A risk-aware improved grey wolf optimizer algorithm for multi-UAV task allocation in spatially constrained environments

2.1 Mission model and assignment representation

Consider a heterogeneous fleet $$ \mathcal{U}=\{1, \ldots, M\} $$ of $$ M $$ UAVs and a set $$ \mathcal{T}=\{1, \ldots, N\} $$ of spatially distributed tasks. The mission environment is a square of side length $$ S $$, and a single base is located at $$ \mathit{\boldsymbol{p}}_b=(S/2, 0) $$ on the lower boundary. Each task $$ j\in\mathcal{T} $$ is described by its position $$ \mathit{\boldsymbol{p}}_j\in[0, S]^2 $$, reward $$ r_j > 0 $$, payload demand $$ q_j > 0 $$, and latest service time $$ \tau_j $$. For tasks without a deadline, $$ \tau_j=\infty $$. Each UAV $$ i\in\mathcal{U} $$ is characterized by cruise speed $$ v_i $$, payload capacity $$ Q_i $$, and an endurance-distance budget $$ L_i $$. Data-driven regression and inference methods have been used to model UAV battery-discharge behavior^[34]. In the present allocation model, these dynamics are abstracted into the maximum allowable flight-distance budget $$ L_i $$.

The environment contains three types of circular zones. A zone $$ z $$ is specified by center $$ \mathit{\boldsymbol{c}}_z $$ and radius $$ \rho_z $$. The sets $$ \mathcal{Z}^{r} $$, $$ \mathcal{Z}^{c} $$, and $$ \mathcal{Z}^{o} $$ denote risk-prone zones, weak-communication zones, and obstacle zones, respectively. Risk-prone and weak-communication zones are modeled as traversable soft-cost regions because they represent increased mission exposure or degraded link reliability rather than absolute no-fly restrictions. Physical obstacles have a different interpretation. In the default formulation, obstacle intersections receive a high finite penalty and are therefore strongly discouraged but not mathematically prohibited. A separate hard-obstacle mode treats an intersecting served segment as infeasible.

The formulation is based on five assumptions. First, each flight route is a straight segment traversed at constant speed $$ v_i $$, so the time to reach a task equals the cumulative outbound service distance divided by $$ v_i $$. Second, each task can be executed by at most one UAV. Third, payload consumption and flight distance accumulate along the route, and service duration is negligible relative to flight time. Fourth, the default endurance model considers the cumulative outbound service distance from the base. The return-guaranteed mode instead requires $$ d_i+\Delta_{\mathrm{prev}, j}+\Delta_{j b}\leq L_i $$ before accepting task $$ j $$, thereby reserving sufficient range to return to base; Section 4.9 evaluates this mode. Finally, all risk, communication, and obstacle maps are static and known. Principal notation is summarized in Table 1.

The decision variable is an assignment vector $$ \mathit{\boldsymbol{x}}=(x_1, \ldots, x_N) $$, with $$ x_j\in\mathcal{U} $$. For population-based real-valued optimization, each individual is represented by a continuous vector $$ \mathit{\boldsymbol{z}}\in[\ell, u]^N $$, where $$ \ell=1 $$ and $$ u=M $$. Rounding and clipping are commonly used when continuous optimization operators are applied to integer-valued decision variables^[35]. The decoding rule is

where $$\operatorname{clip}(\cdot, 1, M)$$ restricts the rounded value to $$[1, M]$$. This mapping places all population-based optimizers in a common Euclidean search space, while the fitness function evaluates decoded task assignments. The mapping is many-to-one: multiple continuous vectors may decode to the same integer assignment, creating flat regions for continuous position updates. Although this representation is retained to provide a common search interface for the continuous baselines, RA-IGWO does not rely exclusively on continuous perturbations because its local-refinement operator acts directly in the integer assignment space. The greedy seed solution in Section 3.2 is encoded into the $$[\ell, u]^N$$ space and subjected to uniform random jitter with an amplitude of 0.3. Boundary clipping is then applied to keep every coordinate within $$[\ell, u]$$.

Motivated by threat-perception-based UAV path planning, candidate paths are evaluated using predefined exposure and feasibility costs^[36]. Several geometric quantities that depend only on the static environment can therefore be precomputed. The augmented node set is $$ \mathcal{V}=\mathcal{T}\cup\{b\} $$, where $$ |\mathcal{V}|=N+1 $$. The pairwise distance matrix $$ \mathit{\boldsymbol{\Delta}}\in\mathbb{R}^{(N+1)\times(N+1)} $$ stores Euclidean distances $$ \Delta_{uv}=\lVert\mathit{\boldsymbol{p}}_u-\mathit{\boldsymbol{p}}_v\rVert_2 $$. The risk-overlap matrix $$ \mathit{\boldsymbol{R}} $$ and obstacle-intersection matrix $$ \mathit{\boldsymbol{H}} $$ are constructed simultaneously.

To calculate the elementwise values of $$ \mathit{\boldsymbol{R}} $$ and $$ \mathit{\boldsymbol{H}} $$, consider the flight segment connecting nodes $$ u $$ and $$ v $$. Let $$ \mathit{\boldsymbol{d}}=\mathit{\boldsymbol{p}}_v-\mathit{\boldsymbol{p}}_u $$ and $$ \mathit{\boldsymbol{f}}=\mathit{\boldsymbol{p}}_u-\mathit{\boldsymbol{c}}_z $$. If $$ \lVert\mathit{\boldsymbol{p}}_v-\mathit{\boldsymbol{p}}_u\rVert_2=0 $$, the overlap is defined as zero. Otherwise, substituting $$ \mathit{\boldsymbol{p}}(\lambda)=\mathit{\boldsymbol{p}}_u+\lambda\mathit{\boldsymbol{d}} $$ into the circle equation gives $$ A_q\lambda^2+B_q\lambda+C_q=0 $$, where $$ A_q=\mathit{\boldsymbol{d}}^{\top}\mathit{\boldsymbol{d}} $$, $$ B_q=2\mathit{\boldsymbol{f}}^{\top}\mathit{\boldsymbol{d}} $$, and $$ C_q=\mathit{\boldsymbol{f}}^{\top}\mathit{\boldsymbol{f}}-\rho_z^2 $$. If the discriminant is negative, the overlap is zero. Otherwise, the two roots are clamped to $$ [0, 1] $$ and sorted as $$ \lambda_1\leq\lambda_2 $$. The segment length within zone $$ z $$ is

Aggregating over zones gives

Weak communication represents degradation in link reliability rather than a direct physical hazard. Its overlap coefficient is therefore set to one half of the unit coefficient assigned to the risk-zone term. The quantity $$ H_{uv} $$ records the number of obstacle regions intersected by segment $$ (u, v) $$. The default scalar objective uses a high but finite obstacle penalty, so obstacle crossings are strongly penalized but not strictly prohibited.

Constructing $$ \mathit{\boldsymbol{R}} $$ and $$ \mathit{\boldsymbol{H}} $$ requires $$ \mathcal{O}\big(N^2(|\mathcal{Z}^{r}|+|\mathcal{Z}^{c}|+|\mathcal{Z}^{o}|)\big) $$ operations, while constructing $$ \mathit{\boldsymbol{\Delta}} $$ requires $$ \mathcal{O}(N^2) $$ time. After preprocessing, retrieving a matrix value requires $$ \mathcal{O}(1) $$ time per flight segment. A complete fitness evaluation then performs matrix lookups proportional to the number of decoded flight segments. This preprocessing avoids repeated geometric intersection calculations and supports comparison under a fixed fitness-evaluation budget.

2.2 Partial-completion service, risk-aware objective, and fitness evaluation

For an allocation vector $$ \mathit{\boldsymbol{x}} $$, let $$\mathcal{T}_i=\{j: x_j=i\}$$ denote the tasks assigned to UAV $$i$$. Because $$ \mathit{\boldsymbol{x}} $$ specifies task ownership but not execution order, a route-generation rule is required before fitness evaluation. The default decoder constructs the visit sequence $$\pi_i$$ by nearest-neighbor ordering: starting from base $$ b $$, UAV $$ i $$ repeatedly selects the unvisited assigned task with the smallest entry in the distance matrix $$ \mathit{\boldsymbol{\Delta}} $$.

For the decoder-sensitivity analysis, the alternative risk-aware ordering selects the next task according to

where $$ u $$ is the current node, $$ \mathcal{T}^{\mathrm{rem}}_i $$ is the set of remaining tasks assigned to UAV $$ i $$, $$ s_D=\max\{S, 1\} $$, $$ s_R=\max\{1, \max_{p, q}R_{pq}\} $$, and $$ s_r=\max\{1, \max_j r_j\} $$. The numerical coefficients in Eq. 4 are fixed across all algorithms and instances, whereas the scale factors $$ s_D $$, $$ s_R $$, and $$ s_r $$ are computed from each instance. If multiple tasks attain the same minimum score, the tie is resolved in favor of the smaller task index. Risk-stop and risk-skip use the same ordering and differ only in their treatment of an infeasible task. The prefix-stop policy represents a conservative pre-committed offline execution assumption and is not intended to model adaptive in-mission rescheduling. Its sensitivity is examined in Section 4.5.

Given the visiting order $$\pi_i$$, UAV $$i$$ proceeds to serve the tasks sequentially while continuously updating its cumulative payload $$W_i$$ and cumulative flight distance $$d_i$$. Specifically, the $$k$$-th task in the sequence, denoted by $$\pi_i^k$$, is accepted only if both the payload and endurance constraints remain satisfied after traversing the next flight segment:

If either condition is violated, the UAV terminates its service, and all subsequent tasks in $$\pi_i$$ are marked as unserved under the default service-prefix policy. Consequently, the served tasks form a prefix $$\mathcal{S}_i\subseteq\mathcal{T}_i$$. This assumption may be conservative when an executable low-demand task follows an infeasible task; the decoder-sensitivity analysis therefore also evaluates skip-and-continue execution. Under skip-and-continue execution, the infeasible task is marked as unserved but the remaining ordered tasks continue to be examined. A skipped task does not update the previous node, cumulative payload, cumulative flight distance, or elapsed time. Consequently, the next task is evaluated from the last successfully served node, and only an accepted task updates the UAV state.

Compared with an all-or-nothing feasibility rule, prefix evaluation preserves credit for tasks completed before the first payload or endurance violation. Under an all-or-nothing rule, a small downstream violation can invalidate an entire assigned route even when several preceding tasks remain physically executable. Therefore, the present model scores the feasible prefix $$\mathcal{S}_i$$ and applies the missed-task penalty only to unserved tasks. This choice yields a more informative evaluation of the executable service. Because downstream tasks are not reconsidered under the default stopping policy, this evaluation remains conservative.

The overall objective function jointly accounts for travel efficiency, environmental exposure, time-window violations, workload balance, and task reward. For each flight segment served within the feasible prefix $$\mathcal{S}_i$$, the associated distance, risk overlap, and obstacle intersections are queried directly from the precomputed matrices $$\{\mathit{\boldsymbol{\Delta}}, \mathit{\boldsymbol{R}}, \mathit{\boldsymbol{H}}\}$$. $$T_j$$ denotes the arrival time at the served task $$j$$, calculated as the cumulative outbound distance of the assigned UAV along its feasible route divided by its constant speed. The individual cost and reward components are defined as

where $$ n_{\mathrm{obs}}=\sum_{\text{served flight segments}}H_{uv} $$ counts obstacle intersections over served flight segments. $$ R_{uv} $$ is defined in Eq. (3); $$ C_{\mathrm{risk}} $$ includes risk-zone exposure and the weighted weak-communication overlap. For a task without a deadline, $$ \tau_j=\infty $$, and its contribution to $$ C_{\mathrm{time}} $$ is interpreted as zero. The balance term $$ C_{\mathrm{bal}} $$ measures workload imbalance in terms of the outbound service distances of the UAVs.

Equations (2)-(3) and (5)-(10) are derived in this study from segment-circle geometry and the stated mission assumptions. Equation (1) follows the cited round-and-clip encoding practice.

Following the standard principle of decomposing multiple criteria into a scalar optimization objective^[37], the objective to be minimized is

The fixed weights are $$ \alpha {\; =\; } 1 $$, $$ \beta\; {=}\; 2 $$, $$ \gamma\; { = }\; 3 $$, $$ \delta\; {=}\; 0.5 $$, $$ \eta{\; =\; }1 $$, $$ \kappa{\; =\; }80 $$, and $$ \kappa_o{\; =\; }500 $$. The negative sign on the task reward $$ R_{\mathrm{task}} $$ converts maximizing the reward into minimizing it. The risk and delay weights are set higher than the distance weight because risk exposure and delayed service represent mission degradation beyond travel length. The missed-task penalty is set higher than the reward of a single task to prevent a solver from sacrificing many tasks only to shorten distance or reduce exposure. The obstacle crossing penalty is a high-cost but non-locking term; a single crossing is more expensive than a moderate detour or reassignment, but the search space remains connected under the soft formulation. Because these terms have different numerical scales, Section 4.6 evaluates weight sensitivity and normalized-objective robustness. Fitness values obtained with different weight sets are interpreted as sensitivity outcomes.

In the experimental scenarios, it may not be possible to achieve 100% completion when the total task demand exceeds the total payload capacity of the UAV swarm. Therefore, a payload-only upper bound on the number of serviceable tasks is obtained by selecting tasks in ascending order of demand:

where $$q_{(1)}\le q_{(2)}\le\cdots$$ are the task demands sorted in ascending order. This theoretical ceiling $$\rho_{\mathrm{cap}}$$ represents the maximum fraction of tasks that can be served if payload were the sole constraint. However, in practice, endurance limitations, time-window requirements, and spatial penalties will degrade the realistically achievable service rate. Therefore, when a solver approaches $$\rho_{\mathrm{cap}}$$, the remaining gap to this theoretical ceiling is more informative than the gap to the ideal $$100%$$ completion rate.

Algorithm 1 summarizes the fitness evaluation. Each call to the objective function increments the global counter $$g$$ by one. Thus, every evaluated candidate consumes one NFE, regardless of whether it is accepted. Route construction dominates the per-evaluation cost. In the worst case, its complexity is $$O(\sum_i |\mathcal{T}_i|^2)$$, and this sum is bounded by $$O(N^2)$$ because $$\sum_i |\mathcal{T}_i|^2 \le (\sum_i |\mathcal{T}_i|)^2 = N^2$$. When the assignment is balanced, the cost is approximately $$O(N^2/M)$$. The subsequent feasibility sweep requires $$O(N)$$ operations. Since all algorithms rely on the same evaluator, NFE provides a common measure of objective-evaluation effort.

3.1 Standard GWO update and quadratic convergence schedule

In standard GWO^[21], the population is ranked by fitness, and the three best agents are denoted by $$ \mathit{\boldsymbol{z}}_{\alpha} $$, $$ \mathit{\boldsymbol{z}}_{\beta} $$, and $$ \mathit{\boldsymbol{z}}_{\delta} $$. For each leader $$ h\in\{\alpha, \beta, \delta\} $$, two random vectors $$ \mathit{\boldsymbol{r}}_{1, h}, \mathit{\boldsymbol{r}}_{2, h}\sim\mathrm{Unif}(0, 1)^N $$ are sampled independently. The corresponding coefficient vectors are $$ \mathit{\boldsymbol{A}}_{h}=2a\mathit{\boldsymbol{r}}_{1, h}-a $$ and $$ \mathit{\boldsymbol{C}}_{h}=2\mathit{\boldsymbol{r}}_{2, h} $$. For a current search agent $$ \mathit{\boldsymbol{z}} $$, the candidate position directed by leader $$ \mathit{\boldsymbol{z}}_{h} $$ is

where $$ \odot $$ and $$ |\cdot| $$ denote elementwise multiplication and elementwise absolute value, respectively. The coefficient pairs $$ (\mathit{\boldsymbol{A}}_{\alpha}, \mathit{\boldsymbol{C}}_{\alpha}) $$, $$ (\mathit{\boldsymbol{A}}_{\beta}, \mathit{\boldsymbol{C}}_{\beta}) $$, and $$ (\mathit{\boldsymbol{A}}_{\delta}, \mathit{\boldsymbol{C}}_{\delta}) $$ are generated independently, matching the implementation in which each leader call draws new random vectors. The final clipping operation implements the boundary handling used by all GWO-based methods.

Since each component of $$ \mathit{\boldsymbol{r}}_{1, h} $$ lies in $$ [0, 1] $$, every component of $$ \mathit{\boldsymbol{A}}_{h} $$ lies in $$ [-a, a] $$. Consequently, $$ |A_{h, k}| > 1 $$, which permits a candidate to move away from a leader, is possible only when $$ a > 1 $$. Conversely, $$ |A_{h, k}| < 1 $$ favors movement toward the corresponding leader.

In standard GWO, $$ a $$ decreases linearly from 2 to 0^[21]. Using normalized evaluation progress $$ p=g/G $$, the linear schedule is $$ a_{\mathrm{lin}}(p)=2(1-p) $$. It reaches $$ a=1 $$ at $$ p=0.5 $$, after which the condition $$ |A_{h, k}| > 1 $$ is no longer possible. This threshold does not eliminate stochastic movement because $$ \mathit{\boldsymbol{C}}_{h} $$ remains random; it only marks the end of the interval in which the magnitude of $$ \mathit{\boldsymbol{A}}_{h} $$ can exceed one. RA-IGWO replaces the linear decay with the quadratic schedule

while retaining the three-leader update in Eq. (13). For $$ 0 < p < 1 $$, $$ a(p)=2(1-p^2) > a_{\mathrm{lin}}(p) $$, so the quadratic schedule retains a larger coefficient range at the same normalized NFE progress. Moreover, $$ a(p) > 1 $$ is equivalent to $$ p < 1/\sqrt{2}\approx0.707 $$. Therefore, the interval during which $$ |A_{h, k}| > 1 $$ is possible increases from $$ 0.5G $$ to approximately $$ 0.707G $$, an absolute increase of $$ 0.207G $$ and a relative increase of $$ \sqrt{2}-1\approx41.4\% $$. The magnitude of its slope increases with $$ p $$, while the convergence factor itself reaches zero at the terminal budget $$ G $$. This schedule is designed to retain exploration-capable updates for a larger fraction of the NFE budget.

3.2 Capacity-aware greedy initialization, OBL, and local refinement

The remaining components are designed around the decoded assignment structure. During initialization, $$\lfloor P/3\rfloor$$ agents are generated using a capacity-aware greedy heuristic. The first greedy pass assigns tasks in descending order of demand; the subsequent passes employ randomized task sequences to maintain population diversity. For each task, the heuristic selects the feasible UAV with the largest remaining payload capacity. If no UAV has sufficient capacity to meet the task requirements, the task will be assigned to the UAV with the largest remaining capacity, thereby providing a near-feasible load distribution for subsequent evolutionary optimization. The obtained integer allocation is then mapped back to the continuous search space $$[\ell, u]^N$$, and a small amount of uniform perturbation is added. This process provides feasible or near-feasible load distributions for a subset of individuals in the population while ensuring that the initial individuals are not completely identical.

OBL^[29] is used to expand the sampling area of the continuous search space. The opposite solution of a proxy $$ \mathit{\boldsymbol{z}} $$ is defined as

After evaluating the opposite candidates at initialization, the best $$ P $$ agents are retained from the merged original and opposite populations. During the early search stage, when $$ g/G < 0.4 $$, generalized OBL is triggered with probability $$ p_{\mathrm{j}} $$. Its coordinate-wise lower and upper bounds are obtained from the current population, and every reflected candidate is clipped to $$ [\ell, u]^N $$. Because UAV labels have no inherent ordinal meaning, OBL is performed in the common continuous encoding space and evaluated after decoding.

The alpha wolf is further refined by first-improvement search in the decoded integer space. The search samples three neighborhood types: load-aware transfer, reassignment, and two-task swapping. The load-aware transfer moves a task from the most capacity-constrained UAV to a UAV with larger remaining capacity to reduce service loss caused by payload limits, while the other operations maintain neighborhood diversity. Each sampled local-refinement candidate is evaluated by Algorithm 1 and consumes one NFE regardless of whether it improves the incumbent solution. This rule prevents local search from receiving an uncounted evaluation advantage over non-memetic baselines.

3.3 Overall RA-IGWO procedure and evaluation-budget accounting

Algorithm 2 summarizes the complete optimizer. Greedy seeding, OBL refresh, GWO position updates, and assignment-level local refinement all follow the same NFE accounting rule. After local refinement, the alpha solution is written back to the population before the leaders are updated again; otherwise, the refined assignment could be lost during the next population ranking.

4.1 Experimental setup and fairness protocol

All methods follow one evaluation protocol. Table 2 lists all settings. Every RA-IGWO candidate produced by a GWO update, OBL refresh, or local refinement consumes one NFE; the final diagnostic call reports metrics only. Reference configurations are used for GA^[9,10], PSO^[38,39,11], GWO^[21], I-GWO^[24], HPSOGWO^[31,32], and UAV multi-task ALNS^[40]. D-ALNS adapts the latter framework to ownership-only decisions. Its random, worst-contribution, overload-oriented, and risk-oriented removal operators are paired with greedy, regret-2, and capacity-balanced repair, adaptive operator weights, and simulated-annealing acceptance. Each repaired assignment consumes one NFE. GWO uses $$ a: 2\rightarrow0 $$. PSO uses $$ w=0.70, c_1=c_2=1.50 $$, close to stability-oriented literature coefficients rather than a claimed optimum, while GA uses the predeclared $$ p_c=0.80, p_m=0.10 $$.

With $$ P=30 < G=4, 000 $$, initial population evaluations are counted first, and every later evaluator call checks the remaining budget. Population and OBL batches are truncated at exhaustion, and local refinement stops at $$ g=G $$. Unchanged local moves are discarded before evaluation; every changed candidate consumes one NFE regardless of acceptance. D-ALNS removal, repair, insertion scoring, and weight updates use assignment data and precomputed matrices; each repaired assignment invokes the full evaluator once. Thus, all seven methods terminate at $$ g=4, 000 $$ without exceeding the budget.

Runtime covers optimizer construction and search. Environment generation, geometric preprocessing, output, visualization, and the post-run diagnostic call are excluded, yielding implementation-level comparisons in a common environment.

Table 3 examines GA, PSO, and GWO parameters under the common population and NFE settings; it is a sensitivity analysis, not independent optimization. Mean fitness spans 2, 198.69-3, 500.42 for GA, 2, 019.49-2, 436.36 for PSO, and 2, 055.63-2, 414.42 for GWO, so the predeclared settings are treated as reproducible rather than optimal. The strongest observed configurations are then tested on 20 new environments with five paired seeds. I-GWO and HPSOGWO retain their reference implementations. In Table 4, RA-IGWO obtains mean fitness 1, 688.91 $$ vs $$. 2, 305.16, 2, 177.37, 2, 293.82, 2, 500.96, and 2, 453.13 for GA, PSO, GWO, I-GWO, and HPSOGWO, respectively, with all FDR-adjusted $$ q < 0.0001 $$.

4.2 Representative main comparison

Table 5 gives one reproducible 4000-NFE run to illustrate distance, risk, completion, balance, and runtime jointly.

RA-IGWO has the lowest fitness and highest or tied-highest completion in all three representative runs. Figure 1 visualizes the large-scale metric decomposition. Because one run is descriptive rather than inferential, robustness is assessed below using independent instances, confidence intervals, effect sizes, and sensitivity tests.

Figure 1. Large-scale representative metric comparison under the shared 4000-NFE budget. Blue bars denote baseline algorithms, and orange bars denote RA-IGWO. Bold value labels denote the best result in each panel, namely the lowest fitness and risk cost and the highest completion rate.

4.3 Convergence behavior

Figure 2 plots best-so-far fitness against NFE for all three representative scales. Capacity-aware seeding structures initial allocations, OBL broadens sampling, and the quadratic schedule prolongs the interval with $$ a > 1 $$ from $$ g/G < 1/2 $$ to $$ g/G < 1/\sqrt{2} $$. Local search changes decoded ownership directly and can repair capacity-related service loss that continuous moves leave unchanged. The curves describe the combined implementation; separate component effects and interactions are evaluated in Section 4.8.

Figure 2. Convergence curves under the shared NFE budget for small, medium, and large-scale scenarios.

4.4 Obstacle-model validation

Because a finite penalty can permit crossings, Table 6 reports mean intersections over 20 environments and five seeds per method and instance. RA-IGWO ranks first in the medium and large cases and second in the small case. These results quantify penalized soft-obstacle behavior and must not be interpreted as strict avoidance.

Table 7 enforces hard obstacles. A task $$ j $$ is rejected before state update when the prospective segment from the last served node $$ u $$ satisfies $$ H_{uj} > 0 $$. Under the tested prefix-stop policy, this terminates only that UAV route and places the blocked and downstream tasks in $$ C_{\mathrm{miss}} $$ without assigning infinite fitness to the complete candidate; already served tasks remain valid. With skip-and-continue, which is not used in this table, only the blocked task would be skipped from an unchanged previous node. All reported crossings are therefore zero. RA-IGWO has the lowest mean fitness and highest completion at every scale, so its population-baseline advantage is not attributable to permissive crossings.

Table 8 varies $$ \kappa_o $$ from 100 to 2, 000 over 20 medium-scale environments. RA-IGWO remains below the strongest population baseline, PSO, at every setting, with all FDR-adjusted comparisons significant. Its mean crossings fall from 0.66 to 0.24 while completion remains within 81.9%-82.8%, indicating stable ranking across the tested finite penalties.

4.5 Decoder sensitivity

Table 9 applies four decoders to all six population methods over 20 medium-scale instances and five seeds. Risk-aware ordering reduces RA-IGWO fitness from 1, 766.47 to 1, 600.78 and eliminates obstacle crossings. For RA-IGWO, skip-and-continue increases completion from 84.8% to 85.3% under nearest-neighbor ordering and leaves completion unchanged at 85.3% under risk-aware ordering. For PSO, completion changes slightly from 78.1% to 78.0% and from 80.7% to 80.3%, respectively. The completion effect of skipping is small and method-dependent. PSO is the strongest population-based baseline for each decoder, while RA-IGWO retains lower fitness than all five population-based baselines under every decoder, with all 20 joint FDR-adjusted comparisons significant ($$ q\leq0.0003 $$). Thus, the ranking is not confined to nearest-neighbor prefix stopping.

4.6 Objective-scale, weight, and normalization sensitivity

Objective-scale robustness is examined through empirical term magnitudes, common weight profiles, and normalization. Across 2, 000 random medium-scale assignments, median absolute weighted contributions for distance, risk, delay, imbalance, reward, missed tasks, and obstacles are 1, 051.25, 727.50, 84.85, 19.82, 196.74, 1, 040.00, and 3, 000.00. The dominant missed-task and obstacle terms motivate the checks below. Because each profile defines a different scalar objective, raw fitness is compared within a common profile.

Table 10 reports objective-weight sensitivity. The three-profile comparison and separate five-level $$ \beta $$ sweep used distinct Monte Carlo batches. Each used 20 independently generated medium-scale instances with five algorithm seeds per instance, giving 100 runs per setting, but the instance and algorithm-seed blocks differed between the two experiments. Although the $$ \beta=0.5 $$ and $$ \beta=8 $$ weight vectors are identical in the two experiments, their results use different instance and algorithm-seed sets and need not coincide numerically. Both show the same completion-risk trade-off: Table 10 reports a risk reduction from 203.70 to 80.71 and a completion reduction from 84.0% to 77.2%, but the separate sweep reports changes from 206.31 to 83.40 and from 85.1% to 78.6%. RA-IGWO remains lower in fitness than all five population-based baselines under every common profile.

For normalized-objective testing, distance, risk, delay, balance, reward, missed-task, and obstacle terms are divided by scales $$ SN $$, $$ N\max_{uv}R_{uv} $$, the largest finite deadline, $$ S $$, $$ \sum_j r_j $$, $$ N $$, and $$ N\max\{|\mathcal{Z}^{o}|, 1\} $$, respectively; the original weights and signs are retained, and each scale is lower-bounded by one. The delay scale is

where tasks with infinite deadlines are excluded and contribute zero delay. If no finite deadline exists, the inner maximum is defined as zero, giving $$ s_{\mathrm{time}}=1 $$ and an identically zero delay term. All instance-specific scales are shared by the compared methods. In Table 11, RA-IGWO is below all population baselines after FDR correction, while its difference from D-ALNS is not significant ($$ q=0.4524 $$), indicating comparable normalized-objective performance.

4.7 Multi-instance generalization and statistical evidence

Each Small-20, Medium-50, and Large-100 setting uses 20 independently generated instances and five seeds per method and instance. The environment is the statistical unit: seeds are averaged within an instance before paired inference. For each environment and seed index, the same integer seed is reset before every algorithm. Table 12 reports the strongest population comparator, D-ALNS, and RA-IGWO; Table 13 covers all seven methods with Benjamini-Hochberg correction jointly applied across 18 comparisons.

Relative to D-ALNS, RA-IGWO reduces mean fitness by 332.78, 87.23, and 78.01 at the three scales, with significant FDR-adjusted signed-rank tests. However, for Large-100, the bootstrap CI $$ [-0.75, 143.83] $$ includes zero, so interval and rank evidence differ, and the large-scale advantage should be interpreted cautiously. D-ALNS has the highest small-case completion and the fewest large-case crossings. RA-IGWO has fewer crossings at the other scales and is 2.7-4.9 times faster. Its small-case fitness advantage mainly reflects lower obstacle-intersection penalties despite lower completion. Therefore, the results indicate that lower default-objective values, not universal dominance across completion, crossings, and runtime.

Table 14 shows scale-dependent hard-obstacle results: D-ALNS is significantly better for Small-30 and Medium-50, but RA-IGWO is better for Large-80. Under normalization they are statistically indistinguishable. Thus, RA-IGWO remains superior to the five population baselines under hard obstacles but not uniformly to the discrete baseline. Separate medium-scale D-ALNS sensitivity checks vary destroy fraction, reaction factor, and cooling rate, and none differs significantly from the predeclared configuration after FDR correction. Figure 3 displays all seven methods across the three multi-instance scales.

Figure 3. Mean fitness over independently generated spatial instances.

4.8 Component ablation

Table 15 uses 20 medium instances and isolates each component. Removing local refinement or accept-only updates raises fitness by 740.46 and 480.55, respectively, with FDR significance. Greedy initialization has mixed evidence: $$ q=0.0551 $$, but its bootstrap CI excludes zero. OBL has no significant independent effect ($$ q=0.0826 $$), with a CI containing zero. Removing the quadratic schedule yields a small significant improvement ($$ \Delta F=-23.90 $$, 95%CI $$ [-45.64, -1.44] $$, $$ q=0.0399 $$). Accordingly, it is retained only as the pre-specified exploration policy.

The interaction study likewise shows no quadratic-schedule gain: fitness changes from 2, 514.06 to 2, 539.65 without local refinement and from 1, 764.72 to 1, 780.56 with it. Accordingly, only local refinement and elitist acceptance receive clear support; greedy evidence is mixed and OBL is independently non-significant.

4.9 Return-to-base validation

Excluding return travel can overestimate endurance feasibility. Table 16 and Figure 4 therefore compare outbound-only and return-guaranteed evaluation for all six population methods. RA-IGWO has the lowest mean fitness and highest completion in both modes, preserving its population-baseline ranking under the more conservative endurance model.

Figure 4. Effect of adding return-to-base feasibility to the endurance evaluation.

4.10 Visualization, practical relevance, and limitations

Figure 5 shows representative allocations and routes and should be read with the intersection statistics; under the soft model, crossings are penalized and reported rather than forbidden. The model addresses offline low-altitude allocation with mixed obstacles, communication degradation, and limited resources^[18,19], focusing on risk-aware ownership, route-prefix feasibility, and unified-NFE evaluation rather than communication-centric power allocation.

Figure 5. RA-IGWO task allocation and routes for the three scenarios. The black box marked "H" denotes the common home base (takeoff and landing pad).

Future work will consider dynamically changing maps, online replanning, adaptive learning of objective weights, and comprehensive automatic tuning under equal budgets.