1, with task priorities of 0.2837, 0.0482, and 0.2984, respectively. The execution sequence is task 4, followed by task 1 and then task 6. Similarly, tasks assigned to UAV 2 can be derived accordingly. For target 1, the second-row values for tasks 1, 2, and 3 are 1.3283, 2.2581, and 1.9564, respectively. Sorting these values in ascending order yields the following task sequence: task 1 (reconnaissance), task 3 (supplies delivery), and task 2 (assessment). The same logic applies to target 2.

4.2.2. Adaptive Penalty Function

The introduction of appropriate constraint-handling methods can effectively enhance the search efficiency of algorithms for feasible solutions. These methods achieve a balance between convergence, diversity, and feasibility by weighting constraint violations and coordinating the exploration of feasible and infeasible regions. Common constraint-handling methods include penalty function methods, constraint dominance methods, and learning-based methods [33]. Each of these methods has its own focus and can leverage its strengths in different aspects. Penalty function methods handle constraints by adding penalty terms for constraint violations to the objective function, ensuring that the solution set meets the constraints. Constraint dominance methods prioritize feasible solutions by comparing the feasibility of solutions with their dominance relationships. Learning-based methods utilize machine learning techniques to predict and improve the search process of the algorithm. In this study, an adaptive penalty function method was adopted, which dynamically adjusts the penalty value based on the degree of constraint conflicts during the search process. This approach avoids the need for parameter tuning and enhances the convergence speed of the algorithm.

The adaptive penalty function method converts the original constrained optimization problem into an unconstrained one by incorporating a penalty term into the objective function. This penalty term adjusts the optimization target value of the model based on the degree of constraint violation of the solution. In this study, an adaptive penalty function was designed to calculate the total number of constraint violations for a given solution. The total number of violations was then multiplied by a fixed value to serve as the penalty term. The modified objective function incorporating the penalty term is as follows:

$$F=\left\{\begin{array}{cc}F\left(\mathit{x}\right),& \mathit{x}\in \mathit{X}\\ F\left(\mathit{x}\right)+rN,& \mathit{x}\notin \mathit{X}\end{array}\right.$$

(24)

where F denotes the objective function, $\mathit{x}$ represents the solution found by the algorithm during the optimization process, $\mathit{X}$ indicates the feasible region, r is the total number of constraint violations, and N is an adaptive value set here to the number of tasks. This allows the algorithm to increase the penalty for infeasible solutions in more complex problems, thereby driving the algorithm to find feasible solutions more rapidly.

Algorithm 6 shows the pseudocode of adaptive penalty function for calculating the total number of violations, which considers the UAV range constraint, UAV supplies-delivery capability constraint, task temporal sequence constraint, and task time window constraint.

Algorithm 6 Adaptive penalty function

Input:  Task allocation solution $\mathit{A}$ Output:  The total number of constraint violations r $r=0$; for i in $N_V$ do     Obtain the task list $\mathit{TL}$ of UAV i through $\mathit{A}$;     Calculate the flight distance d of UAV i using $\mathit{TL}$;     if $d>D_i^U$ then         $r=r+1$;     end if     Obtain the number of supplies-delivery tasks l executed by UAV i through $\mathit{TL}$;     if $l>L_i$ then         $r=r+1$;     end if end for for j in $N_T$ do     Obtain the task list $\mathit{TL}$ on target j through $\mathit{A}$;     if $t_{reconnaissance}^e>t_{suppliesdelivery}^s$ then         $r=r+1$;     end if     if $t_{suppliesdelivery}^e>t_{assessment}^s$ then         $r=r+1$;     end if     for task k in $\mathit{TL}$ do         if $E{T}_k>t_k^s$ then            $r=r+1$;         end if         if $t_k^e>L{T}_k$ then            $r=r+1$;         end if     end for end for

4.2.3. Task-Allocation Model Configuration and Experimental Results

In the simulation, two scenarios were established. The first scenario simulated the process of six UAVs performing tasks at 18 targets, while the second scenario simulated the process of eight UAVs performing tasks at 24 targets. The positions of the targets and UAVs were simplified as points. The total coverage area of the operational zone was 90,000 km², represented by the x-axis coordinates (0, 300) and y-axis coordinates (0, 300). To reflect realistic conditions, UAVs with lower flight speeds were assigned a greater maximum range. The task capability parameter of a UAV was positively correlated with its value, and the probability of task failure was positively correlated with its value. The reconnaissance task was required to be completed within the first 4000 s to ensure the UAV swarm’s grasp of global information. The supplies-delivery task was required to be completed within the first 8000 s to avoid the issue of reconnaissance information becoming obsolete due to excessive task intervals. Additionally, a minimum time interval of 300 s was required to be maintained between the assessment and supplies-delivery tasks to ensure that personnel at the target location have received the supplies. Reconnaissance and assessment tasks did not consume any onboard resources, whereas every supplies-delivery task consumed exactly one unit. The speed, task capability parameter, and UAV value varied among individuals. Limited by their range and onboard resources, the UAVs were restricted in the number of tasks they can perform.

Table 9. Details of UAVs (Scenario 1).

UAV	Position/km	Speed/(km/s)	Task Capability Parameter	Max Range/km	Value	Onboard Resources
1	(34.45, 34.43)	0.24	[0.90, 0.87, 0.79]	3200	0.85	15
2	(9.72, 50.85)	0.28	[0.77, 0.82, 0.75]	3000	0.78	13
3	(50.30, 95.56)	0.28	[0.78, 0.83, 0.87]	3000	0.83	15
4	(60.83, 46.54)	0.28	[0.80, 0.83, 0.75]	3000	0.79	13
5	(38.90, 7.09)	0.18	[0.76, 0.82, 0.81]	3400	0.80	10
6	(29.80, 1.13)	0.16	[0.87, 0.84, 0.83]	3400	0.84	13

and

Table 10. Details of targets and tasks (Scenario 1).

Target	Position/km	Task Values	Task Execution Time/s	Probability of Task Failure
1	(273.22, 131.11)	[0.76, 0.76, 0.84]	[16.79, 11.07, 20.94]	[0.38, 0.38, 0.42]
2	(136.10, 177.79)	[0.78, 0.76, 0.89]	[19.36, 21.43, 28.78]	[0.39, 0.38, 0.44]
3	(252.56, 224.05)	[0.83, 0.90, 0.76]	[27.06, 29.83, 13.00]	[0.41, 0.45, 0.38]
4	(245.53, 164.42)	[0.81, 0.79, 0.85]	[27.90, 12.99, 11.29]	[0.40, 0.39, 0.43]
5	(205.65, 290.38)	[0.78, 0.77, 0.84]	[25.76, 13.22, 27.95]	[0.39, 0.38, 0.42]
6	(175.38, 175.20)	[0.89, 0.90, 0.84]	[29.07, 27.94, 12.55]	[0.45, 0.45, 0.42]
7	(154.83, 245.09)	[0.81, 0.80, 0.85]	[15.41, 11.67, 20.64]	[0.41, 0.40, 0.43]
8	(256.92, 235.26)	[0.84, 0.80, 0.87]	[12.90, 11.02, 23.62]	[0.42, 0.40, 0.44]
9	(295.72, 171.83)	[0.84, 0.89, 0.75]	[11.85, 14.65, 17.31]	[0.42, 0.45, 0.38]
10	(218.45, 238.61)	[0.82, 0.76, 0.87]	[25.45, 12.67, 18.29]	[0.41, 0.38, 0.43]
11	(233.40, 230.03)	[0.85, 0.90, 0.84]	[19.09, 22.49, 18.54]	[0.43, 0.45, 0.42]
12	(185.68, 281.01)	[0.83, 0.79, 0.85]	[16.06, 19.26, 12.87]	[0.41, 0.39, 0.43]
13	(127.31, 128.02)	[0.83, 0.89, 0.84]	[24.77, 20.16, 16.84]	[0.42, 0.44, 0.42]
14	(192.93, 117.56)	[0.87, 0.86, 0.85]	[24.34, 16.50, 18.01]	[0.44, 0.43, 0.43]
15	(233.82, 281.93)	[0.88, 0.81, 0.76]	[24.68, 27.43, 28.66]	[0.44, 0.40, 0.38]
16	(154.15, 263.87)	[0.80, 0.77, 0.78]	[11.83, 10.10, 23.88]	[0.40, 0.39, 0.39]
17	(263.45, 158.86)	[0.85, 0.87, 0.77]	[12.50, 15.12, 13.35]	[0.43, 0.43, 0.38]
18	(264.43, 146.68)	[0.79, 0.77, 0.89]	[12.31, 19.44, 23.24]	[0.39, 0.38, 0.45]

list the detailed information of the UAVs, targets, and the tasks on these targets under Scenario 1.

Table 11. Details of UAVs (Scenario 2).

UAV	Position/km	Speed/(km/s)	Task Capability Parameter	Max Range/km	Value	Onboard Resources
1	(34.45, 34.43)	0.24	[0.81, 0.79, 0.79]	3200	0.80	24
2	(87.00, 36.35)	0.28	[0.79, 0.82, 0.83]	3000	0.81	14
3	(6.08, 19.43)	0.26	[0.90, 0.81, 0.77]	3000	0.83	13
4	(43.15, 51.61)	0.29	[0.88, 0.85, 0.80]	3000	0.84	17
5	(52.12, 1.67)	0.21	[0.78, 0.84, 0.76]	3200	0.79	13
6	(46.45, 41.52)	0.19	[0.89, 0.88, 0.87]	3400	0.88	12
7	(56.76, 50.16)	0.28	[0.87, 0.82, 0.75]	3000	0.81	14
8	(26.68, 82.16)	0.27	[0.83, 0.89, 0.81]	3000	0.84	19

and

Table 12. Details of targets and tasks (Scenario 2).

Target	Position/km	Task Values	Task Execution Time/s	Probability of Task Failure
1	(175.20, 154.83)	[0.87, 0.77, 0.88]	[23.53, 13.54, 12.03]	[0.43, 0.39, 0.44]
2	(245.09, 256.92)	[0.89, 0.88, 0.77]	[22.01, 29.15, 29.46]	[0.45, 0.44, 0.38]
3	(235.26, 295.72)	[0.79, 0.76, 0.83]	[21.78, 18.60, 17.09]	[0.40, 0.38, 0.41]
4	(171.83, 218.45)	[0.77, 0.76, 0.85]	[23.66, 21.47, 17.15]	[0.39, 0.38, 0.43]
5	(238.61, 233.40)	[0.76, 0.78, 0.80]	[26.23, 21.64, 29.05]	[0.38, 0.39, 0.40]
6	(230.03, 185.68)	[0.87, 0.77, 0.81]	[10.30, 19.37, 10.98]	[0.43, 0.39, 0.41]
7	(281.01, 127.31)	[0.82, 0.84, 0.81]	[25.36, 23.86, 29.54]	[0.41, 0.42, 0.41]
8	(128.02, 192.93)	[0.80, 0.82, 0.77]	[22.64, 20.55, 14.92]	[0.40, 0.41, 0.39]
9	(117.56, 233.82)	[0.86, 0.83, 0.80]	[23.37, 21.02, 28.38]	[0.43, 0.41, 0.40]
10	(281.93, 154.15)	[0.86, 0.80, 0.81]	[22.46, 26.24, 24.51]	[0.43, 0.40, 0.41]
11	(263.87, 263.45)	[0.86, 0.88, 0.89]	[23.64, 27.72, 17.99]	[0.43, 0.44, 0.44]
12	(158.86, 264.43)	[0.76, 0.75, 0.85]	[11.43, 16.89, 13.13]	[0.38, 0.38, 0.43]
13	(146.68, 167.94)	[0.77, 0.79, 0.78]	[14.34, 23.84, 25.53]	[0.38, 0.39, 0.39]
14	(108.40, 110.73)	[0.77, 0.82, 0.85]	[12.29, 15.27, 12.53]	[0.38, 0.41, 0.42]
15	(116.92, 209.40)	[0.79, 0.90, 0.88]	[28.76, 10.79, 22.95]	[0.40, 0.45, 0.44]
16	(225.76, 193.61)	[0.79, 0.85, 0.83]	[18.21, 22.86, 11.13]	[0.39, 0.42, 0.42]
17	(139.28, 214.32)	[0.88, 0.76, 0.81]	[24.36, 12.23, 19.57]	[0.44, 0.38, 0.40]
18	(108.47, 287.79)	[0.90, 0.80, 0.79]	[23.63, 15.54, 26.86]	[0.45, 0.40, 0.39]
19	(282.10, 270.60)	[0.77, 0.85, 0.80]	[20.78, 11.26, 25.89]	[0.39, 0.43, 0.40]
20	(203.66, 298.34)	[0.89, 0.87, 0.84]	[25.67, 12.93, 22.59]	[0.44, 0.44, 0.42]
21	(299.17, 130.00)	[0.79, 0.77, 0.82]	[10.16, 16.60, 11.15]	[0.39, 0.39, 0.41]
22	(107.05, 279.01)	[0.88, 0.79, 0.79]	[15.89, 25.08, 18.77]	[0.44, 0.40, 0.39]
23	(175.96, 129.86)	[0.75, 0.77, 0.84]	[21.82, 19.09, 27.25]	[0.38, 0.39, 0.42]
24	(151.47, 112.94)	[0.90, 0.77, 0.83]	[16.10, 19.74, 27.79]	[0.45, 0.38, 0.41]

list the detailed information of the UAVs, targets, and the tasks on these targets under Scenario 2.

The population size was 100, the maximum number of iterations was 100, and the PS archive capacity was 100. MMSALO, MOALO, IMOPSO, INSGA-II, and INSGA-III were executed independently 20 times. The reference point was set to (108, 108, 108) for Scenario 1 and (192, 192, 192) for Scenario 2. The resulting HV values for the five algorithms are compared in

Table 13. HV values of algorithms on the task-allocation model.

Scenario	HV Value	MMSALO	MOALO	IMOPSO	INSGA-II	INSGA-III
1	Avg.	$\mathbf{6}.\mathbf{22}\times {\mathbf{10}}^{\mathbf{5}}$	$6.14\times {10}^5$	$6.20\times {10}^5$	$6.19\times {10}^5$	$6.19\times {10}^5$
	Std.	$3.71\times {10}^3$	$6.68\times {10}^3$	$1.83\times {10}^3$	$5.86\times {10}^3$	$4.38\times {10}^3$
2	Avg.	$\mathbf{4}.\mathbf{18}\times {\mathbf{10}}^{\mathbf{6}}$	$4.11\times {10}^6$	$4.16\times {10}^6$	$4.16\times {10}^6$	$4.16\times {10}^6$
	Std.	$1.96\times {10}^4$	$3.61\times {10}^4$	$1.49\times {10}^4$	$2.54\times {10}^4$	$1.59\times {10}^4$

Bold denotes the optimal result.

. To facilitate HV computation, the objective values of task execution time were linearly scaled to [0, 108] in Scenario 1 and to [0, 192] in Scenario 2.

Table 13 and Figure 6 show that MMSALO achieves the highest average HV across all scenarios, demonstrating its superior capability in tackling the multi-task, multi-constraint assignment problem for UAV swarms. Moreover, MMSALO exhibits a significantly smaller HV standard deviation than does ƒMOALO, indicating enhanced stability. Because operational environments preclude repeated algorithmic runs to cherry-pick favorable solutions, this increased robustness renders MMSALO more valuable in practice.

Table 14. Wilcoxon rank-sum test analysis of HV results on the task-allocation model.

Scenario	MOALO	IMOPSO	INSGA-II	INSGA-III
1	$\mathbf{2}.\mathbf{34}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{3}.\mathbf{98}\times {\mathbf{10}}^{-\mathbf{2}}$	$\mathbf{3}.\mathbf{26}\times {\mathbf{10}}^{-\mathbf{2}}$	$\mathbf{2}.\mathbf{65}\times {\mathbf{10}}^{-\mathbf{2}}$
2	$\mathbf{3}.\mathbf{67}\times {\mathbf{10}}^{-\mathbf{7}}$	$\mathbf{2}.\mathbf{65}\times {\mathbf{10}}^{-\mathbf{2}}$	$\mathbf{1}.\mathbf{28}\times {\mathbf{10}}^{-\mathbf{2}}$	$\mathbf{2}.\mathbf{48}\times {\mathbf{10}}^{-\mathbf{2}}$

Bold denotes statistical significance.

presents the p-values of the Wilcoxon rank-sum test. It can be observed that when $\alpha$ is set to 0.05, all p-values are significantly lower than $\alpha$. This indicates that the performance of MMSALO is statistically significantly better than that of the comparison algorithms.

Table 15. Runtime performance comparison on the task-allocation model.

Scenario	MMSALO	MOALO	IMOPSO	INSGA-II	INSGA-III
1	$6.58\times {10}^1$	$6.62\times {10}^1$	$1.44\times {10}^2$	$1.81\times {10}^1$	$\mathbf{1}.\mathbf{67}\times {\mathbf{10}}^{\mathbf{1}}$
2	$1.05\times {10}^2$	$8.66\times {10}^1$	$2.02\times {10}^2$	$2.89\times {10}^1$	$\mathbf{2}.\mathbf{66}\times {\mathbf{10}}^{\mathbf{1}}$

Bold denotes the shortest runtime.

presents the average computation time required for each algorithm to solve the task-allocation problem in a single run. As can be observed, the computation times of MMSALO and MOALO remain within the same order of magnitude. Due to the significantly higher complexity of the task-allocation problem compared to the test functions, the computation times of IMOPSO, INSGA-II, and INSGA-III increase by one order of magnitude. In contrast, the computation times of MMSALO and MOALO show no significant change. This demonstrates the favorable scalability of MMSALO and its efficient performance in addressing complex problems.

Table 16. Results from ablation studies on the task-allocation model.

Scenario	MMSALO	SS	RB	APU	PBES	MOALO
1	$6.2235\times {10}^5$	$6.1626\times {10}^5$	$6.1409\times {10}^5$	$6.1517\times {10}^5$	$6.1415\times {10}^5$	$6.1408\times {10}^5$
Improvement	1.0135	1.0036	1.0000	1.0018	1.0001	1.0000
Runtime	$6.5766\times {10}^1$	$6.1644\times {10}^1$	$7.9269\times {10}^1$	$5.9756\times {10}^1$	$6.8892\times {10}^1$	$6.6172\times {10}^1$

presents the HV and runtime results obtained from the ablation experiments conducted by MMSALO on the task-allocation problem. It can be observed that “SS” and “APU” contribute significantly when addressing complex problems.

Collectively, MMSALO attains the highest HV value and outperforms the competing algorithms by a statistically significant margin. These results demonstrate that the sparsity selection mechanism, random boundary strategy, adaptive position update rule, and preference-based elite selection mechanism collectively enhance algorithmic performance, enabling MMSALO to provide an effective reference for analyzing multi-task, multi-constraint task-allocation problems in UAV swarms.

5. Conclusions

The UAV swarm collaborative task-allocation problem considered in this study is a multi-objective, multi-constrained optimization problem. The main conclusions drawn are as follows:

A multi-task, multi-constraint task-allocation model for UAV swarms was established. To meet the demands of practical applications, more detailed constraints need to be considered. Three optimization objectives were introduced to effectively evaluate various allocation schemes. A double-layer encoding mechanism and an adaptive penalty function method were designed to handle the key constraints. Compared with traditional penalty methods, this approach can dynamically adjust the penalty values based on the degree of constraint violations during the search process, thereby avoiding the need for parameter tuning and accelerating the convergence speed of the algorithm.

The MMSALO algorithm is proposed to effectively solve the model. MMSALO integrates a sparsity selection mechanism that significantly improves global exploration. A random boundary strategy increases the stochasticity and diversity of the ant walks around antlions, thereby enhancing population diversity. An adaptive position update strategy balances exploration in the early phase and exploitation in the latter phase. A preference-based elitist selection mechanism further strengthens the search capability and ensures better solution distribution. Simulation results demonstrate the feasibility of the allocation model and confirm that the resulting schedules accurately capture the characteristics of both UAVs and tasks. Comparative experiments against state-of-the-art algorithms verify the superior performance of the proposed MMSALO.

This study was limited to the task allocation for stationary targets using a centralized strategy and did not account for potential collisions among UAVs. In future work, we plan to design larger-scale scenarios and account for uncertainties in real-world environments. A distributed task0allocation approach will be adopted and dynamic task-reallocation strategies developed. Corresponding models and algorithms will also be investigated.