Abstract
In disaster-relief logistics, disrupted ground transportation networks and the limited payload and endurance of UAVs make it difficult to allocate emergency materials across geographically dispersed demand points. This study formulates a capacity-constrained bi-objective multi-UAV material distribution problem that simultaneously minimizes total flight distance and workload imbalance. Each demand point is served by one UAV in a single-trip route, and route feasibility is evaluated under payload-capacity and maximum route-distance constraints. To solve this constrained discrete optimization problem, we propose an immune-enhanced NSGA-II algorithm, referred to as INSGA-II. The algorithm represents each solution using an integer assignment vector and a priority vector for route decoding, and integrates immune cloning, stimulation-guided clone allocation, and a linearly decreasing mutation probability to improve the exploration of non-dominated allocation-routing solutions. Comparative experiments were conducted over 30 independent runs against standard NSGA-II, MOEA/D, Weighted-GA, and Weighted-ACO using hypervolume (HV), inverted generational distance (IGD), and runtime as evaluation metrics. INSGA-II achieved the highest mean HV of 0.895 and the lowest mean IGD of 0.184 among the compared algorithms, showing favorable average Pareto-front approximation performance in the tested scenario. Its average runtime was higher than NSGA-II and MOEA/D due to additional immune and repair operations, but lower than Weighted-GA and Weighted-ACO in the tested setting. The results suggest that INSGA-II provides quality-oriented Pareto trade-off solutions for capacity-constrained multi-UAV disaster-relief material distribution.
1 Introduction
In recent years, natural disasters such as floods, earthquakes, landslides, and wildfires have frequently damaged road networks, reduced ground-transport accessibility, and delayed emergency logistics and post-disaster reconstruction (Wang et al., 2025; Rojas Trejos et al., 2023). In such post-disaster environments, unmanned aerial vehicles (UAVs) provide a flexible and rapidly deployable aerial delivery platform. Owing to their maneuverability, low personnel risk, and ability to access areas where ground vehicles may be delayed or blocked, UAVs have been increasingly considered for emergency reconnaissance, medical supply delivery, search and rescue support, and temporary communication assistance.
However, disaster-relief material distribution using UAVs remains a challenging optimization problem. A single UAV is usually limited by payload capacity, battery endurance, and maximum flight range, making it difficult to satisfy multiple spatially distributed demand points in large-scale emergency scenarios. Multi-UAV collaboration can improve delivery capability, but it also introduces new decision-making challenges, including how to assign demand points to different UAVs, how to determine the visiting sequence of each UAV, and how to balance route efficiency with workload equity across the fleet.
Traditional path planning and scheduling methods have been widely used to improve UAV route generation and task execution. For example, curve-based path planning methods can improve path smoothness and obstacle avoidance performance (Wang et al., 2022), while swarm-intelligence-based scheduling methods can enhance search efficiency and coordination ability in complex task environments (Meng et al., 2024). Nevertheless, disaster-relief delivery usually involves multiple conflicting objectives. A route with a shorter total distance may lead to unbalanced UAV workloads, whereas a more balanced allocation may require longer travel distance. Scalarized single-objective methods usually require predefined weights and provide one preferred solution under a given weight setting. To obtain different trade-off solutions, multiple weight configurations or repeated runs are generally required.
To improve multi-objective search and decision flexibility, different optimization mechanisms have been explored in recent studies. Decomposition-based strategies can divide complex multi-objective problems into subproblems to improve scalability (Yu, 2025). Particle-swarm-based multi-objective methods have been used to enhance global search capability in complex optimization tasks (Singh et al., 2025). Hybrid meta-heuristic mechanisms further combine the advantages of different search operators to improve solution exploration (Abualigah, 2024). In addition, learning-based adaptive planning methods have shown potential for dynamic route decision-making in changing environments (Wang and Hu, 2023). These studies indicate that combining multi-objective optimization with adaptive search mechanisms is promising for complex UAV coordination problems.
Nevertheless, in post-disaster UAV material distribution, it remains necessary to jointly model material demand quantity, UAV payload capacity, route-distance constraints, total flight distance, workload imbalance, and discrete task-allocation decisions. A Pareto-based optimization framework is therefore suitable for providing multiple trade-off solutions for decision-makers under different emergency priorities.
To address this problem, this paper formulates a capacity-constrained bi-objective multi-UAV material distribution model. The model considers heterogeneous demand quantities, UAV payload capacity, maximum route distance, total flight distance, and workload imbalance. Based on this formulation, an immune-enhanced NSGA-II algorithm, referred to as INSGA-II, is developed. The proposed method combines integer task-assignment encoding, priority-based route decoding, immune cloning, stimulation-guided clone allocation, and dynamic mutation probability to explore high-quality non-dominated task-allocation and routing solutions in a constrained discrete search space.
The main contributions of this article include:
A capacity-constrained bi-objective multi-UAV material distribution model is formulated by jointly considering heterogeneous demand quantities, UAV payload capacity, maximum flight range, total flight distance, and workload imbalance.
An integer chromosome encoding and route-decoding strategy is designed to map each task-assignment solution into complete single-trip UAV delivery routes. The feasibility of each decoded route is further evaluated according to payload and flight-range constraints.
An immune-enhanced NSGA-II algorithm is developed by integrating immune cloning, stimulation-guided clone allocation, and dynamic mutation into the evolutionary search process, enabling the algorithm to explore high-quality non-dominated task-allocation and routing solutions.
Extensive comparative experiments are conducted against NSGA-II, MOEA/D, Weighted-GA, and Weighted-ACO to evaluate the optimization performance of the proposed method in terms of HV, IGD, runtime, convergence behavior, and routing results.
The structure of this article is as follows. Section 2 reviews related studies on UAV path planning, task collaboration, and multi-objective optimization. Section 3 introduces the proposed INSGA-II method and elaborates on its principle and implementation in detail. Section 4 presents the experimental results and compares the proposed method with the benchmark method. Finally, Section 5 summarizes the full text and discusses the potential development directions in the future.
2 Related work
2.1 Path planning
Path planning is a fundamental component of multi-UAV collaborative distribution, inspection and monitoring tasks. It aims to generate feasible flight trajectories while reducing flight distance, energy consumption, task completion time, and operational risk. For complex and high-dimensional environments, various graph-search, heuristic, and hybrid optimization methods have been developed (Gao et al., 2024). improved the traditional A* algorithm by integrating weighted heuristic search, node reduction, and cubic B-spline smoothing, thereby improving path continuity and search efficiency (Song et al., 2020). proposed an improved sparse A* three-dimensional path search method by incorporating environmental and task information into the evaluation function (Lindqvist et al., 2024). designed a tree-based next-best trajectory method for three-dimensional UAV exploration, balancing trajectory cost and exploration benefit. In disaster-related or constrained environments, hybrid heuristic methods have also been widely used to improve path-search efficiency and robustness (Dong et al., 2024). proposed an improved ant colony–genetic hybrid algorithm for dynamic evacuation route planning in dam-break flood scenarios, which is relevant to emergency route search under damaged transportation conditions (Wu et al., 2018). combined particle swarm optimization, the Metropolis criterion, and RTS smoothing for path planning of UAVs/AUVs, demonstrating that hybrid search mechanisms can help avoid local optima in complex path-planning problems. In terms of regional coverage and route coordination (Cho et al., 2021), proposed a multi-UAV coverage path generation strategy for maritime search and rescue based on hexagonal grid decomposition and mixed-integer linear programming (Chen et al., 2022). combined linear programming and ant colony optimization to construct a heterogeneous multi-UAV coverage path model, while (Li et al., 2023) investigated exact and heuristic Dubins coverage path planning for known environments.
Overall, these studies provide important foundations for UAV trajectory generation, coverage routing, and search efficiency improvement. However, most of them primarily focus on geometric path feasibility, coverage efficiency, or route smoothness. They do not fully address emergency material distribution problems in which each demand point has a specific demand quantity and each UAV is constrained by payload capacity and route distance. Therefore, path planning methods alone are insufficient for modelling capacity-constrained post-disaster material delivery.
2.2 Task collaboration
Task allocation and collaborative scheduling determine how missions or demand points are assigned to different UAVs. In multi-UAV systems, task allocation is often coupled with energy constraints, time windows, vehicle heterogeneity, communication limitations, and route feasibility (Zhou et al., 2020). investigated energy-aware task scheduling under real-time and task-priority constraints, showing the importance of constraint-aware scheduling in heterogeneous systems (Wang et al., 2023). considered dynamic factors such as UAV performance differences, task area size, and time-window constraints, and used the Contract Net Protocol to allocate tasks in multi-UAV cooperative reconnaissance (Li et al., 2022). optimized three-dimensional UAV trajectories from an energy-efficiency perspective, while (Liu et al., 2024) used deep reinforcement learning for three-dimensional trajectory planning of cellular-connected UAVs (Li et al., 2025). introduced reinforcement learning into logistics task allocation, providing useful insights for intelligent scheduling in goods-transportation scenarios.
In emergency logistics settings, accessibility loss and road-network disruption directly affect the feasibility of ground-based delivery and increase the need for aerial material distribution. In this context, UAV task allocation should not only determine which UAV serves which point, but also account for the demand quantity of each point, the payload capacity of each UAV, the maximum route distance, and the resulting workload distribution across the fleet. Existing studies on UAV task collaboration have made progress in allocation strategies, trajectory planning, and intelligent scheduling, but many of them still emphasize reconnaissance, coverage, communication, or general task execution. Compared with these settings, emergency material distribution requires a stronger coupling between demand magnitude, vehicle capacity, route length, and decision flexibility. This motivates the capacity-constrained task-allocation and route-decoding model developed in this study.
2.3 Multi-objective optimization
Multi-objective optimization provides a natural framework for multi-UAV coordination because practical missions often require simultaneous consideration of route efficiency, energy consumption, task completion, safety, and workload equity. Classical multi-objective evolutionary algorithms such as NSGA-II (Verma et al., 2021), MOEA/D (Shen et al., 2016), and SPEA2 (Jiang et al., 2022) have been widely used to approximate Pareto fronts and generate multiple trade-off solutions for decision-makers. In UAV-related applications (Yunchong et al., 2025), considered constraints such as payload, task time window, task sequencing and range, and developed an improved NSGA-II-based method for heterogeneous UAV task assignment (Gao et al., 2018). applied an improved multi-objective particle swarm optimization method to multi-UAV task allocation (Yafei and Liang, 2023). proposed an adaptive angular-region division strategy to improve the global search ability of multi-objective particle swarm optimization for multi-UAV task assignment (Ruan and Duan, 2020). developed a multi-objective social-learning pigeon-inspired optimization method for UAV formation obstacle avoidance (Gao et al., 2022). further proposed a unified multi-objective optimization framework for UAV cooperative task assignment and reassignment.
Beyond UAV systems, multi-objective optimization has also been used in other scheduling and operation problems where multiple conflicting objectives must be balanced (Hong et al., 2024). Applied multi-objective genetic algorithms to reservoir collaborative operations to balance hydropower generation, water supply and ecological benefits. Such studies indicate that Pareto-based methods are suitable for decision scenarios where no single objective can fully represent operational performance. For emergency UAV distribution, minimizing total flight distance and reducing workload imbalance are potentially conflicting objectives: a shorter route may concentrate tasks on a few UAVs, while a more balanced workload may require longer total travel distance. Therefore, a Pareto-based optimization framework is more appropriate than a fixed-weight single-objective formulation.
Nevertheless, existing multi-objective UAV studies still leave three gaps for disaster-relief material distribution. First, many works focus on coverage, reconnaissance, obstacle avoidance, or general task allocation, whereas fewer explicitly model material demand quantities together with UAV payload and route-distance constraints. Second, scalarized single-objective baselines are sensitive to weight settings and may hide the trade-off structure between distance and workload imbalance. Third, standard multi-objective algorithms may not sufficiently explore local variants of promising allocation structures in a constrained discrete search space. To address these gaps, this study formulates a capacity-constrained bi-objective UAV material distribution model and develops an immune-enhanced NSGA-II algorithm with integer assignment encoding, priority-based route decoding, immune cloning, stimulation-guided clone allocation, and dynamic mutation probability.
3 Materials and methods
This study addresses a capacitated multi-UAV material distribution problem in a post-disaster emergency logistics scenario. A single warehouse is used as the dispatch depot, and a set of geographically dispersed demand points require emergency supplies with heterogeneous demand quantities. Each UAV is constrained by its maximum payload capacity and a predefined maximum closed-route distance. The optimization task is to assign demand points to UAVs and determine the corresponding visiting sequence for each UAV, while minimizing the total flight distance and reducing workload imbalance across the UAV fleet.
To make the model consistent with the implemented chromosome structure and routing results, the following assumptions are adopted. First, all UAVs depart from the warehouse and return to the same warehouse after completing their assigned delivery tasks. Second, each demand point is served by exactly one UAV, and split delivery among multiple UAVs is not considered in the current model. Third, each UAV performs one closed delivery route in one planning cycle; multiple sorties are not included in the current formulation. Fourth, the total demand assigned to each UAV must not exceed its payload capacity, and the decoded route length must not exceed its maximum flight range. Under these assumptions, the problem is formulated as a static, single-depot, single-sortie, non-splittable multi-UAV delivery optimization problem.
To solve the above bi-objective optimization problem, this study proposes an immune-enhanced version of NSGA-II, referred to as INSGA-II. Standard NSGA-II selects non-dominated solutions through Pareto ranking and crowding-distance comparison. However, during iterative evolution, promising individuals may still be lost or insufficiently explored in local search regions. Therefore, INSGA-II introduces immune-inspired cloning, stimulation-guided clone, crossover, and dynamic mutation into the NSGA-II framework. As shown in Figure 1, the proposed framework contains demand-point initialization, chromosome encoding, route decoding, objective-function evaluation, immune operations, non-dominated sorting, and Pareto-solution output.
FIGURE 1
3.1 Mathematical formulation
This study establishes a bi-objective optimization model that simultaneously minimizes the total flight distance and the workload imbalance among UAVs. Let M denote the number of UAVs and denote the set of demand tasks. Each task corresponds to a demand point with a projected Cartesian position and a strictly positive demand quantity , where . The local coordinate system is centered at the warehouse depot, such that . The other node coordinates satisfy and are not restricted to the positive quadrant. All projected coordinates and route distances are expressed in kilometers. The task allocation result is denoted as {}, where represents the task subset assigned to UAV m, for .
For the m-th UAV, the decoded route starts from the warehouse , visits the demand points in according to the route decoding rule described in Section 3.2, and finally returns to the warehouse. The corresponding coordinate sequence is denoted as , where . The route distance of the m-th UAV is defined as follows:
The summation contains route segments, including the segment from the depot to the first demand point, the segments between consecutive demand points, and the final return segment from the last demand point to the depot. Where denotes the position of the node visited at route position . The distance between two points is calculated as:and is expressed in kilometers. The total flight-distance objective is defined as:
The second objective is workload imbalance. The total load assigned to the m-th UAV is calculated as:
The average load of all UAVs is:
Where is the total number of UAVs, and is the total load of the m-th UAV.
The workload imbalance objective is then defined as:
Because is fixed across experiments, normalization by would only rescale without affecting solution rankings or Pareto dominance; therefore, the unnormalized formulation is adopted.
In addition to the two objective functions, the following feasibility constraints are considered. Let be a binary assignment variable. If demand point i is assigned to , then ; otherwise, . Each demand point must be assigned to exactly one UAV:
The total demand assigned to each UAV must not exceed its maximum payload capacity:
The decoded route length of each UAV must not exceed its maximum route distance:
Where denotes the maximum payload capacity of . The model permits UAV-specific capacities , whereas the tested fleet is homogeneous. denotes the maximum allowable closed-route distance of , expressed in kilometers, denotes the demand quantity of demand point i, denotes the total load assigned to , and denotes its decoded closed-route length in the same unit.
A smaller indicates a more balanced distribution of delivery workload among UAVs. The squared-deviation formulation emphasizes large workload imbalances, discourages workload concentration on individual UAVs, and provides a variance-based measure of fleet-wide workload dispersion. In disaster-relief logistics, load balance is meaningful because excessive concentration of payload or route burden on a single UAV may increase battery depletion risk and reduce fleet-level resilience. However, load balance is not always the dominant objective. For time-critical emergency delivery, a solution with shorter flight distance and larger load imbalance may still be preferred. Therefore, and are optimized simultaneously to provide decision-makers with multiple Pareto trade-off solutions.
3.2 Chromosome encoding and route decoding
To represent the assignment relationship between demand points and UAVs, this study employs an integer chromosome encoding scheme. The assignment vector is defined aswhere is the zero-based code of the UAV assigned to task . Its relationship with is
Each chromosome consists of N genes, where N is the total number of demand points. Each gene corresponds to a demand point, and its integer value represents the UAV assigned to serve that point. Each assignment gene takes an integer value in , where is the total number of UAVs. The chromosome encoding process is illustrated in Figure 2.
FIGURE 2
The integer entries indicate UAV assignment, and the associated priority values are used to decode the visiting order within each UAV route. For example, the chromosome [0,1,2,0,1,2] indicates that demand points 1 and 4 are assigned to UAV 0, demand points 2 and 5 are assigned to UAV 1, and demand points 3 and 6 are assigned to UAV 2. Therefore, the chromosome first determines the task subset of each UAV.
Since a solution contains both task assignment and route-priority information, a route decoding procedure is required to determine the visiting sequence of the assigned demand points. For each UAV, the assigned demand points are extracted from the chromosome. Starting from the warehouse , the UAV visits the assigned demand points according to the decoding rule and then returns to the warehouse. In this study, Each chromosome is represented as , where is the assignment vector and is the priority vector used to determine the relative visiting order of demand points assigned to the same UAV. The assignment vector determines which UAV serves each demand point, while the priority vector determines the relative visiting order of demand points assigned to the same UAV. During route decoding, the demand points assigned to each UAV are sorted according to their priority values, and the sorted sequence is used as the visiting order of the corresponding UAV route. Under normal route decoding, the priority vector is the sole criterion used to determine the visiting order. A nearest-neighbor ordering procedure is invoked only during feasibility repair when the inherited priority order results in a route-distance violation. This decoding process converts each chromosome into a complete set of closed UAV routes, enabling the calculation of , , , and .
After decoding, a two-stage repair procedure is applied. In the payload-repair stage, the UAV with the largest current load is selected as the source UAV. Its assigned demand points are examined in descending order of demand, while candidate destination UAVs are examined in ascending order of current load. The first transfer satisfying the destination payload constraint is accepted. This procedure is repeated until all UAV loads satisfy their payload limits.
After payload repair, the routes decoded from the inherited priority values are evaluated. If a route-distance violation exists, the visiting sequences are first reconstructed using a nearest-neighbor rule. If reordering alone does not eliminate the violation, the UAV with the longest route is selected as the source UAV. Its demand points are examined in descending order of their Euclidean distance from the depot, while candidate destination UAVs are examined in ascending order of current load. The first transfer satisfying the destination payload constraint is accepted. After each transfer, the corresponding routes are rebuilt and all payload and route-distance constraints are re-evaluated. The procedure terminates when all constraints are satisfied or when no admissible transfer can be found.
Algorithm 1
Input: Solution x = (a, π), demands di, capacities Qm, route limits Rm, number of points N, number of UAVs M
Output: Repaired solution x or FAILURE
1 Calculate UAV loads
2 while any > do
3 s ← UAV with the largest load
4 ← first feasible pair, with points of s ordered by decreasing demand and destination UAVs by increasing load
5 if no feasible pair exists then return FAILURE
6 Reassign from s to m and update and
7 end while
8 Decode the routes; if all , return x
9 Rebuild the routes by nearest-neighbor ordering
10 for t = 1 to N × M do//task–UAV search-space cap
if all then return x
11 s ← the UAV with the longest route
12 ←first feasible task–destination-UAV pair, with points of s ordered by decreasing depot distance and destinations by increasing load
13 if no feasible pair exists then return FAILURE
14 Reassign from s to m; rebuild and re-evaluate the routes
15 end for
16 return x if all constraints are satisfied; otherwise return FAILURE
The route-distance repair loop is capped at NM iterations. Since each of the N demand points has at most M-1 alternative destination UAVs, the number of distinct reassignment candidates is N (M-1)<NM; hence, NM provides a conservative termination safeguard. Let s and m denote the source and destination UAVs, respectively. A transfer pair is feasible if , , and . After the transfer, the loads are updated as and . Ties between UAVs are resolved by the UAV index. During payload repair, ties between points with equal demand are resolved using a seeded random tie-breaker. During route-distance repair, ties between points with equal depot distance are resolved by the demand-point index. The seeded tie-breaker ensures reproducibility across repeated runs. If no admissible reassignment can be identified, the solution is marked as infeasible and excluded from the final non-dominated solution set.
3.3 Immune operations
To improve the exploration ability of standard NSGA-II, this study introduces immune-inspired operations into the evolutionary process. As shown in Figure 3, the immune operations include immune cloning, crossover, mutation, stimulation evaluation, and stimulation-guided clone allocation. Different from a single scalar fitness value, the quality of an individual in INSGA-II is evaluated according to both objective performance and distribution characteristics in the population.
FIGURE 3
Immune cloning is applied to parent individuals located in the first two Pareto fronts. Each parent represents a complete task-assignment and route-priority chromosome. The original parent population is retained in the candidate pool, while additional copies of the eligible parents are generated and subjected to the generation-dependent mutation operator. This process locally explores promising assignment structures without discarding their original versions.
Before clone-derived candidates are generated, a stimulation score is calculated for each eligible parent individual according to its Pareto rank and crowding distance. Let denote the i-th eligible parent. Its stimulation score is defined as follows:
Where denotes the zero-based Pareto rank of , as determined by non-dominated sorting, and denotes its crowding distance. Only individuals satisfying , corresponding to the first two Pareto fronts, are eligible for immune cloning. The indicator function equals 1 when the specified condition is satisfied and 0 otherwise. The crowding-distance threshold is fixed at in experiments, while boundary individuals with automatically receive the diversity bonus.
The number of clone-derived candidates generated from is determined by:where is the number of clone-derived candidates generated from parent , and is the clone factor. The rank-dependent term assigns a higher stimulation score to individuals in better Pareto fronts, whereas the crowding-distance term provides an additional cloning opportunity for boundary or relatively sparsely distributed individuals. Therefore, the stimulation score controls the allocation of cloning effort rather than directly determining whether an individual survives to the next-generation.
Standard offspring are generated through crossover and mutation, whereas clone-derived candidates are generated as copies of eligible parents and subjected to generation-dependent mutation, as shown in Figure 4. Two crossover points are randomly selected, and the gene segment between them is exchanged between two parent chromosomes. This operation recombines different task-assignment structures. For mutation, each selected assignment gene is reassigned to another valid UAV code in . Since each gene always takes a valid UAV index, the mutated chromosome remains a valid task-assignment representation. Its feasibility with respect to payload and flight-range constraints is further checked during objective-function evaluation.
FIGURE 4
The stimulation score determines the number of clone-derived candidates. The parent population, standard crossover–mutation offspring, and mutated clone-derived candidates are combined into one candidate population. Non-dominated sorting and crowding-distance comparison are then applied to retain individuals for the next-generation. Pareto rank guides convergence, and crowding distance together with stimulation-based cloning helps preserve solutions near the boundary and in sparsely populated regions of the Pareto front.
3.4 Dynamic mutation probability
The dynamic mutation probability is designed to balance global exploration and local exploitation during the evolutionary process. In the early generations, a relatively high mutation probability is used to explore different UAV-task assignment structures and route-priority orders. As the generation number increases, the mutation probability gradually decreases to preserve high-quality solution structures and improve local refinement. The mutation probability at generation (t) is defined as:where = 0.30 is the initial mutation probability, = 0.05 is the final mutation probability, is the current generation index, and is the maximum number of generations. Therefore, decreases linearly from 0.30 to 0.05 during evolution. A linear decay schedule is adopted because it provides a simple and stable balance between early-stage exploration and late-stage exploitation in combinatorial evolutionary search.
The mutation operator is applied to both the assignment vector and the priority vector. For the assignment vector, a selected demand point may be reassigned to another valid UAV index. For the priority vector, two priority values may be exchanged to modify the visiting order within the decoded UAV route. After mutation, the chromosome is repaired and evaluated according to the payload and route-distance constraints. If no feasible repair can be found, the chromosome is treated as infeasible and is not retained in the final non-dominated solution set.
3.5 Overall algorithm procedure
The overall INSGA-II procedure consists of six main steps. First, an initial population is generated using integer chromosome encoding, after which each chromosome is decoded, repaired when necessary, and evaluated using the two objective functions. Second, non-dominated sorting and crowding-distance calculation are performed on the current population. Third, standard offspring are generated through binary-tournament selection, crossover, and generation-dependent mutation. Fourth, individuals in the first two Pareto fronts are identified as immune-cloning parents, and their stimulation scores and clone numbers are calculated. Fifth, the resulting clone copies undergo generation-dependent mutation, repair, and evaluation. Finally, the parent population, standard offspring, and clone-derived candidates are combined and subjected to environmental selection. This process continues until the maximum number of generations is reached, after which the final non-dominated solution set is returned.
The overall process of the INSGA-II algorithm is as follows:
Algorithm 2
Input: demand point set T, UAV number M, population size , maximum generation G, initial mutation probability , final mutation probability , clone factor
Output: Pareto-optimal task assignment set
1 Initialize population using integer chromosome encoding
2 Decode, repair, and evaluate all individuals in
4 Set t = 0
5 while (t < G) do
6 Perform non-dominated sorting and calculate crowding distances for
7 Calculate the generation-dependent mutation probability
8 Generate standard offspring through binary-tournament selection, crossover, and mutation using
9 Decode, repair, and evaluate all individuals in
10 Identify immune-cloning parents
11 for each do
12 Calculate its stimulation score and clone number
13 Generate clone-derived candidates and mutate them using
14 end for
15 Decode, repair, and evaluate all clone-derived candidates
16 Form the combined population
17 Perform non-dominated sorting and crowding-distance calculation on
18 Select the best individuals from Ut to form
19
20 end while
21 Return the final non-dominated solution set
4 Experiments and results
4.1 Study area and data acquisition
The spatial data consist of the latitude and longitude coordinates of the warehouse and demand points. To facilitate route-distance calculation, these geographic coordinates were converted into a depot-centered local two-dimensional Cartesian coordinate system. Let denote the latitude and longitude of the warehouse and those of demand point i. The local Cartesian coordinates of demand point i were calculated aswhere all angular coordinates are expressed in radians and R = 6371 km. Accordingly, the warehouse is located at (0,0), and route distances are calculated in kilometers. Table 1 lists six representative points from the complete instance.
TABLE 1
| Point | Demand | Position | Point | Demand | Position |
|---|---|---|---|---|---|
| 1 | 32 | (30.5508, 114.3226) | 4 | 18 | (30.5089, 114.3756) |
| 2 | 47 | (30.5662, 114.3458) | 5 | 41 | (30.4837, 114.3668) |
| 3 | 23 | (30.5407, 114.3524) | 6 | 36 | (30.5126, 114.4145) |
Representative demand points and geographic coordinates.
The geographic coordinates reported in Table 1 were converted using the above equations, and the resulting local Cartesian coordinates were used as spatial inputs to the optimization code.
The main simulation scenario parameters and algorithmic hyperparameters are summarized in Table 2. The scenario parameters are taken from the implemented single-trip UAV delivery instance, including the number of UAVs, demand quantity, payload capacity, maximum route distance, delivery mode, and route-decoding rule. In the route representation, the assignment vector determines the UAV assigned to each demand point, while the priority vector determines the visiting order of demand points within each UAV route. The algorithmic hyperparameters were selected based on preliminary trial runs and commonly used settings in evolutionary optimization.
TABLE 2
| Category | Parameter | Value |
|---|---|---|
| Scenario | Number of demand points | 30 |
| Number of UAVs | 3 | |
| Depot location | (0.0, 0.0) | |
| Demand range | 15–49 | |
| Total demand | 930 | |
| Maximum payload capacity per UAV | 400 | |
| Maximum route distance per UAV | 10 km | |
| Delivery mode | Non-splittable | |
| Sortie setting | Single-trip | |
| Replenishment | Not allowed | |
| Route decoding | Priority-based sorting | |
| Algorithm | Population size | 100 |
| Evolutionary generations | 300 | |
| Independent runs | 30 | |
| INSGA-II | Initial mutation rate | 0.30 |
| Final mutation rate | 0.05 | |
| Clone factor | 2 | |
| NSGA-II/MOEA/D | Mutation rate | 0.15 |
| Weighted baselines | Scalarization weights | 0.1–0.9 |
| Objective scales | (16.67 km,3000.0) |
Simulation scenario and algorithm parameter settings.
Although the UAV can travel up to 30 km without payload, a conservative maximum closed-route distance of 10 km, corresponding to the full-payload condition, is adopted in the current model. This limit is applied independently to the decoded closed route of each UAV.
During route decoding, demand points assigned to the same UAV are sorted according to their priority values to form a closed route from the depot to the assigned demand points and back to the depot. The same population size, generation number, and independent-run setting were used for INSGA-II, NSGA-II, MOEA/D, Weighted-GA, and Weighted-ACO where applicable.
4.2 Reformulation of benchmarking framework
To address the methodological limitations of scalarized single-objective comparisons, this study establishes a rigorous benchmarking framework. The proposed INSGA-II is evaluated against two distinct paradigms:
Multi-Objective Baselines: Standard NSGA-II and MOEA/D are implemented to assess the non-dominated border mapping performance under identical multi-objective conditions.
Single-Objective Baseline Reformulations: Classical Genetic Algorithm (Weighted-GA) and Ant Colony Optimization (Weighted-ACO) are deployed using a scalarized fitness function:where and denote normalized objective values, , and . For each weight configuration, Weighted-GA and Weighted-ACO are run separately, and the obtained solutions are merged for Pareto comparison and metric evaluation.
4.3 Multi-objective performance and statistical evaluation
To evaluate the multi-objective optimization performance of INSGA-II, each algorithm was independently executed 30 times. The same set of random seeds was used across the algorithms to enable paired statistical comparisons. INSGA-II was benchmarked against two multi-objective algorithms, NSGA-II and MOEA/D, and two weighted single-objective algorithms, Weighted-GA and Weighted-ACO. Hypervolume (HV) and inverted generational distance (IGD) were used to assess Pareto-front approximation quality, while runtime was recorded to evaluate computational cost.
Table 3 reports the mean, standard deviation, and computational runtime obtained over the 30 runs. To determine whether the observed performance differences were statistically significant, Friedman tests were first conducted separately for HV and IGD across the five algorithms. Two-sided Wilcoxon signed-rank tests were subsequently used to compare INSGA-II with each baseline. The resulting pairwise p-values were adjusted using the Holm procedure to control the family-wise error rate, with the significance level set to . The Friedman average ranks and Holm-adjusted pairwise results are presented in Table 4.
TABLE 3
| Algorithm | HV (mean ± std) | IGD (mean ± std) | Runtime/s (mean ± std) |
|---|---|---|---|
| INSGA-II (proposed) | 0.895444 ± 0.167818 | 0.183655 ± 0.091418 | 93.1942 ± 2.6952 |
| NSGA-II | 0.874276 ± 0.244595 | 0.187520 ± 0.132764 | 30.1977 ± 0.8614 |
| MOEA/D | 0.512041 ± 0.203706 | 0.510134 ± 0.152257 | 14.6358 ± 0.2442 |
| Weighted-GA | 0.484697 ± 0.227263 | 0.527383 ± 0.163729 | 104.3408 ± 2.0124 |
| Weighted-ACO | 0.087560 ± 0.053583 | 0.992212 ± 0.076762 | 258.9382 ± 5.3483 |
Statistical comparison of multi-objective performance and computational runtime over 30 independent runs.
Higher HV, values indicate better Pareto-front coverage, whereas lower IGD, and runtime values indicate better performance.
TABLE 4
| Algorithm | HV average rank | IGD average rank | Holm-adjusted p vs. INSGA-II (HV) | Holm-adjusted p vs. INSGA-II (IGD) |
|---|---|---|---|---|
| INSGA-II (proposed) | 1.667 | 1.600 | Reference | Reference |
| NSGA-II | 1.700 | 1.633 | 0.887 | 0.952 |
| MOEA/D | 3.300 | 3.400 | <0.001 | <0.001 |
| Weighted-GA | 3.367 | 3.367 | <0.001 | <0.001 |
| Weighted-ACO | 4.967 | 5.000 | <0.001 | <0.001 |
Friedman average ranks and Holm-adjusted Wilcoxon signed-rank test results over 30 paired runs.
Lower average ranks indicate better performance. Friedman tests were significant for HV , , and IGD , . Pairwise p-values are from Holm-adjusted two-sided Wilcoxon tests against INSGA-II.
As shown in Tables 3 and 4, INSGA-II obtained the highest mean HV, the lowest mean IGD, and the best average rank for both performance indicators. Compared with NSGA-II, the mean HV increased from 0.874276 to 0.895444, representing an improvement of approximately 2.42%, while the mean IGD decreased from 0.187520 to 0.183655, representing an improvement of approximately 2.06%. Across the 30 runs, INSGA-II and NSGA-II exhibited broadly comparable HV and IGD performance, with INSGA-II retaining a modest numerical advantage in both indicators.
The Friedman analysis confirmed overall performance differences among the five algorithms. A clearer separation was observed between INSGA-II and the other three baselines. The Holm-adjusted comparisons with MOEA/D, Weighted-GA, and Weighted-ACO were significant for both HV and IGD . These results indicate that INSGA-II is competitive with NSGA-II, while the differences relative to MOEA/D, Weighted-GA, and Weighted-ACO are statistically significant for both HV and IGD in the tested scenario.
The additional immune operations increased the mean runtime of INSGA-II to 93.1942 s, compared with 30.1977 s for NSGA-II and 14.6358 s for MOEA/D. This increase is associated with the cloning, stimulation evaluation, generation-dependent mutation, and feasibility-repair operations performed during the evolutionary process. INSGA-II also required less runtime than Weighted-GA and Weighted-ACO while achieving higher mean HV and lower mean IGD in the present experiment. The additional computation may therefore be acceptable in planning applications where Pareto-front quality and the availability of alternative trade-off solutions are prioritized over minimum runtime.
Figure 5 presents the distributions of HV, IGD, and runtime over the 30 runs, showing the performance variability and computational cost of the five algorithms.
FIGURE 5
Figure 6 further illustrates the HV convergence profiles of the multi-objective algorithms. INSGA-II shows faster HV growth in the early generations and maintains a higher final HV than standard NSGA-II in the tested scenario. This convergence pattern may be associated with the immune-inspired cloning and dynamic mutation mechanisms. However, as reported in Table 3, this improvement is obtained at the cost of additional runtime compared with standard NSGA-II and MOEA/D.
FIGURE 6
4.4 Pareto front and post-disaster distribution routing analysis
To elucidate the explicit decision-making trade-offs, Figure 7 charts the aggregated Pareto fronts mapped by each algorithm. For operational interpretation, the objective functions are expressed in terms of total flight distance and fleet-load imbalance, where is calculated as the sum of squared deviations from the average UAV load.
FIGURE 7
Figure 7 shows that the INSGA-II Pareto front contains several competitive solutions in the lower-distance region. When greater priority is assigned to minimizing total flight distance , these solutions provide additional distance-oriented alternatives in the tested instance. To visualize the specific routing configurations under different tactical priorities, Figure 8 displays three representative deployment profiles derived from the INSGA-II Pareto front, alongside the baseline routing topologies mapped in Figure 9.
FIGURE 8
FIGURE 9
Minimum Distance Profile (Figure 8a): This solution prioritizes route efficiency and obtains a total flight distance of 11.235 km. It represents a distance-oriented option, although its load imbalance is relatively high (Imbalance = 366.0).
Minimum Load Imbalance Profile (Figure 8b): This solution achieves a load-imbalance value of 0.0, indicating an even distribution of assigned demand among the three UAVs. Such a solution may support more balanced fleet utilization.
Knee Solution Profile (Figure 8c): This representative solution provides a balanced compromise between the two objectives, with a total distance of 11.405 km and a load imbalance of 14.0. Compared to the single-objective Weighted-GA topology (Distance = 11.982 km, Imbalance = 26.0), the INSGA-II Knee solution simultaneously trims the logistics footprint by 4.8% and the load variance by 46.1%, suggesting that the immune-enhanced search mechanism can help identify more favorable routing structures in this scenario.
Figure 9 presents representative routing structures generated by the baseline algorithms under the same spatial configuration. The results illustrate how different optimization formulations and selection mechanisms produce different routing patterns under the current parameter settings.
Weighted-ACO Representative Solution (Figure 9e): Under the current scalarization setting, the representative Weighted-ACO solution contains more route crossings and obtains a total flight distance of 11.865 km and a load imbalance of 23,702.0. The assigned loads are less evenly distributed among the UAVs in this solution, with one UAV carrying a relatively large share of the demand. This pattern may lead to less balanced fleet utilization in the tested instance.
Weighted-GA Representative Solution (Figure 9d): The representative Weighted-GA solution obtains a load-imbalance value of 26.0 and a total flight distance of 11.982 km. As shown in Figure 9d, its routes contain more overlapping and cross-regional segments than the selected INSGA-II knee solution, which may increase operational complexity in this particular solution.
Standard Multi-Objective Baselines (Figure 9b,c): Standard NSGA-II and MOEA/D also generate feasible routing solutions, although differences can be observed in the distribution and boundary coverage of their Pareto fronts in the tested instance. For the representative solutions shown, the INSGA-II knee solution provides a favorable trade-off between total flight distance and load imbalance, particularly relative to the two weighted baselines.
4.5 Discussion
Algorithm Performance and Mechanism Analysis. The comparative results suggest that the immune-enhanced mechanism contributes to Pareto-front approximation in the tested scenario. INSGA-II achieves the highest mean HV and the lowest mean IGD among the compared algorithms, indicating favorable average performance in preserving non-dominated solutions and exploring the trade-off boundary between flight distance and load imbalance. This performance may be associated with the immune cloning and stimulation-guided clone allocation mechanisms, which allow promising individuals to generate local variants while maintaining candidate diversity during the evolutionary process.
At the same time, the runtime comparison reveals a clear quality–cost trade-off. INSGA-II is slower than standard NSGA-II and MOEA/D because the immune cloning process increases the number of candidate solutions evaluated in each generation. Therefore, the proposed algorithm should not be interpreted as the fastest method in terms of wall-clock time. In the tested scenario, INSGA-II yields favorable mean Pareto-front approximation metrics and a diverse set of decision alternatives, reflecting its quality-oriented search characteristics.
Practical Implications for Disaster Relief Logistics. The practical value of INSGA-II lies in providing decision-makers with a set of non-dominated routing alternatives rather than a single scalarized solution. In disaster-relief logistics, different operational stages may require different priorities. For time-critical first-aid delivery, decision-makers may select solutions with shorter total flight distance even if the workload is less balanced. For sustained material distribution, a lower load-imbalance solution may be preferred because excessive concentration of payload or route burden on a single UAV can increase battery depletion risk and reduce fleet-level resilience. Therefore, the Pareto front generated by INSGA-II supports flexible decision-making by explicitly presenting the trade-off between route efficiency and workload equity.
5 Conclusion
With the frequent occurrence of natural disasters and the continuous growth in the demand for emergency medical transportation and material delivery in remote areas, multi-UAV collaborative delivery has become a research hotspot due to its flexible deployment and rapid response. However, in practical applications, the flight range and battery endurance of UAVs are limited, their battery life is insufficient, and the number of delivery points is large with complex geographical distribution, which increases the difficulty of path planning and task allocation for UAVs.
In this study, a bi-objective optimization model for multi-UAV material distribution was constructed to simultaneously minimize total flight distance and workload imbalance. Based on this model, an immune-enhanced NSGA-II algorithm was developed by incorporating immune cloning, stimulation-guided clone, and dynamic mutation into the standard NSGA-II framework. Comparative experiments over 30 independent runs show that INSGA-II achieves competitive mean HV and IGD performance among the tested algorithms. The additional immune operations result in a longer runtime than standard NSGA-II and MOEA/D, reflecting a trade-off between computational efficiency and Pareto-front exploration. Overall, INSGA-II provides a quality-oriented alternative for the investigated multi-UAV disaster-relief material distribution scenario, particularly when solution diversity and decision flexibility are prioritized.
The Pareto-front solutions obtained by INSGA-II illustrate the trade-off between flight distance and load balancing in the investigated multi-UAV distribution problem. Each Pareto optimal solution represents a compromise between flight distance and load balancing, providing multiple optimization options for actual mission planning and path allocation.
Future work will focus on improving the computational efficiency of the proposed algorithm and extending the current static single-depot setting to more realistic disaster-relief scenarios. In particular, split-demand delivery, multiple sorties, heterogeneous UAV fleets, dynamic obstacles, no-fly zones, and real-time re-planning will be incorporated into future models. In addition, air–ground cooperative distribution based on road-network accessibility will be investigated to better represent complex post-disaster logistics operations.
Statements
Data availability statement
The datasets presented in this study can be found in online repositories. The names of the repository/repositories and accession number(s) can be found below: The source code for this study can be obtained and downloaded from https://github.com/byq817/UVA.
Author contributions
JS: Methodology, Writing – review and editing, Validation, Conceptualization, Investigation, Writing – original draft, Formal Analysis. HL: Funding acquisition, Resources, Conceptualization, Writing – review and editing, Supervision, Writing – original draft. EL: Conceptualization, Supervision, Methodology, Writing – original draft, Investigation. ZZ: Visualization, Software, Writing – original draft, Resources, Investigation. LW: Writing – review and editing, Formal Analysis, Validation. YJ: Investigation, Writing – review and editing, Methodology, Project administration. YB: Supervision, Investigation, Writing – review and editing, Project administration. HK: Writing – review and editing, Validation, Investigation, Supervision. CX: Validation, Investigation, Supervision, Writing – review and editing. YZ: Investigation, Writing – review and editing, Validation, Supervision. MD: Writing – review and editing, Investigation, Validation, Supervision. JZ: Writing – review and editing, Supervision, Investigation, Validation.
Funding
The author(s) declared that financial support was received for this work and/or its publication. This research was funded by Information System Project of Linhai Reservoir Project in Mudanjiang City, Heilongjiang Province, grant number DCZX2023-HLJ-0766.
Conflict of interest
Authors JS, EL, and ZZ were employed by Heilongjiang Shuitou Linhai Reservoir Water Supply Engineering Management Co. Authors LW, YJ, YB, HK, CX, YZ, MD, and JZ were employed by Spacewill Info. Co.
The remaining author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Generative AI statement
The author(s) declared that generative AI was not used in the creation of this manuscript.
Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.
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Summary
Keywords
intelligent algorithms, material distribution, multi-objective optimization, task assignments, unmanned aerial vehicles
Citation
Shang J, Li H, Li E, Zhang Z, Wang L, Jiang Y, Bai Y, Kang H, Xiao C, Zhu Y, Dong M and Zhang J (2026) Immune-enhanced NSGA-II for multi-objective multi-UAV disaster relief material distribution. Front. Future Transp. 7:1801962. doi: 10.3389/ffutr.2026.1801962
Received
02 February 2026
Revised
17 July 2026
Accepted
21 July 2026
Published
12 August 2026
Volume
7 - 2026
Edited by
Qiang Bai, Chang’an University, China
Reviewed by
Jose Luis Moura, University of Cantabria, Spain
Michail-Antisthenis Tsompanas, University of the West of England, United Kingdom
Youssri Hassan Youssri, Egypt University of Informatics, Egypt
Updates
Copyright
© 2026 Shang, Li, Li, Zhang, Wang, Jiang, Bai, Kang, Xiao, Zhu, Dong and Zhang.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Heng Li, Li821099463@qq.com
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.