A Multi-Swarm Dynamic Crow Search Algorithm for Multi-UAV Dynamic Task Allocation

Highlights

What are the main findings?

• A dynamic multi-objective optimization model is formulated for multi-UAV dynamic task allocation, incorporating state inheritance and diverse dynamic events.
• A multi-swarm dynamic crow search algorithm with tailed strategies is proposed to solve the problem efficiently.

What is the implication of the main finding?

• The model enables dynamic task allocation that reflects the continuous scenario information and covers a broader range of real-world events.
• The algorithm achieves superior performance with acceptable runtime, shows effectiveness in handling dynamic task allocation problems.

Abstract

Efficient cooperative task allocation is essential for multiple unmanned aerial vehicles (UAVs) performing complex missions. However, diverse dynamic events in real-world scenarios require rapid response through dynamic task allocation (DTA). Although evolutionary algorithms have been widely adopted for DTA, existing methods often fail to maintain consistency between allocation decisions and actual operational states, consider only limited classes of dynamic events, and still leave room for performance improvement. This paper formulates multi-UAV DTA as a dynamic multi-objective optimization problem (DMOP) that jointly minimizes the residual target value and mission makespan, incorporating a state inheritance mechanism and a comprehensive set of dynamic events covering multiple facets of disruptions in observation task scenarios. To solve this DMOP, a multi-swarm dynamic crow search algorithm for task allocation (MDCSATA) is proposed, which integrates five strategies: violation-tolerant multi-swarm co-evolution for feasibility and diversity; objective-oriented heuristic initialization to accelerate convergence; an adaptive position update for better exploration and exploitation; stagnation and elite guided perturbation for intensified local exploitation; and an event-aware change response for rapid adaptation to dynamic events. Experiments on three constructed scenarios against seven state-of-the-art algorithms show that MDCSATA achieves superior performance on the evaluation metrics with acceptable runtime. It obtains the best MHV and MIGD in all scenarios, improving MHV by at least 0.93% and reducing MIGD by at least 12.92% across scenarios. These results confirm its effectiveness for DTA.

1. Introduction

Unmanned aerial vehicles (UAVs) have been widely adopted across various domains, such as remote sensing, plant protection, and damage detection, owing to their flexibility and low operational cost [1,2,3]. Research has increasingly shifted from single-UAV platforms to multi-UAV cooperative swarms, which can undertake complex tasks beyond the capability of individual units—examples include search and rescue, area surveillance, and supplies delivery [4,5,6]. Compared with a single-UAV system, a multi-UAV swarm offers distinct advantages in mission efficiency and robustness, because it can execute tasks in parallel and maintain overall mission integrity even if some UAVs are lost [7]. To fully realize these advantages, efficient cooperative task allocation is essential: it assigns specific tasks to individual UAVs according to decision demands and governs how resources are distributed to meet mission objectives, which has made it an extensively studied topic [8,9,10]. In practice, decision-makers often seek plans that balance diverse operational priorities, for example, maximizing mission reward while minimizing UAV cost. These multi-faceted requirements turn task allocation into a multi-objective optimization problem that requires simultaneous optimization of conflicting objectives [11].

In real-world deployments, however, dynamic events exist that change the scenario information, giving rise to the dynamic task allocation (DTA) problem. Traditional static allocation methods lack the flexibility to adapt to such changes [12]. To address DTA, researchers have proposed heuristic-based, machine learning-based, market mechanism-based, and evolutionary algorithm (EA)-based methods [13,14,15,16]. The first three categories mainly rely on problem-specific rules, negotiation, or learned policies, whereas EAs provide flexible search for complex multi-objective allocation problems. Among these, EA-based approaches are widely favored for their scalability, global search capability, and multi-objective optimization performance. For instance, Deng et al. [17] modeled heterogeneous UAV task allocation within a digital twin framework and proposed an improved dynamic multi-objective adaptive weight particle swarm optimization (DMOAWPSO) algorithm, which quantifies environmental severity and adaptively selects population restructuring strategies to track the moving Pareto front (PF). Zhang et al. [18] formulated search and track missions as dynamic multi-objective problems (DMOPs) with varying dimensions and objectives, and proposed the dynamic constrained two-archive evolutionary algorithm (DCTAEA) with a self-adaptive penalty mechanism to reuse information from infeasible solutions. Wang et al. [19] addressed reconnaissance task allocation with moving targets via the dynamic multi-objective particle swarm optimization algorithm based on centroid-moving prediction (DCMPSO), which uses the movement of the population centroid in the decision space between environments to predict the PF location. Moreover, some studies address the DTA problem by combining multiple objectives into one through linear weighting. Zhang et al. [16] modeled forest fire rescue assignment as a dynamic optimization problem driven by stochastic fire spread and proposed a variable-length particle swarm optimization with a dynamic encoding scheme that maps particles to evolving fire-edge cells, enabling adaptive reallocation within a simulation-integrated ensemble framework. Huang et al. [20] established a fuzzy multi-constraint model and employed a four-dimensional information grey wolf optimizer with dynamic time slices to handle environmental uncertainties.

Despite these advances, existing EA-based research still exhibits three major shortcomings. First, most models fail to account for the state continuity of mission execution: they overlook how ongoing tasks continuously alter UAV or task states, leading to a decoupling between decision information and the working environment. Second, the dynamic events considered are relatively narrow in scope, typically focusing only on structural changes such as platform failures or new target appearances, while neglecting events such as sensor damage, weather-induced capability fluctuations, and decision variations. Third, there still remains room for improvement of the methods in both event response speed and allocation performance. Therefore, existing methods do not simultaneously address execution-state consistency, heterogeneous dynamic events, and rapid adaptation in the constrained multi-UAV DTA scenario considered in this study.

To address these challenges, this paper proposes the multi-swarm dynamic crow search algorithm for task allocation (MDCSATA). On the modeling side, critical information such as UAV coordinates and task completion status is inherited from actual execution results, enhancing the consistency of the dynamic allocation scheme. Diverse dynamic events are introduced, covering UAVs, targets, environment, and decision preferences, constructing a more realistic and comprehensive task allocation scenario. On the algorithmic side, considering the key components of dynamic multi-objective evolutionary algorithms (DMOEAs)—the static optimizer and the change response strategy [21]—we adopt the crow search algorithm (CSA) for its simple search pattern, low parameter count, and competitive optimization performance [22,23]. For change response, event severity is quantified to guide the rapid response to the dynamic events, thereby improving convergence and speed in handling DTA problems. Our main contributions are:

(1)

We formulate a DMOP framework for multi-UAV dynamic task allocation. By establishing a state inheritance mechanism and introducing diverse dynamic events, this framework can better mirror real-world operational complexities.

(2)

We propose MDCSATA with five coordinated strategies tailored to DTA problems: violation-tolerant multi-swarm co-evolution, objective-oriented heuristic initialization, adaptive position update, stagnation and elite guided perturbation, and event-aware change response.

(3)

Three benchmark scenarios with diverse dynamic events are constructed, and MDCSATA is evaluated against seven state-of-the-art DMOEAs using two performance metrics. Experimental results verify that MDCSATA generally achieves superior task allocation performance with acceptable runtime overhead.

The remainder of this paper is organized as follows. Section 2 reviews the background of DMOP and the CSA. Section 3 formulates the multi-UAV DTA problem. Section 4 introduces the proposed MDCSATA algorithm. Section 5 reports the experimental evaluation. Finally, Section 6 concludes the paper and discusses future research directions.

2. Background

2.1. Dynamic Multi-Objective Optimization Problems

Without loss of generality, a minimization DMOP can be defined as

$$\left\{\begin{array}{cc}\hfill \underset{\mathit{x}}{min}& \mathit{F}(\mathit{x},t)=\left(f_1(\mathit{x},t),f_2(\mathit{x},t),\cdots ,f_{M_t}(\mathit{x},t)\right)\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& g_i(\mathit{x},t)\ge 0,i=1,\cdots ,n_g\left(t\right)\hfill \\ \hfill & h_j(\mathit{x},t)=0,j=1,\cdots ,n_h\left(t\right)\hfill \\ \hfill & \mathit{x}\in {\Omega }_x,t\in {\Omega }_t\hfill \end{array}\right.$$

(1)

where $\mathit{x}$ is a decision vector and ${\Omega }_x\subseteq {\mathbb{R}}^n$, t is the time variable and ${\Omega }_t\subseteq \mathbb{R}$. At time t, $\mathit{F}(\mathit{x},t)$ is the objective function vector that evaluates the decision vector $\mathit{x}$, $M_t$ is the number of objectives, $g_i(\mathit{x},t)$ and $h_j(\mathit{x},t)$ are inequality constraints and equality constraints, and $n_g\left(t\right)$ and $n_h\left(t\right)$ are the numbers of inequality constraints and equality constraints, respectively. A DMOP can be regarded as a time series of static multi-objective optimization problems. Therefore, solving DMOPs involves tracking the Pareto set (PS) and PF of the multi-objective optimization problem series, and the PS and PF of a DMOP at time t are denoted as $P{S}_t$ and $P{F}_t$.

2.2. Crow Search Algorithm

CSA is inspired by the intelligent behavior of crows that store their own food at secret places and also search for food hidden by other birds to steal it [24]. Owing to its simple search mechanism and computational efficiency, CSA has been extended to various optimization problems [25,26,27]. CSA assumes each crow’s current position is $\mathit{x}$, and each crow hides its food at its best position so far as $\mathit{m}$. At iteration $\tau$, crow i targets crow j to attempt to steal its food. Then, the position update of crow i depends on the awareness probability $AP$ and the flight length $fl$ of crows as follows:

$${\mathit{x}}_i^{\tau +1}=\left\{\begin{array}{cc}{\mathit{x}}_i^{\tau }+r_i\times fl\times \left({\mathit{m}}_j^{\tau }-{\mathit{x}}_i^{\tau }\right),\hfill & r_j\ge AP\hfill \\ \mathrm{a}\mathrm{random}\mathrm{position},\hfill & r_j<AP\hfill \end{array}\right.$$

(2)

where $r_i$ and $r_j$ are independent random numbers with uniform distributions between 0 and 1. In CSA, the memory ${\mathit{m}}_j^{\tau }$ represents the best food-hiding position found by crow j. If crow j is unaware of being followed, crow i moves toward this memory for exploitation; otherwise, crow i is relocated randomly for exploration. Thus, $AP$ controls the probability of random exploration, whereas $fl$ controls the movement step size. A larger $fl$ encourages broader exploration, while a smaller $fl$ supports local exploitation. CSA therefore balances exploration and exploitation through a simple memory-guided update with few control parameters and low computational overhead, making it suitable for repeated re-optimization in time-sensitive DTA scenarios.

3. Scenario Description and Problem Modeling

3.1. Scenario Description

We consider scenarios in which S UAVs $\mathcal{U}=\{U_1,U_2,\dots ,U_S\}$ observe Z targets $\mathcal{T}=\{T_1,T_2,\dots ,T_Z\}$ inside a three-dimensional (3D) mission space $[0,L_x]\times [0,L_y]\times [0,L_z]\subset {\mathbb{R}}^3$. The observation of targets is motivated by a wide range of practical UAV applications that require acquiring real-time information about ground objects or areas of interest, such as environmental monitoring, disaster damage assessment, and military reconnaissance. In these applications, each target carries an observational value that reflects the importance or urgency of obtaining information from it, and the collected value often decays over time or is degraded by sensor limitations and environmental factors. The main attributes of UAV $U_i$ and target $T_j$ are listed in

Table 1. Attributes of UAV $U_i$.

Attribute	Symbol
Position at time t	${\mathit{p}}_i^t$
Flight speed	$v_i$
Remaining energy percentage at time t	$E_i^t$
Maximum endurance time	$t_{\mathrm{endu},i}$
Vulnerability factor	${\xi }_i$
Sensing capability at time t	${\eta }_i\left(t\right)$
Sensor availability vector at time t	${\mathit{I}}_i^t$

and

Table 2. Attributes of target $T_j$.

Attribute	Symbol
Location at time t	${\mathit{p}}_j^t$
Observational value at time t	$V_j^t$
Required observation time	$t_{\mathrm{obse},j}$
Completion status at time t	$c_j^t$

. To mathematically formulate the operational logic, we adopt the following assumptions:

• Each UAV and target is treated as a point mass, and the flight route is simplified to the Euclidean distance. While observing a target, the UAV is assumed to hover at the target location.
• A UAV may carry visible-light (VL), infrared (IR), and radar (RD) sensors. Environmental factors that degrade sensing performance are unified into sensor-specific effectiveness factors, which affect the overall capability through multi-sensor fusion.
• A centralized control center monitors the scenario, collects external intelligence, and updates the environmental state for dynamic task allocation.

The control center triggers dynamic task allocation immediately upon detecting any of the following four types of dynamic events:

• UAV events: sensor malfunctions or UAV crashes.
• Target events: new target appearances, target value decreases, or target movements.
• Environmental events: weather changes that degrade sensor performance.
• Preference events: shifts in decision-maker preferences.

3.2. Task Allocation Problem Modeling

Let ${\mathcal{A}}^t$ denote the task-to-UAV allocation scheme at time t, which specifies the ordered task sequences of all UAVs. For UAV $U_i$, ${\mathcal{A}}_i^t=(T_{i,1},T_{i,2},\dots ,T_{i,|{\mathcal{A}}_i^t|})$ is the ordered list of targets assigned to it at time t. Let ${\mathcal{T}}_{\mathrm{unfin}}^t=\{T_j\in \mathcal{T}\mid c_j^t=0\}$ be the set of unfinished targets, where $c_j^t=1$ indicates that target $T_j$ has been completed and $c_j^t=0$ otherwise. The DTA problem is formulated through the following objectives and constraints.

3.2.1. Objectives

The optimization problem is

$$\underset{{\mathcal{A}}^t}{min}\mathit{F}({\mathcal{A}}^t,t)=\left(f_1({\mathcal{A}}^t,t),f_2({\mathcal{A}}^t,t)\right),$$

(3)

where $f_1$ is the remaining target value and $f_2$ is the mission makespan. The two objectives are selected to capture the main operational trade-off. The remaining target value reflects the mission reward, while the makespan reflects the mission cost in terms of time.

Remaining target value. The primary goal of multi-UAV observation missions is to maximize the total observational value collected from targets. The observational value of a target decays with time as information becomes outdated, and its acquisition through UAVs is degraded by sensor limitations and environmental factors. To formulate this as a minimization problem, we define the objective as the remaining unobserved target value:

$$f_1({\mathcal{A}}^t,t)=\sum _{T_j\in {\mathcal{T}}_{\mathrm{unfin}}^t}{V}_j^t-{\displaystyle \sum _{i=1}^S}\sum _{k\in {\mathcal{A}}_i^t}{V}_k^t{\eta }_i\left(t\right)p_{\mathrm{surv},i}^{\left(k\right)}{e}^{-{\lambda }_{\mathrm{obse}}\Delta t_{i,k}},$$

(4)

where $e^{-{\lambda }_{\mathrm{obse}}\Delta t_{i,k}}$ models the timeliness decay of target $T_k$’s observational value with decay constant ${\lambda }_{\mathrm{obse}}$, and $\Delta t_{i,k}$ is the time elapsed since UAV $U_i$ reached target $T_k$. The sensing capability ${\eta }_i\left(t\right)$ follows a probabilistic multi-sensor fusion model:

$${\eta }_i\left(t\right)=1-\prod _{m\in \mathcal{M}}\left(1-I_{m,i}^t·{\omega }_m^t\right),$$

(5)

where $\mathcal{M}\subseteq \{\mathrm{VL},\mathrm{IR},\mathrm{RD}\}$ is the set of sensors on UAV $U_i$, $I_{m,i}^t\in \{0,1\}$ is the availability status of sensor m, and ${\omega }_m^t\in (0,1]$ is the time-varying weather effectiveness factor for sensor m. This fusion model improves sensing capability and reliability in complex environments [28]. The weather factors ${\omega }_m^t$ are non-increasing during deterioration events, simulating worsening conditions such as dense fog or electromagnetic interference.

The survival probability $p_{\mathrm{surv},i}^{\left(k\right)}$ is derived from a cumulative path-dependent risk to provide an approximation. For a flight segment of length $d_{i,k}$ flown by UAV $U_i$ toward target $T_k$ according to its task sequence, the segment risk factor ${\theta }_{i,k}$ is defined as

$${\theta }_{i,k}=1-e^{-{\lambda }_{\mathrm{risk}}{\xi }_i{d}_{i,k}},$$

(6)

where ${\lambda }_{\mathrm{risk}}$ is a flight risk constant and ${\xi }_i$ is the vulnerability factor of UAV $U_i$; it increases with payload weight, as heavier payloads enlarge the physical footprint and signatures. The cumulative survival probability is then updated recursively:

$$p_{\mathrm{surv},i}^{\left(k\right)}=p_{\mathrm{surv},i}^{(k-1)}·\left(1-{\theta }_{i,k}\right),\mathrm{with}{p}_{\mathrm{surv},i}^{\left(0\right)}=1.$$

(7)

Mission makespan. In time-critical missions, an unbalanced workload can severely delay the overall response. To promote load balance and rapid mission completion, we minimize the makespan, i.e., the maximum individual completion time, across the swarm:

$$f_2({\mathcal{A}}^t,t)=\underset{i\in \{1,\dots ,S\}}{max}{t}_{\mathrm{comp},i},$$

(8)

where $t_{\mathrm{comp},i}=t+{\sum }_{k\in {\mathcal{A}}_i^t}\left(d_{i,k}/v_i+t_{\mathrm{obse},k}\right)$ is the absolute completion time of UAV $U_i$ after starting from its current position at time t, and $d_{i,k}$ denotes the Euclidean distance from its previous position to target $T_k$.

3.2.2. Constraints

Completeness constraint. To ensure no task is abandoned, every unfinished target at time t must be assigned to a UAV:

$${\displaystyle \bigcup _{i=1}^S}{\mathcal{A}}_i^t={\mathcal{T}}_{\mathrm{unfin}}^t.$$

(9)

Collaboration constraint. To avoid redundant assignments and ensure resource efficiency, each target is handled by exactly one UAV; the assigned target sets are pairwise disjoint:

$${\mathcal{A}}_i^t\cap {\mathcal{A}}_j^t=\varnothing ,\forall i\ne j.$$

(10)

Energy constraint. To prevent mid-air energy depletion, each UAV must retain positive residual energy after completing its remaining tasks:

$$g_i({\mathcal{A}}^t,t)=E_i^t-{\gamma }_i(t_{\mathrm{comp},i}-t)>0,$$

(11)

where ${\gamma }_i=1/t_{\mathrm{endu},i}$ is the energy consumption rate (percentage per unit time) of UAV $U_i$.

4. Multi-Swarm Dynamic Crow Search Algorithm for Task Allocation

To solve the DTA problem formulated in Section 3, we propose the multi-swarm dynamic crow search algorithm for task allocation (MDCSATA). Built upon a random-key encoding that maps discrete task assignments to a continuous search space compatible with CSA, the algorithm integrates five coordinated strategies to address the problem’s dynamic, multi-objective, and constrained nature: (i) a violation-tolerant multi-swarm co-evolution balances feasibility and diversity; (ii) an objective-oriented heuristic initialization provides an informed start; (iii) an adaptive position update combines stagnation awareness, Lévy flights, and multi-source leaders; (iv) a stagnation and elite guided perturbation intensifies local search; and (v) an event-aware change response quantifies severity and adaptively regenerates solutions. Details of the encoding and each strategy are given in the following subsections.

Algorithm 1 first summarizes the overall workflow, after which the following subsections explain its main components.

Algorithm 1 Pseudo-code of MDCSATA

Input:  Total swarm size N, objective count M, parameters $AP$, $f{l}_{min}$, $fl$, ${ϵ}_0$ Output:  Approximated PS in ${\mathit{A}}_{\mathrm{feas}}$   1: Initialize M sub-swarms with size of $\lfloor N/M\rfloor$ via the heuristic initialization;   2: Evaluate all crows and initialize their memories with $mt=1$;   3: Initialize ${\mathit{A}}_{\mathrm{feas}}$ and ${\mathit{A}}_{\inf }$ and perform the multi-swarm co-evolution;   4: while termination criterion not met do   5:       if dynamic events triggered then   6:             Execute the event-aware change response;   7:       end if   8:       for each sub-swarm do   9:             for each crow i in the sub-swarm do 10:                 Select a leader j via the multi-source rule; 11:                 Update the position of crow i via (15); 12:             end for 13:       end for 14:       Execute stagnation-triggered perturbation on crows with $mt>1$; 15:       Evaluate new positions and update memories and $mt$ counters; 16:       Update ${\mathit{A}}_{\mathrm{feas}}$ and ${\mathit{A}}_{\inf }$ using crows with $mt=1$; 17:       Execute elite perturbation on ${\mathit{A}}_{\mathrm{feas}}$; 18:       Perform the multi-swarm co-evolution; 19: end while

4.1. Random-Key Task Allocation Encoding

Task allocation is inherently combinatorial, yet CSA operates in a continuous search space. We adopt the random-key representation [29,30] to bridge this gap: the fractional parts of real-valued variables serve as sort keys for the discrete assignment, so any real-valued vector maps to a structurally valid allocation. By design, this construction satisfies the completeness and collaboration constraints in Section 3.2.2, eliminates repair operators, and makes the search space fully continuous, thereby fitting continuous optimizers such as CSA.

Each candidate allocation is represented as a variable-length random-key vector $\mathit{x}=(x_1,x_2,\dots ,x_Z)\in {[0,S)}^Z$, where $Z=|{\mathcal{T}}^t|$ is the total number of targets appeared up to time t; Z grows monotonically as new targets appear. The j-th variable $x_j$ encodes both the UAV assigned to target $T_j$ and the target’s execution priority within that UAV: the integer part $\lfloor x_j\rfloor \in \{0,1,\dots ,S-1\}$ identifies the assigned UAV, and the fractional part $\left\{x_j\right\}$ gives the within-UAV execution order that smaller value denotes earlier execution.

Figure 1 illustrates the encoding and decoding process with six variables: five existing targets ${\mathrm{T}}_1$–${\mathrm{T}}_5$ and one newly appeared target ${\mathrm{NT}}_1$. Among them, completed target ${\mathrm{T}}_1$ acts as a placeholder and is skipped during decoding; ongoing task ${\mathrm{T}}_5$ is locked to ${\mathrm{U}}_2$ with its fractional part set to zero, ensuring execution precedence; new target ${\mathrm{NT}}_1$ is appended to the vector; and the remaining unstarted targets ${\mathrm{T}}_2$, ${\mathrm{T}}_3$, ${\mathrm{T}}_4$ are free variables. The integer part of each variable identifies the assigned UAV, and the fractional part determines the within-UAV execution priority. The allocation scheme is obtained by discarding placeholders, grouping the remaining variables by assigned UAV, and sorting each group in ascending fractional order. The resulting allocation assigns $({\mathrm{T}}_4,{\mathrm{NT}}_1)$ to ${\mathrm{U}}_1$ and $({\mathrm{T}}_5,{\mathrm{T}}_3,{\mathrm{T}}_2)$ to ${\mathrm{U}}_2$. This encoding ensures that the vector never shrinks, existing index mappings remain stable, the completeness and collaboration constraints are satisfied, and the continuity of in-progress observations is preserved without repair operators.

In Figure 1, the completed target colored in gray is skipped during decoding, the ongoing target colored in light blue is locked because it is already under execution, and the free targets colored in black together with the newly appeared target colored in red can be reassigned and reordered. The encoding ensures that each unfinished target is assigned exactly once; UAVs with no allocated target may be temporarily idle.

4.2. Violation-Tolerant Multi-Swarm Co-Evolution

MDCSATA adopts a multi-population-for-multi-objective paradigm, which maintains selection pressure along each Pareto direction while sharing information across populations [31,32,33]. Let N be the total swarm size; the algorithm maintains M sub-swarms of size $\lfloor N/M\rfloor$, each dedicated to one objective. In this study, $M=2$, so the two sub-swarms are designated to $f_1$ and $f_2$, respectively. The term co-evolution refers to their interaction through swarm merging, shared feasible and infeasible archives, and objective-specific reallocation. In each iteration, solutions from all sub-swarms are merged and reallocated via a three-tier hierarchical sort that enforces relaxed feasibility:

1.

Feasible solutions: solutions that satisfy all constraints;

2.

Violation tolerated solutions: infeasible solutions whose constraint violation is within the tolerance threshold $ϵ$;

3.

Strictly infeasible solutions: infeasible solutions whose constraint violation exceeds the tolerance threshold $ϵ$.

The feasible and violation-tolerated candidates are pooled and ranked independently for each sub-swarm. They are sorted by $f_1$ to refill the $f_1$-oriented sub-swarm and by $f_2$ to refill the $f_2$-oriented sub-swarm, with the best $\lfloor N/M\rfloor$ candidates retained in each case. For example, between two candidates, the one with the lower $f_1$ is ranked first for the $f_1$-oriented sub-swarm, whereas the one with the lower $f_2$ is ranked first for the $f_2$-oriented sub-swarm. If the first two tiers together yield fewer than $\lfloor N/M\rfloor$ solutions, the remaining slots are filled with strictly infeasible solutions from the third tier. The tolerance threshold decays with iteration progress to encourage early exploration and late strict feasibility:

$$ϵ\left(\tau \right)={ϵ}_0·{\left(1-\tau /{\tau }_t\right)}^2$$

(12)

where ${ϵ}_0=\left|\mathcal{U}\right|$ is the initial tolerance, $\tau$ is the current iteration, and ${\tau }_t$ is the maximum iteration count at the current time t.

Instead of a single archive, MDCSATA uses a dual-archive structure: a feasible archive ${\mathit{A}}_{\mathrm{feas}}$ and an infeasible archive ${\mathit{A}}_{\inf }$, maintained by crowding-distance truncation [34]. ${\mathit{A}}_{\mathrm{feas}}$ stores non-dominated feasible solutions and guides the main search, while ${\mathit{A}}_{\inf }$ stores infeasible non-dominated solutions. During leader selection in Section 4.4, infeasible individuals have a 50% chance of following a leader from ${\mathit{A}}_{\inf }$, which helps explore along the constraint boundary to discover feasible regions.

Based on the above, the memory update adopts a violation-tolerant rule that replaces standard Pareto dominance. For a crow i, let $m{t}_i$ be its stagnation counter, which records the number of consecutive iterations without memory improvement. The update proceeds as follows:

• If $\mathit{x}$ $ϵ$-dominates $\mathit{m}$, the memory is replaced with $\mathit{x}$ and $m{t}_i$ is reset to 1.
• If $\mathit{x}$ and $\mathit{m}$ are $ϵ$-equivalent, the designated objective of the crow’s corresponding sub-swarm acts as a tiebreaker: if $\mathit{x}$ wins, it replaces the memory, and $m{t}_i$ is reset to 1. Regardless of the tiebreaker, $\mathit{x}$ is always added to the appropriate archive—${\mathit{A}}_{\mathrm{feas}}$ if feasible, or ${\mathit{A}}_{\inf }$ otherwise.
• Otherwise, $\mathit{m}$ $ϵ$-dominates $\mathit{x}$, the memory is retained, and $m{t}_i$ is increased by 1.

4.3. Objective-Oriented Heuristic Initialization

To accelerate convergence toward the PF and preserve diversity across the objective space, a composite initialization strategy is designed. Each sub-swarm is initialized by combining one space-filling sampler with two problem-aware heuristics, yielding a more informed start than random initialization.

Fifty percent of the individuals are generated via Latin hypercube sampling (LHS). Unlike uniform random sampling, LHS stratifies each dimension into equal-probability intervals and draws exactly one sample per interval, reducing clustering and improving decision-space coverage for a given sample size [35]. This offers a stronger foundation for global exploration.

Another 25% are generated by a spatiotemporal greedy heuristic: each target is assigned to its currently nearest UAV, and the within-UAV task order is sorted in descending order of the value-to-distance ratio $V_k^t/max(d_{i,k},1)$. The nearest-neighbor assignment shortens flight paths, reducing energy consumption and path-dependent risk. It also prioritizes high-value targets that can be reached quickly, mitigating the temporal decay of observational value in (4) and steering the swarms toward lower $f_1$.

The remaining 25% are constructed by a load-balanced heuristic that distributes unfinished targets across available UAVs in a round-robin fashion. Since the makespan $f_2$ in (8) is determined by the most heavily loaded UAV, balancing workload directly reduces $f_2$ and provides guiding solutions near its lower bound.

4.4. Adaptive Position Update

As discussed in Section 2, $AP$ and $fl$ are important CSA parameters that control exploitation and exploration. Accordingly, several studies have adaptively adjusted these parameters and reported improved performance [36]. Inspired by this, MDCSATA adapts the awareness probability $AP$ and the flight length $fl$, updating the crows’ positions as follows.

Multi-source leader selection. For each crow, a leader is drawn from one of three sources: $3/8$ probability by objective-based roulette-wheel selection from the relevant archive, which favors better fitness on the sub-swarm’s designated objective; $3/8$ probability by binary tournament selection on crowding distance from the relevant archive, which favors diversity; and $1/4$ probability by random selection within its own sub-swarm to enhance diversity through non-elite solutions. These probabilities sum to one. The equal archive-based probabilities balance objective-oriented convergence and crowding-distance-based diversity, while the smaller intra-swarm probability preserves exploration through non-elite information without weakening the guidance of archived solutions. The relevant archive, either ${\mathit{A}}_{\mathrm{feas}}$ or ${\mathit{A}}_{\inf }$, is determined by the feasibility rule in Section 4.2. This composition provides both selection pressure and diversity preservation.

Stagnation-aware awareness probability. The adaptive awareness probability is

$$A{P}_i=AP·\left(0.5+ln(1+m{t}_i)\right),$$

(13)

so that crows trapped in local optima gradually increase their random search probability, facilitating a self-regulated recovery from stagnation.

Lévy flight length. When the following behavior is triggered, then the flight length is sampled from a Lévy distribution via Mantegna’s algorithm with $\beta =1.5$, then modulated by a cosine envelope to gradually transition the search from exploration to exploitation:

$$f{l}^{\tau }=f{l}_{min}+min\left(cos\left(\frac{\tau mod{\tau }_t}{{\tau }_t}·\frac{\pi }{2}\right)·\left|L\left(\beta \right)\right|,fl\right)$$

(14)

where $f{l}_{min}$ is the base flight length, $fl$ is the flight length parameter used to truncate the Lévy-modulated increment, and $L\left(\beta \right)$ is a Lévy flight variable. Compared to Gaussian or uniform random walks, the heavy-tailed Lévy step produces frequent short moves with occasional long jumps. This foraging-like pattern is widely used in EAs to explore the search space more thoroughly and escape local optima [35]. The cosine envelope gradually reduces the size of long jumps as $\tau$ approaches ${\tau }_t$, so the search smoothly transitions from broad exploration to exploitation.

The selected leader affects crow i through its memory, which corresponds to the food-hiding position followed in CSA. Combining the above components, the position update formula is

$${\mathit{x}}_i^{\tau +1}=\left\{\begin{array}{cc}{\mathit{x}}_i^{\tau }+r_i·f{l}^{\tau }·\left({\mathit{m}}_j^{\tau }-{\mathit{x}}_i^{\tau }\right),\hfill & r_j\ge A{P}_i\hfill \\ \mathrm{a}\mathrm{random}\mathrm{position},\hfill & r_j<A{P}_i\hfill \end{array}\right.$$

(15)

Moreover, the variables exceeding the search bounds are clamped to the boundary after the position update.

4.5. Stagnation and Elite Guided Perturbation

To intensify local exploitation, MDCSATA incorporates two perturbation operators that target stagnated and elite solutions, respectively.

Stagnation-triggered perturbation. For each crow with $mt>1$, two variables that are not part of any ongoing task are randomly selected, and their integer parts are swapped while the fractional parts are kept unchanged. The perturbed solution is immediately re-evaluated. If the new solution $ϵ$-dominates the crow’s memory, the memory is replaced and $mt$ is reset to 1. If the two are non-dominated, the new solution is added to the appropriate archive. Otherwise, the variable changes are rolled back to prevent degradation. This mechanism provides targeted repair for stagnated solutions without undirected disruption.

Elite perturbation. At the end of each iteration, $\lceil N/\left(5M\right)\rceil$ solutions are randomly selected from the feasible archive ${\mathit{A}}_{\mathrm{feas}}$ for elite perturbation [31]. For each selected elite, we apply a Gaussian perturbation with standard deviation $0.1·(x_j^{max}-x_j^{min})$ to a randomly chosen free variable $x_j$. Any mutant that is not worse than the original under $ϵ$-dominance is added to the archive. Since most perturbations only alter the fractional part, this operator primarily fine-tunes the execution order within each UAV, thereby helping to search along the PF.

4.6. Event-Aware Change Response

In DTA problems, dynamic events induce abrupt changes in the problem landscape. A complete restart would discard the evolutionary information from previous problem environments, whereas a simple warm start may fail to adapt to severe changes. MDCSATA therefore employs an event-aware severity quantification and an adaptive regeneration strategy.

After an environmental change, each solution is first dimension-adapted: newly appeared targets are appended to the vector, completed targets are retained only as skipped placeholders, ongoing targets are locked to their executing UAVs, and assignments to crashed UAVs are released for reallocation. Then $K=min(10,|{\mathit{A}}_{\mathrm{feas}}\left|\right)$ elite solutions are randomly selected from ${\mathit{A}}_{\mathrm{feas}}$ and re-evaluated. The severity $\delta$ is defined as

$$\delta ={\delta }_{\mathrm{base}}+{\displaystyle \frac{1}{K·M}}{\displaystyle \sum _{i=1}^K}{\displaystyle \sum _{\ell =1}^M}{\displaystyle \frac{\left|f_{\ell }{\left({\mathit{x}}_i\right)}^{\prime }-f_{\ell }\left({\mathit{x}}_i\right)\right|}{|f_{\ell }\left({\mathit{x}}_i\right)|}}$$

(16)

where $f_{\ell }\left({\mathit{x}}_i\right)$ and $f_{\ell }{\left({\mathit{x}}_i\right)}^{\prime }$ are the pre-change and post-change objective values, and ${\delta }_{\mathrm{base}}=0.15$ is added when the event set includes new target appearances or UAV crashes, which may not be fully captured by objective changes from re-evaluation alone. This base term represents a minimal structural-change penalty, while the actual extent of the change is still reflected by the re-evaluation term.

Based on $\delta$, each sub-swarm’s new population is composed of three adaptive components:

1.

Elite retention ($R_{\mathrm{keep}}=max(0.2,0.6-min(3\delta ,0.4))$): the best $R_{\mathrm{keep}}$ fraction of the union of the sub-swarm and the re-evaluated elites is retained for maintaining solution diversity.

2.

Heuristic injection ($R_{\mathrm{heur}}=min(0.5,0.2+min(3\delta ,0.3))$): balanced and greedy heuristics are applied to generate solutions that help guide the search after dynamic events.

3.

Global exploration ($R_{\mathrm{lhs}}=1-R_{\mathrm{keep}}-R_{\mathrm{heur}}$): LHS solutions are generated to distribute in the reshaped search space.

These ratios retain historical information when the change severity is small and increase guided reconstruction and global exploration as the estimated severity grows. Ongoing-task variables are locked within the heuristic and LHS components to preserve the continuity of in-progress observations. After that, both archives are re-initialized and the multi-swarm co-evolution is performed.

4.7. Overall Procedure and Time Complexity

The main optimization steps are summarized in Algorithm 1. Here is a simple example of the process: consider two UAVs and three targets. A crow at $\mathit{x}=(0.2,1.6,1.3)$ is decoded as ${\mathrm{U}}_1:\left({\mathrm{T}}_1\right)$ and ${\mathrm{U}}_2:({\mathrm{T}}_3,{\mathrm{T}}_2)$ and then evaluated using the two objectives. Suppose its selected leader has memory $\mathit{m}=(1.1,0.4,1.8)$ and $r_{i}f{l}^{\tau }=0.5$. The following behavior produces ${\mathit{x}}^{\prime }=(0.65,1.0,1.55)$, which is decoded and evaluated as ${\mathrm{U}}_1:\left({\mathrm{T}}_1\right)$ and ${\mathrm{U}}_2:({\mathrm{T}}_2,{\mathrm{T}}_3)$. If a new target subsequently appears, its variable is appended, while completed targets are skipped and ongoing targets remain locked before regeneration.

Now, we analyze the time complexity of MDCSATA with M objectives, swarm size N, problem dimension n, maximum iteration $\Gamma$, dynamic event count $\Theta$, and objective evaluation cost F. Both archives are bounded by N. The complexities of the main stages are:

• Initialization: heuristic sorting costs $O(N·nlogn)$ and the evaluation costs $O(N·F)$.
• Position update: $O(\Gamma ·N·n)$.
• Evaluation: $O(\Gamma ·N·F)$.
• Multi-swarm co-evolution with violation-tolerant sorting: $O(\Gamma ·MNlogN)$.
• Dual-archive maintenance via crowding-distance truncation: $O(\Gamma ·N^2)$.
• Stagnation-triggered perturbation and elite perturbation: $O(\Gamma ·N(n+F\left)\right)$.
• Event-aware change response: re-evaluation and regeneration cost $O(\Theta ·N(nlogn+F\left)\right)$.

The overall complexity is $O\left((\Gamma +\Theta )·N·F+\Gamma ·N^2+(\Gamma +\Theta )·N·nlogn\right)$. The objective evaluation is dominated by the survival-probability integration along flight segments, giving $F\gg nlogn$. Consequently, the effective complexity simplifies to $O\left((\Gamma +\Theta )·N·F\right)$, indicating that the auxiliary mechanisms impose limited overhead compared with the indispensable objective evaluations.

5. Experiments and Results

5.1. Experimental Setup

5.1.1. Scenarios Settings

Three 3D scenarios are constructed to evaluate MDCSATA comprehensively: S₁, S₂, and S₃. All share an $8000\mathrm{m}\times 8000\mathrm{m}\times 200\mathrm{m}$ operational airspace. Following a top-down hierarchical derivation, the states of S₂ and S₁ are extracted from S₃, ensuring spatial consistency and physical inheritance across scenarios. ${\lambda }_{\mathrm{obse}}$ and ${\lambda }_{\mathrm{risk}}$ are set to 0.001 and 0.0002, respectively, to align with the scale of the operational airspace.

Scenarios S₁, S₂, and S₃ involve 3, 4, and 5 UAVs, and 10, 14, and 18 initial targets, respectively. There are two types of UAVs in the scenarios, and detailed attributes are listed in

Table 3. Attribute of UAVs across the scenarios.

ID	Type	Speed (m/s)	Endurance (s)	VL	IR	Radar	Vulnerability	Position (x, y, z)	Involved Scenarios
U1	A	15	1800	1	1	0	0.9	(1466, 73, 80)	S1/S2/S3
U2	B	15	2400	1	1	1	1.0	(1373, 545, 76)	S1/S2/S3
U3	A	15	1800	1	1	0	0.9	(992, 1091, 73)	S1/S2/S3
U4	B	15	2400	1	1	1	1.0	(557, 1390, 72)	S2/S3
U5	A	15	1800	1	1	0	0.9	(70, 1400, 75)	S3

. Target configurations are provided in

Table 4. Attribute of targets across the scenarios.

ID	Observational Value	Required Time (s)	Position (x, y, z)	Involved Scenarios
T1	67	12	(1786, 3385, 82)	S1/S2/S3
T2	52	11	(5761, 1668, 105)	S1/S2/S3
T3	84	16	(910, 5741, 78)	S1/S2/S3
T4	78	14	(5359, 7562, 78)	S1/S2/S3
T5	92	17	(1948, 1934, 82)	S1/S2/S3
T6	56	11	(4517, 2712, 90)	S1/S2/S3
T7	57	11	(3873, 5182, 89)	S1/S2/S3
T8	59	10	(4886, 6545, 140)	S1/S2/S3
T9	91	18	(3267, 1243, 86)	S1/S2/S3
T10	99	19	(7238, 2633, 82)	S1/S2/S3
T11	97	19	(2549, 4163, 88)	S2/S3
T12	71	14	(5627, 5859, 134)	S2/S3
T13	89	17	(2874, 3164, 80)	S2/S3
T14	90	17	(4406, 3981, 81)	S2/S3
T15	71	13	(2025, 5425, 88)	S3
T16	98	19	(7516, 6138, 111)	S3
T17	99	20	(1533, 1018, 77)	S3
T18	68	14	(5192, 233, 82)	S3
NT1	76	15	(2184, 6889, 83)	S1/S2/S3
NT2	60	11	(6792, 7246, 80)	S3

. A sequence of dynamic events is injected across the scenarios, following a subset hierarchy: S₁ experiences fewer and sparser events than S₂, which in turn has fewer events than S₃, while all share the same trigger times as S₃. All events are detailed in

Table 5. Dynamic events across the scenarios.

No.	Time (s)	Description	Involved Scenarios
1	100	UAV U5 crashed	S3
2	100	New target NT1 appears at (2184, 6889, 83)	S1/S2/S3
3	170	Decision preference shifted from $[0.5,0.5]$ to $[0.75,0.25]$	S3
4	170	Target T12 value decreased from 71 to 35	S2/S3
5	250	UAV U2 sensor VL damaged	S3
6	250	UAV U3 crashed	S1/S2/S3
7	330	Weather deteriorated sensor effectiveness factor (VL: 0.94, IR: 0.58, RD: 0.54)	S3
8	330	UAV U1 sensor IR damaged	S2/S3
9	440	Target T5 moved from (1948, 1934, 82) to (1537, 1365, 125)	S3
10	440	Decision preference shifted from $[0.5,0.5]$ to $[0.75,0.25]$	S1/S2/S3
11	540	Target T13 value decreased from 89 to 44	S3
12	540	Target NT1 moved from (2184, 6889, 83) to (1206, 6405, 129)	S2
13	540	Target T15 moved from (2025, 5425, 88) to (1206, 6405, 129)	S3
14	660	New target NT2 appears at (6792, 7246, 80)	S3
15	660	Weather deteriorated sensor effectiveness factor (VL: 0.75, IR: 0.76, RD: 0.91)	S1/S2/S3

.

5.1.2. Comparison Algorithms

MDCSATA is compared with seven DMOEAs. In addition to DMOAWPSO, DCTAEA, and DCMPSO introduced in Section 1, four classical DMOEAs are employed:

• Dynamic non-dominated sorting genetic algorithm II (DNSGA-II) [37]: extends NSGA-II with a dynamic response that replaces a fraction of the population by randomly generated or mutated individuals upon detecting a change.
• Steady-state and generational evolutionary algorithm (SGEA) [38]: combines steady-state and generational updates; upon a change, half of the population estimates the PS movement trend, and the remainder is reinitialized using the predicted trend.
• Multi-objective evolutionary algorithm based on decomposition with support vector regression (MOEA/D-SVR) [39]: trains support vector regression models on historical solution data to predict the decision variables of new solutions when a problem change is detected.
• Distributed bi-behaviors crow search algorithm II (DBCSA-II) [27]: a CSA variant for DMOPs that applies Gaussian-beta functions to enhance search performance, retaining high-quality elites and randomly re-initializing low-quality solutions upon change.

5.1.3. Parameter Settings

All algorithms share the population size N = 100, the archive size of 100, and the same total number of iterations per scenario. Each run begins with an initial optimization phase of $2{\tau }_t$ iterations; thereafter, a re-optimization phase of ${\tau }_t$ iterations is triggered upon each dynamic event, with ${\tau }_t=20$. At the end of each phase, the solution to be executed is selected from the current PS according to the decision-maker’s preference weights, which are initially set to $[0.5,0.5]$ for equal priority to value and makespan. Each algorithm is independently executed for 30 runs. Parameters specific to each comparison algorithm follow their original papers, except for population size, archive size, and iteration numbers.

The specific parameters of MDCSATA are listed in

Table 6. Parameter settings of MDCSATA.

Parameter	Symbol	Value
Total swarm size	N	100
Sub-swarm size	$\lfloor N/M\rfloor$	50
Awareness probability	$AP$	0.05
Flight length	$fl$	1.5
Minimum flight length	$f{l}_{min}$	1.0

.

5.1.4. Performance Metrics

Two standard DMOP metrics, mean hypervolume (MHV) and mean inverted generational distance (MIGD), are adopted [21]. Let ${{\rm Y}}_t=\{1,2,\dots ,|{{\rm Y}}_t\left|\right\}$ denote the ordered sequence of environmental time steps in a single run, ${\widehat{PF}}_t$ the PF approximation obtained at time t, and $P{F}_t^{*}$ the corresponding reference PF constructed from the classic DNSGA-II with large population and iteration budgets at that step. MHV and MIGD are defined as:

The hypervolume $\mathrm{HV}({\widehat{PF}}_t,{\mathit{r}}_t)$ measures the volume of the objective space dominated by ${\widehat{PF}}_t$ relative to a reference point ${\mathit{r}}_t$. MHV averages this quantity across all time steps:

$$\mathrm{MHV}={\displaystyle \frac{1}{|{{\rm Y}}_t|}}\sum _{t\in {{\rm Y}}_t}\mathrm{HV}({\widehat{PF}}_t,{\mathit{r}}_t)$$

(17)

The per-step reference point ${\mathit{r}}_t$ is set to 1.1 times the per-objective maximum over the union of $P{F}_t^{*}$ and all algorithms’ PFs at time t. A higher MHV indicates better performance.

IGD at time t measures the average distance from each point on $P{F}_t^{*}$ to its nearest point in ${\widehat{PF}}_t$, jointly reflecting convergence and diversity. MIGD averages this across all time steps:

$$\mathrm{MIGD}={\displaystyle \frac{1}{|{{\rm Y}}_t|}}\sum _{t\in {{\rm Y}}_t}{\displaystyle \frac{1}{|P{F}_t^{*}|}}\sum _{\mathit{v}\in P{F}_t^{*}}\underset{\mathit{u}\in {\widehat{PF}}_t}{min}\parallel \mathit{v}-\mathit{u}\parallel$$

(18)

A lower MIGD indicates better tracking of the moving PF.

5.2. Results and Analysis

5.2.1. Performance Comparison

Table 7. MHV results across different scenarios.

Algorithm	S1	S2	S3
MDCSATA	$\mathbf{2.279}\times {\mathbf{10}}^{\mathbf{5}}$ ± $1.626\times {10}^4$	$\mathbf{4.453}\times {\mathbf{10}}^{\mathbf{5}}$ ± $2.491\times {10}^4$	$\mathbf{5.759}\times {\mathbf{10}}^{\mathbf{5}}$ ± $4.453\times {10}^4$
DMOAWPSO	$2.059\times {10}^5$ ± $1.920\times {10}^4$	$3.855\times {10}^5$ ± $\mathbf{2.442}\times {\mathbf{10}}^{\mathbf{4}}$	$4.603\times {10}^5$ ± $4.326\times {10}^4$
DCTAEA	$2.116\times {10}^5$ ± $\mathbf{8.890}\times {\mathbf{10}}^{\mathbf{3}}$	$4.411\times {10}^5$ ± $3.121\times {10}^4$	$5.510\times {10}^5$ ± $4.513\times {10}^4$
DCMPSO	$1.890\times {10}^5$ ± $1.873\times {10}^4$	$3.213\times {10}^5$ ± $3.331\times {10}^4$	$3.838\times {10}^5$ ± $4.582\times {10}^4$
DNSGA-II	$2.101\times {10}^5$ ± $2.229\times {10}^4$	$4.211\times {10}^5$ ± $2.509\times {10}^4$	$5.406\times {10}^5$ ± $4.237\times {10}^4$
SGEA	$2.024\times {10}^5$ ± $1.398\times {10}^4$	$4.038\times {10}^5$ ± $3.595\times {10}^4$	$5.130\times {10}^5$ ± $4.908\times {10}^4$
MOEADSVR	$1.838\times {10}^5$ ± $2.365\times {10}^4$	$3.638\times {10}^5$ ± $4.085\times {10}^4$	$4.536\times {10}^5$ ± $5.036\times {10}^4$
DBCSA-II	$1.333\times {10}^5$ ± $2.195\times {10}^4$	$2.135\times {10}^5$ ± $3.661\times {10}^4$	$2.352\times {10}^5$ ± $\mathbf{3.631}\times {\mathbf{10}}^{\mathbf{4}}$

Note: Bold denotes the best result.

and

Table 8. MIGD results across different scenarios.

Algorithm	S1	S2	S3
MDCSATA	$\mathbf{8.010}\times {\mathbf{10}}^{\mathbf{1}}$ ± $\mathbf{2.285}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{8.342}\times {\mathbf{10}}^{\mathbf{1}}$ ± $\mathbf{1.521}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{1.297}\times {\mathbf{10}}^{\mathbf{2}}$ ± $\mathbf{2.699}\times {\mathbf{10}}^{\mathbf{1}}$
DMOAWPSO	$1.250\times {10}^2$ ± $4.043\times {10}^1$	$1.524\times {10}^2$ ± $2.829\times {10}^1$	$2.502\times {10}^2$ ± $4.628\times {10}^1$
DCTAEA	$1.178\times {10}^2$ ± $2.515\times {10}^1$	$9.580\times {10}^1$ ± $2.226\times {10}^1$	$1.579\times {10}^2$ ± $3.814\times {10}^1$
DCMPSO	$1.666\times {10}^2$ ± $3.715\times {10}^1$	$2.273\times {10}^2$ ± $4.228\times {10}^1$	$3.610\times {10}^2$ ± $5.793\times {10}^1$
DNSGA-II	$1.333\times {10}^2$ ± $3.668\times {10}^1$	$1.043\times {10}^2$ ± $2.068\times {10}^1$	$1.674\times {10}^2$ ± $4.802\times {10}^1$
SGEA	$1.429\times {10}^2$ ± $3.736\times {10}^1$	$1.432\times {10}^2$ ± $4.565\times {10}^1$	$2.042\times {10}^2$ ± $5.299\times {10}^1$
MOEADSVR	$2.022\times {10}^2$ ± $5.846\times {10}^1$	$2.018\times {10}^2$ ± $5.626\times {10}^1$	$2.963\times {10}^2$ ± $6.919\times {10}^1$
DBCSA-II	$3.133\times {10}^2$ ± $6.046\times {10}^1$	$4.296\times {10}^2$ ± $6.473\times {10}^1$	$6.218\times {10}^2$ ± $6.328\times {10}^1$

Note: Bold denotes the best result.

report the mean ± std of MHV and MIGD over 30 runs of all algorithms across the three scenarios. Statistical significance is assessed by the Wilcoxon rank-sum test, and the p-value results are presented in

Table 9. The p-value results of comparative Wilcoxon rank-sum test.

Scenario	Metric	DMOAWPSO	DCTAEA	DCMPSO	DNSGA-II	SGEA	MOEA/D-SVR	DBCSA-II
	MHV	$1.868\times {10}^{-05}$ **	$2.959\times {10}^{-05}$ **	$3.474\times {10}^{-10}$ **	$1.236\times {10}^{-03}$ **	$1.596\times {10}^{-07}$ **	$1.547\times {10}^{-09}$ **	$3.020\times {10}^{-11}$ **
S1	MIGD	$1.385\times {10}^{-06}$ **	$5.600\times {10}^{-07}$ **	$7.389\times {10}^{-11}$ **	$1.359\times {10}^{-07}$ **	$1.698\times {10}^{-08}$ **	$5.494\times {10}^{-11}$ **	$3.020\times {10}^{-11}$ **
	MHV	$1.695\times {10}^{-09}$ **	$4.643\times {10}^{-01}$	$3.020\times {10}^{-11}$ **	$4.218\times {10}^{-04}$ **	$3.592\times {10}^{-05}$ **	$3.825\times {10}^{-09}$ **	$3.020\times {10}^{-11}$ **
S2	MIGD	$4.504\times {10}^{-11}$ **	$2.068\times {10}^{-02}$ *	$3.020\times {10}^{-11}$ **	$3.006\times {10}^{-04}$ **	$1.254\times {10}^{-07}$ **	$8.993\times {10}^{-11}$ **	$3.020\times {10}^{-11}$ **
	MHV	$1.206\times {10}^{-10}$ **	$6.353\times {10}^{-02}$	$3.020\times {10}^{-11}$ **	$1.123\times {10}^{-02}$ *	$1.996\times {10}^{-05}$ **	$6.722\times {10}^{-10}$ **	$3.020\times {10}^{-11}$ **
S3	MIGD	$6.696\times {10}^{-11}$ **	$5.322\times {10}^{-03}$ **	$3.020\times {10}^{-11}$ **	$6.203\times {10}^{-04}$ **	$3.256\times {10}^{-07}$ **	$6.696\times {10}^{-11}$ **	$3.020\times {10}^{-11}$ **

Note: * and ** denotes the p-value is lower than the significance level 0.05 and 0.05/7 for Bonferroni correction, respectively.

.

MDCSATA achieves the best MHV and MIGD on all three scenarios, with relatively low standard deviations that indicate stable performance across independent runs. In S₁, all comparisons are highly significant after Bonferroni correction, with MDCSATA outperforming DCTAEA and DNSGA-II by 7.69% and 8.49% in MHV, and by 32.00% and 39.92% in MIGD, respectively. In S₂, DCTAEA is the only baseline whose MHV does not significantly differ from MDCSATA after Bonferroni correction, although MDCSATA still achieves a 0.93% improvement; DNSGA-II remains significantly worse in MHV. In MIGD, MDCSATA outperforms DCTAEA by 12.92% and DNSGA-II by 20.05%, demonstrating better preservation of front diversity. In S₃, MDCSATA’s MHV advantage over DCTAEA is not statistically significant, but the 17.89% MIGD improvement over DCTAEA is significant. The advantage over DNSGA-II in S₃ is significant for MHV at the 0.05 level and highly significant for MIGD. Across all settings, the only comparisons that fail to reach statistical significance are the MHV comparisons against DCTAEA in S₂ and S₃; every other pairwise test confirms the superiority of MDCSATA.

To further quantify the advantage, the per-algorithm improvement ratios on MHV and MIGD across the three scenarios are calculated. MDCSATA outperforms every baseline on both metrics in all scenarios. The largest gains are observed against DBCSA-II, where MHV improves by 71.00% in S₁, 108.54% in S₂, and 144.84% in S₃, while MIGD improves by 74.44%, 80.58%, and 79.14%, respectively. Against DCMPSO, MHV gains range from 20.55% in S₁ to 50.06% in S₃, and MIGD gains range from 51.93% to 64.08%. Even against the strongest competitors, MDCSATA maintains consistent advantages: MHV improvements over DCTAEA and DNSGA-II reach 0.93%–8.49%, and MIGD improvements reach 12.92–39.92%, with only two comparisons lacking statistical significance. These consistent margins across metrics and scenarios underscore the superiority of MDCSATA.

Among the baselines, DBCSA-II and DCMPSO exhibit the worst performance. DMOAWPSO, SGEA, and MOEA/D-SVR all rank consistently below MDCSATA, with significant gaps on both metrics. These results collectively validate that the tailored strategies of MDCSATA enable efficient task allocation across diverse dynamic scenarios.

Figure 2 and Figure 3 show the per-iteration HV and IGD convergence curves for all three scenarios. MDCSATA leads all algorithms from the earliest iterations, confirming that the objective-oriented heuristic initialization provides a well-informed starting population. It also converges faster within each environmental time step, reflecting the combined effect of the adaptive position update and the stagnation and elite guided perturbation. After each dynamic event, MDCSATA’s metric value recovers rapidly. This resilience is driven by the event-aware change response: the severity metric $\delta$ quantifies each event’s impact and adjusts the regeneration ratios accordingly. As a result, MDCSATA reaches a higher performance level within each problem environment faster than the other algorithms.

By contrast, DNSGA-II uses a fixed-ratio random injection, which fails to redirect the population efficiently after disruptions such as UAV crashes, leading to slower performance recovery. SGEA and MOEA/D-SVR rely on prediction-based reinitialization, which becomes inaccurate under compound events and incurs a high change-response cost. DBCSA-II and DCMPSO exhibit curves that differ from others, consistent with the results reported above.

5.2.2. Ablation Study

To examine the contribution of each proposed strategy, an incremental ablation study is conducted with all settings identical to those described in Section 5.1. The baseline MOCSA is a dynamic MOCSA core that includes a single-swarm CSA update, Pareto archive, and restart-based dynamic response. Starting from MOCSA, the proposed strategies are added sequentially. MOCSA_M introduces the violation-tolerant multi-swarm co-evolution mechanism; MOCSA_MH further adds objective-oriented heuristic initialization; MOCSA_MHA adds the adaptive position update strategy; MOCSA_MHAP adds stagnation and elite guided perturbation; and MDCSATA further activates the event-aware change response. The MHV and MIGD results are reported in

Table 10. MHV results of the ablation study.

Variant	S1	S2	S3
MOCSA	$1.838\times {10}^5$ ± $2.014\times {10}^4$	$2.861\times {10}^5$ ± $3.006\times {10}^4$	$3.283\times {10}^5$ ± $\mathbf{2.801}\times {\mathbf{10}}^{\mathbf{4}}$
MOCSAM	$1.982\times {10}^5$ ± $2.018\times {10}^4$	$3.415\times {10}^5$ ± $3.359\times {10}^4$	$3.565\times {10}^5$ ± $4.527\times {10}^4$
MOCSAMH	$2.267\times {10}^5$ ± $1.732\times {10}^4$	$4.097\times {10}^5$ ± $3.342\times {10}^4$	$5.052\times {10}^5$ ± $3.053\times {10}^4$
MOCSAMHA	$2.182\times {10}^5$ ± $\mathbf{1.524}\times {\mathbf{10}}^{\mathbf{4}}$	$4.270\times {10}^5$ ± $3.326\times {10}^4$	$5.536\times {10}^5$ ± $4.285\times {10}^4$
MOCSAMHAP	$2.274\times {10}^5$ ± $1.591\times {10}^4$	$\mathbf{4.461}\times {\mathbf{10}}^{\mathbf{5}}$ ± $3.371\times {10}^4$	$5.659\times {10}^5$ ± $4.691\times {10}^4$
MDCSATA	$\mathbf{2.279}\times {\mathbf{10}}^{\mathbf{5}}$ ± $1.626\times {10}^4$	$4.453\times {10}^5$ ± $\mathbf{2.491}\times {\mathbf{10}}^{\mathbf{4}}$	$\mathbf{5.759}\times {\mathbf{10}}^{\mathbf{5}}$ ± $4.453\times {10}^4$

Note: Bold denotes the best result.

and

Table 11. MIGD results of the ablation study.

Variant	S1	S2	S3
MOCSA	$1.811\times {10}^2$ ± $4.711\times {10}^1$	$2.843\times {10}^2$ ± $4.948\times {10}^1$	$4.473\times {10}^2$ ± $4.125\times {10}^1$
MOCSAM	$1.571\times {10}^2$ ± $4.044\times {10}^1$	$2.061\times {10}^2$ ± $4.345\times {10}^1$	$3.853\times {10}^2$ ± $5.465\times {10}^1$
MOCSAMH	$9.382\times {10}^1$ ± $3.128\times {10}^1$	$1.114\times {10}^2$ ± $2.203\times {10}^1$	$1.854\times {10}^2$ ± $2.736\times {10}^1$
MOCSAMHA	$9.882\times {10}^1$ ± $2.395\times {10}^1$	$9.929\times {10}^1$ ± $1.876\times {10}^1$	$1.482\times {10}^2$ ± $2.968\times {10}^1$
MOCSAMHAP	$8.174\times {10}^1$ ± $\mathbf{1.978}\times {\mathbf{10}}^{\mathbf{1}}$	$8.946\times {10}^1$ ± $1.535\times {10}^1$	$1.316\times {10}^2$ ± $2.799\times {10}^1$
MDCSATA	$\mathbf{8.010}\times {\mathbf{10}}^{\mathbf{1}}$ ± $2.285\times {10}^1$	$\mathbf{8.342}\times {\mathbf{10}}^{\mathbf{1}}$ ± $\mathbf{1.521}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{1.297}\times {\mathbf{10}}^{\mathbf{2}}$ ± $\mathbf{2.699}\times {\mathbf{10}}^{\mathbf{1}}$

Note: Bold denotes the best result.

.

The complete MDCSATA consistently improves over the dynamic MOCSA core by a large margin. Compared with MOCSA, MDCSATA increases MHV by 24.0%, 55.6%, and 75.4% in S₁, S₂, and S₃, respectively, while reducing MIGD by 55.8%, 70.7%, and 71.0%. These results confirm that the performance advantage of MDCSATA does not come merely from the problem-specific encoding, but from the proposed algorithmic strategies.

The first increment from MOCSA to MOCSA_M yields improvements in all scenarios, indicating that the violation-tolerant multi-swarm co-evolution mechanism helps preserve feasible search directions and front diversity under dynamic constraints. The most pronounced gain is observed from MOCSA_M to MOCSA_MH. After objective-oriented heuristic initialization is introduced, MHV increases by 14.4%, 20.0%, and 41.7%, and MIGD decreases by 40.3%, 45.9%, and 51.9% in the three scenarios. This demonstrates that initialization tailored to the remaining target value and makespan objectives provides high-quality starting solutions and the resulting PF approximation.

The adaptive position update further improves the performance in the scenarios. From MOCSA_MH to MOCSA_MHA, MHV increases by 4.2% in S₂ and 9.6% in S₃, and MIGD decreases by 10.9% and 20.1%, respectively. In S₁, MOCSA_MHA is slightly worse than MOCSA_MH, this may suggest that the smaller search space already benefits sufficiently from heuristic initialization and that additional exploratory movement may introduce minor fluctuations. The stagnation and elite guided perturbation in MOCSA_MHAP then improves both indicators in all scenarios, showing that local reallocation around stagnated and elite solutions is effective for refining PF approximations.

Finally, the event-aware change response provides smaller but generally favorable improvements over MOCSA_MHAP. MDCSATA obtains the best MIGD at all three scales and the best MHV in S₁ and S₃; its MHV in S₂ is only 0.2% lower than that of MOCSA_MHAP. This indicates that the event-aware change response mainly stabilizes adaptation across changes rather than serving as the dominant source of improvement.

Overall, the ablation results support that all proposed components contribute to the performance of MDCSATA, validating their effectiveness.

5.2.3. Parameter Sensitivity Analysis

To evaluate the robustness of MDCSATA to the CSA search control parameters, a full-factorial sensitivity analysis is performed using $AP\in \{0.05,0.20\}$, $fl\in \{1.5,2.5\}$, and $f{l}_{min}\in \{1.0,2.0\}$. The resulting eight combinations are evaluated under the same settings as in Section 5.1. This design examines both the marginal influence of each parameter and whether their effects depend on the other parameter settings. In addition, the sensitivity of the event-aware change response is examined using two fixed settings corresponding to the bounds of the adaptive rule: a conservative setting with $(R_{\mathrm{keep}},R_{\mathrm{heur}},R_{\mathrm{lhs}})=(0.6,0.2,0.2)$ and an aggressive setting with $(R_{\mathrm{keep}},R_{\mathrm{heur}},R_{\mathrm{lhs}})=(0.2,0.5,0.3)$. The MHV and MIGD results are shown in

Table 12. MHV results of the search parameter sensitivity analysis.

$(\mathit{AP},\mathit{fl},{\mathit{fl}}_{min})$	S1	S2	S3
$(0.20,1.5,1.0)$	$2.284\times {10}^5$ ± $1.504\times {10}^4$	$4.510\times {10}^5$ ± $2.740\times {10}^4$	$5.424\times {10}^5$ ± $4.416\times {10}^4$
$(0.20,1.5,2.0)$	$2.238\times {10}^5$ ± $1.661\times {10}^4$	$4.338\times {10}^5$ ± $3.191\times {10}^4$	$5.341\times {10}^5$ ± $4.493\times {10}^4$
$(0.20,2.5,1.0)$	$\mathbf{2.285}\times {\mathbf{10}}^{\mathbf{5}}$ ± $\mathbf{1.300}\times {\mathbf{10}}^{\mathbf{4}}$	$4.468\times {10}^5$ ± $2.326\times {10}^4$	$5.628\times {10}^5$ ± $5.138\times {10}^4$
$(0.20,2.5,2.0)$	$2.211\times {10}^5$ ± $1.520\times {10}^4$	$4.318\times {10}^5$ ± $2.378\times {10}^4$	$5.267\times {10}^5$ ± $4.068\times {10}^4$
$(0.05,1.5,1.0)$	$2.279\times {10}^5$ ± $1.626\times {10}^4$	$4.453\times {10}^5$ ± $2.491\times {10}^4$	$5.759\times {10}^5$ ± $4.453\times {10}^4$
$(0.05,1.5,2.0)$	$2.227\times {10}^5$ ± $1.517\times {10}^4$	$4.489\times {10}^5$ ± $3.080\times {10}^4$	$5.594\times {10}^5$ ± $\mathbf{3.996}\times {\mathbf{10}}^{\mathbf{4}}$
$(0.05,2.5,1.0)$	$2.229\times {10}^5$ ± $1.519\times {10}^4$	$\mathbf{4.529}\times {\mathbf{10}}^{\mathbf{5}}$ ± $\mathbf{2.235}\times {\mathbf{10}}^{\mathbf{4}}$	$\mathbf{5.821}\times {\mathbf{10}}^{\mathbf{5}}$ ± $4.011\times {10}^4$
$(0.05,2.5,2.0)$	$2.280\times {10}^5$ ± $1.505\times {10}^4$	$4.396\times {10}^5$ ± $2.655\times {10}^4$	$5.493\times {10}^5$ ± $5.173\times {10}^4$

Note: Bold denotes the best result.

,

Table 13. MIGD results of the search parameter sensitivity analysis.

$(\mathit{AP},\mathit{fl},{\mathit{fl}}_{min})$	S1	S2	S3
$(0.20,1.5,1.0)$	$8.177\times {10}^1$ ± $2.712\times {10}^1$	$8.316\times {10}^1$ ± $1.411\times {10}^1$	$1.519\times {10}^2$ ± $3.575\times {10}^1$
$(0.20,1.5,2.0)$	$8.711\times {10}^1$ ± $2.771\times {10}^1$	$9.437\times {10}^1$ ± $1.936\times {10}^1$	$1.721\times {10}^2$ ± $3.901\times {10}^1$
$(0.20,2.5,1.0)$	$8.394\times {10}^1$ ± $\mathbf{2.050}\times {\mathbf{10}}^{\mathbf{1}}$	$8.735\times {10}^1$ ± $1.626\times {10}^1$	$1.426\times {10}^2$ ± $3.554\times {10}^1$
$(0.20,2.5,2.0)$	$9.006\times {10}^1$ ± $2.584\times {10}^1$	$9.668\times {10}^1$ ± $2.072\times {10}^1$	$1.736\times {10}^2$ ± $3.091\times {10}^1$
$(0.05,1.5,1.0)$	$\mathbf{8.010}\times {\mathbf{10}}^{\mathbf{1}}$ ± $2.285\times {10}^1$	$8.342\times {10}^1$ ± $1.521\times {10}^1$	$1.297\times {10}^2$ ± $2.699\times {10}^1$
$(0.05,1.5,2.0)$	$8.877\times {10}^1$ ± $2.495\times {10}^1$	$8.497\times {10}^1$ ± $1.744\times {10}^1$	$1.448\times {10}^2$ ± $3.437\times {10}^1$
$(0.05,2.5,1.0)$	$8.324\times {10}^1$ ± $2.603\times {10}^1$	$\mathbf{7.986}\times {\mathbf{10}}^{\mathbf{1}}$ ± $\mathbf{1.278}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{1.252}\times {\mathbf{10}}^{\mathbf{2}}$ ± $\mathbf{2.500}\times {\mathbf{10}}^{\mathbf{1}}$
$(0.05,2.5,2.0)$	$8.812\times {10}^1$ ± $2.433\times {10}^1$	$8.797\times {10}^1$ ± $1.818\times {10}^1$	$1.524\times {10}^2$ ± $4.103\times {10}^1$

Note: Bold denotes the best result.

,

Table 14. MHV results of the event-response sensitivity analysis.

Response Setting	S1	S2	S3
Adaptive	$\mathbf{2.279}\times {\mathbf{10}}^{\mathbf{5}}$ ± $1.626\times {10}^4$	$4.453\times {10}^5$ ± $2.491\times {10}^4$	$\mathbf{5.759}\times {\mathbf{10}}^{\mathbf{5}}$ ± $\mathbf{4.453}\times {\mathbf{10}}^{\mathbf{4}}$
Conservative	$2.270\times {10}^5$ ± $\mathbf{1.588}\times {\mathbf{10}}^{\mathbf{4}}$	$4.435\times {10}^5$ ± $2.771\times {10}^4$	$5.692\times {10}^5$ ± $4.782\times {10}^4$
Aggressive	$2.278\times {10}^5$ ± $1.625\times {10}^4$	$\mathbf{4.463}\times {\mathbf{10}}^{\mathbf{5}}$ ± $\mathbf{2.476}\times {\mathbf{10}}^{\mathbf{4}}$	$5.736\times {10}^5$ ± $4.512\times {10}^4$

Note: Bold denotes the best result.

and

Table 15. MIGD results of the event-response sensitivity analysis.

Response Setting	S1	S2	S3
Adaptive	$\mathbf{8.010}\times {\mathbf{10}}^{\mathbf{1}}$ ± $\mathbf{2.285}\times {\mathbf{10}}^{\mathbf{1}}$	$8.342\times {10}^1$ ± $1.521\times {10}^1$	$\mathbf{1.297}\times {\mathbf{10}}^{\mathbf{2}}$ ± $\mathbf{2.699}\times {\mathbf{10}}^{\mathbf{1}}$
Conservative	$8.274\times {10}^1$ ± $2.402\times {10}^1$	$8.324\times {10}^1$ ± $1.832\times {10}^1$	$1.337\times {10}^2$ ± $3.110\times {10}^1$
Aggressive	$8.062\times {10}^1$ ± $2.341\times {10}^1$	$\mathbf{8.198}\times {\mathbf{10}}^{\mathbf{1}}$ ± $\mathbf{1.428}\times {\mathbf{10}}^{\mathbf{1}}$	$1.309\times {10}^2$ ± $3.021\times {10}^1$

Note: Bold denotes the best result.

.

The factorial results show that the effect of $AP$ becomes clearer as the problem scale increases. The marginal differences between $AP=0.05$ and $AP=0.20$ are negligible in S₁, whereas $AP=0.05$ improves average MHV by 1.3% and 4.6% and reduces average MIGD by 7.0% and 13.8% in S₂ and S₃, respectively. A large $AP$ triggers random relocation more frequently and may disrupt well-structured allocations in larger search spaces. The selected value $AP=0.05$ consequently provides the more reliable cross-scenario setting.

The $f{l}_{min}$ parameter has a more consistent influence. Averaged over $AP$ and $fl$, $f{l}_{min}=1.0$ improves MHV by 1.4%, 2.4%, and 4.3% and reduces MIGD by 7.1%, 8.3%, and 14.5% from S₁ to S₃. This suggests that an excessively large minimum movement can force substantial changes even when a crow should exploit a promising position. In contrast, the marginal influence of $fl$ is small: $fl=1.5$ is slightly better on average in S₁ and S₂, while $fl=2.5$ is slightly better in S₃. The default combination $(0.05,1.5,1.0)$ gives the best MIGD in S₁, whereas $(0.05,2.5,1.0)$ gives the best MHV and MIGD in S₂ and S₃. Thus, the default $fl=1.5$ is retained as a conservative cross-scenario setting, while a larger $fl$ can assist exploration in larger search spaces.

The event-response sensitivity results in Table 14 and Table 15 further show that the $\delta$-based adaptive response is stable. It obtains the best MHV and MIGD in S₁ and S₃, while the aggressive fixed response is slightly better in S₂. The small gaps indicate that no fixed response intensity dominates across all scenarios. The adaptive rule therefore provides a robust compromise by adjusting regeneration according to the estimated severity of each environmental change.

Overall, the sensitivity study supports the selected parameter configuration and shows that the adaptive position update and the $\delta$-based response remain stable under moderate parameter variations.

5.2.4. Preference-Weighted Solution Quality

Figure 4 presents the preference-weighted objective value of the executed solution at each environmental time step, selected under the decision-maker’s preference weights. Across all three scenarios, MDCSATA delivers the lowest or near-lowest weighted objective value at most time steps, reflecting well-distributed PFs.

DCTAEA and DNSGA-II are the most competitive baselines and occasionally edge ahead of MDCSATA at certain time steps, which is consistent with their MHV and MIGD results. In S₁, MDCSATA achieves a 3.56% improvement over DNSGA-II and a 9.05% improvement over DCTAEA across all time steps. In S₂, MDCSATA holds a 1.09% improvement over DCTAEA and a 1.81% improvement over DNSGA-II. In S₃, MDCSATA attains a 1.70% improvement over DNSGA-II and a 3.79% improvement over DCTAEA. This overall advantage indicates that MDCSATA sustains both front diversity and convergence more effectively as the problem environment evolves, whereas the competitors tend to lose solution quality.

SGEA also performs well among the algorithms and occasionally matches MDCSATA, but its solution quality fluctuates across time steps and rarely surpasses the top two. The remaining algorithms operate at larger disadvantages, suggesting that their PFs are either poorly converged or insufficiently diverse to provide strong candidates under the given preference weights. DMOAWPSO trails MDCSATA by 5.63% in S₁, 8.41% in S₂, and 9.53% in S₃. Against DBCSA-II, the weakest baseline, the improvement of MDCSATA reaches 24.05% in S₁, 28.58% in S₂, and 26.48% in S₃. These consistent margins indicate that MDCSATA not only produces high-quality PF approximations but also translates this advantage into superior executed-solution quality under shifting decision-maker preferences.

5.2.5. Task Allocation Visualization

Figure 5 presents the Gantt chart of MDCSATA’s execution schedule across the scenarios, and Figure 6, Figure 7, and Figure 8 show the per-time step spatial allocation maps for S₁, S₂, and S₃, respectively. Target colors are mapped from observational values using a cool-to-warm scale. For example, T₂ with value 52 appears blue, NT₁ with value 76 appears light gray, and T₁₀ with value 99 appears red.

These visualizations highlight several advantages of the proposed model formulation and encoding design. Three validity properties are consistently maintained across all dynamic events. First, ongoing tasks are never interrupted, as the random-key encoding locks in-progress assignments to the executing UAV, preventing mid-task reassignment. Second, completed targets are never re-allocated, because the variable-length encoding grows monotonically as new targets appear while skipping completed targets when decoding. Third, newly appeared targets are integrated within the time steps, appearing immediately in the schedule of the following re-optimization phase. These properties are guaranteed by design, without relying on post-processing repair. The spatial allocation maps further illustrate the workload redistribution behavior under dynamic events. After a UAV crash, the surviving UAVs absorb the remaining unassigned targets with minimal makespan increase, driven by the $f_2$-biased heuristic initialization that seeds the swarms with load-balanced assignments. After a sensor damage event, the affected UAV’s reduced sensing capability ${\eta }_i\left(t\right)$ lowers its contribution to $f_1$, causing high-value targets to be naturally shifted to UAVs with better sensing capability. These adaptive behaviors emerge directly from the objective formulation and MDCSATA’s framework, requiring no post-processing rules.

5.2.6. Runtime Efficiency

Table 16. Average run time of algorithms (unit: seconds).

Scenario	MDCSATA	DMOAWPSO	DCTAEA	DCMPSO	DNSGA-II	SGEA	MOEA/D-SVR	DBCSA-II
S1	0.414	1.480	2.139	0.291	1.665	19.813	1.449	0.289
S2	0.792	2.823	3.824	0.565	2.773	31.271	3.080	0.574
S3	0.941	2.826	3.521	0.683	2.999	28.156	3.429	0.641

Note: Results in bold font represent the least average runtime.

gives the average runtime of all algorithms. Benefiting from the inherently simple search mechanism of CSA, both DBCSA-II and MDCSATA require very short optimization time. Three algorithms—DBCSA-II, DCMPSO, and MDCSATA—form a distinctly faster group across all scenarios. Among them, DBCSA-II and DCMPSO rank first or second, yet both exhibit severely degraded MHV and MIGD; in S₃, DBCSA-II fails to maintain meaningful PF tracking. MDCSATA ranks third in speed, with its additional overhead relative to the other two attributable to the integrated strategies, which is consistent with the complexity analysis in Section 4.7. Despite this, MDCSATA achieves the best MHV and MIGD, striking a favorable balance between solution quality and efficiency. The remaining algorithms require obviously more runtime than the above group, with SGEA incurring overhead roughly 30 times that of MDCSATA, yet their allocation quality does not exceed that of MDCSATA. These results indicate that MDCSATA provides a highly effective trade-off between optimization performance and computational cost, making it well-suited for time-critical operations where both accuracy and responsiveness are essential.

In summary, MDCSATA consistently surpasses all seven comparison algorithms across the three scenarios on both MHV and MIGD, while maintaining third runtime. The advantage confirms that the integrated strategies deliver rapid PF tracking in the dynamically changing problem environment, which validates MDCSATA’s effectiveness in multi-UAV DTA for time-critical missions.

6. Conclusions

This paper tackled cooperative task allocation for UAV swarms in dynamic scenarios. Existing methods often fail to maintain consistency between allocation decisions and actual operational states, consider only limited classes of dynamic events, and still leave room for performance improvement. To overcome these, we formulated a DMOP with a state inheritance mechanism that continuously synchronizes allocation with actual UAV and target states. A comprehensive event model encompassing platform failures, sensor degradation, target dynamics, weather deterioration, and decision-maker preference shifts was built across three observation scenarios.

To solve this DMOP, we proposed MDCSATA. The algorithm employs a random-key encoding that maps discrete assignments to a continuous space and enforces task continuity through variable designs. Five tailored strategies are integrated: a violation-tolerant multi-swarm co-evolution preserves feasibility and diversity; objective-oriented heuristic initialization seeds the population toward each objective; an adaptive position update strengthens exploration and exploitation; a stagnation and elite guided perturbation intensifies local exploitation; and an event-aware change response quantifies severity and adaptively regenerates solutions for rapid response to dynamic events.

Extensive experiments against seven state-of-the-art DMOEAs across three scenarios show that MDCSATA achieves the best MHV and MIGD in all settings, with statistically significant improvements in 53 out of 56 pairwise comparisons. Moreover, MDCSATA maintains an acceptable runtime overhead; therefore, it is well suited for time-critical missions.

Future work can explore several directions. First, the current allocation-level model represents UAVs as point masses with a common cruise speed and Euclidean flight segments. These assumptions may produce optimistic estimates of the mission makespan, residual energy, and flight risk. Future work will couple task allocation with trajectory planning to incorporate platform-specific speeds, turning-radius constraints, obstacle avoidance, inter-UAV safe distances, detailed flight-energy models, and spatial risk fields describing threat distributions, terrain risks, and target-area risks. Their effects on makespan, energy feasibility, and survival probability will then be quantitatively evaluated. Second, extending MDCSATA to distributed coordination under intermittent communication would address practical deployment scenarios where a centralized control center is unavailable. Third, physical experiments on real UAV platforms would further validate our approach and help bridge the gap between algorithm development and real-world deployment.