A Guided Collaborative Optimization Framework for the Stability-Constrained UAV Routing and Three-Dimensional Loading Problem

Abstract

The joint optimization of routing and three-dimensional loading is a highly complex NP-hard combinatorial problem, particularly when stringent center-of-gravity (CoG) stability constraints are required for unmanned aerial vehicle (UAV) operations. Existing algorithms typically adopt a route-first, load-second evaluation strategy for these interconnected components, often yielding distance-optimal yet physically infeasible solutions. To address this bottleneck, this paper formulates the Three-Dimensional Loading-Constrained UAV Routing Problem (3DLC-UAVRP), integrating unloading sequence consistency, spatial packing feasibility, and CoG deviation control into the routing decision process. A guided collaborative optimization framework, GLS-WSCPA, is proposed, coupling an Improved White Shark Optimization (IWSO) algorithm for global route exploration with a Human-like Divide-and-Conquer Packing Strategy (HLDCPS) for spatial arrangement. Unlike conventional decoupled approaches that treat loading feasibility as a post hoc filter, a Center-of-Gravity-Guided Path Adjustment (CGPA) and Local Loading Repair (LLR) mechanism is introduced to establish a dynamic feedback loop between routing search and loading evaluation, so that CoG violations are actively translated into guided routing perturbations rather than simply triggering solution rejection. Experimental results demonstrate that GLS-WSCPA generally achieves better solutions than the compared algorithms across the tested problem scales, with the performance gap tending to widen as the instance size increases within the tested range. Ablation studies verify the complementary roles of CGPA and LLR, and sensitivity analysis confirms that moderately relaxing payload and CoG constraints reduces routing distance within safety boundaries. Case analysis shows that the proposed method reduces fleet size by 20% and total delivery distance by 6.85% compared to traditional decoupled strategies.

1. Introduction

The unmanned aerial vehicle routing problem (UAVRP) represents a highly complex class of NP-hard combinatorial optimization problems. As operational paradigms shift from single-target dispatch to multi-destination integrated missions, the computational complexity of determining optimal visit sequences expands exponentially. Furthermore, in practical deployments, theoretical routing configurations are tightly coupled with strict physical constraints, particularly three-dimensional spatial packing and dynamic center of gravity (CoG) stability.

Unlike traditional vehicle routing formulations that typically abstract payloads as simple scalar weights, integrating CoG stability necessitates the simultaneous evaluation of discrete sequence structures and continuous spatial mass distribution. This structural conflict between routing optimization and spatial feasibility significantly alters the search space. Consequently, bridging the gap between distance-optimal routing and physical flight balance constitutes a primary computational bottleneck, requiring advanced algorithmic paradigms capable of handling tightly coupled, multi-domain constraints [1,2,3].

The computational difficulty of this problem is further exacerbated by the strict interdependence between discrete routing sequences and spatial loading configurations. A routing solution that is theoretically optimal in terms of distance may prove physically infeasible if the corresponding three-dimensional packing violates dynamic CoG stability boundaries. Furthermore, the multi-parcel delivery paradigm enforces a strict last in, first out (LIFO) spatial accessibility constraint, meaning that the sequential unloading process dynamically alters the onboard mass distribution and CoG state at each visited node. Consequently, developing a coordinated optimization algorithm that jointly resolves graph-based path sequencing, 3D spatial packing, and continuous stability constraints has emerged as a significant computational challenge.

Traditionally, the UAVRP has been treated as an extension of the classical vehicle routing problem (VRP), initially focusing on single-destination metrics like flight time [4], travel distance [5], or operational cost [6], and progressively expanding to multi-UAV collaborative planning [7,8,9]. Most exact and heuristic methods abstract the payload as a one-dimensional scalar demand and primarily consider total weight constraints [10]. While this abstraction reduces the dimensionality of the search space and accelerates algorithmic convergence, it entirely disregards the geometric and spatial constraints inherent in three-dimensional bin packing (3DBP).

Integrated routing and 3D packing problems (3L-VRP) have been investigated in ground and maritime logistics to better capture practical feasibility [11,12]. Computationally, solving these joint models requires navigating a hybrid search space containing both discrete permutations (routing sequences) and continuous spatial variables (packing coordinates). In broader transportation domains, such as aircraft cargo loading [13] and container stowage [14,15], load balance and CoG positioning are treated as critical constraints affecting stability. However, existing metaheuristic frameworks for routing typically adopt a route-first, load-second evaluation strategy. When applied to UAV operations requiring strict CoG control, this route-first, load-second evaluation frequently generates distance-optimal routes that are physically infeasible, resulting in significant computational waste.

Unlike large-scale platforms with broad stability margins, UAV platforms operate within highly constrained spatial and payload limits. Algorithmically, introducing tight CoG constraints drastically fragments the feasible solution space. Minor permutations in the routing sequence trigger substantial shifts in the spatial packing layout and subsequent CoG deviations. Conventional search operators fail to capture this non-linear interdependence, leading to extremely high rejection rates of candidate solutions and premature convergence to local optima. Despite these advances, a fundamental gap remains in the literature: existing UAVRP studies abstract cargo as scalar demand quantities, neglecting three-dimensional geometric feasibility, while existing 3L-VRP frameworks primarily target ground transportation platforms with generous stability margins and do not address the stringent CoG constraints inherent to UAV operations. Furthermore, existing metaheuristic frameworks universally adopt a route-first, load-second evaluation strategy which is fundamentally inadequate for problems where loading feasibility and routing optimality are structurally interdependent.

Therefore, embedding dynamic CoG control directly into the metaheuristic search trajectory remains a critical research gap. Ignoring the intrinsic coupling among sequence generation, 3D spatial packing, and continuous stability evaluation causes profound computational inefficiencies. A coordinated algorithmic mechanism that establishes a dynamic feedback loop between routing structure adjustments and loading feasibility is highly desirable to effectively explore this heavily constrained search space.

To bridge this computational gap, this paper formulates the Three-Dimensional Loading-Constrained UAV Routing Problem (3DLC-UAVRP) as an np-hard combinatorial optimization model. This formulation explicitly incorporates unloading sequence consistency, three-dimensional spatial packing feasibility, and continuous CoG deviation boundaries into the routing decision variables. By embedding dynamic stability evaluation directly into the route generation phase, the proposed model prevents the exploration of physically infeasible regions within the search space and enhances operational reliability.

To efficiently solve this heavily constrained model, a guided collaborative optimization framework, denoted as GLS-WSCPA, is developed. This framework tightly couples an Improved White Shark Optimization (IWSO) algorithm, designed for global discrete route exploration, with a Human-like Divide-and-Conquer Packing Strategy (HLDCPS) for spatial arrangement generation. Crucially, a Center-of-Gravity-Guided Path Adjustment (CGPA) and Local Loading Repair (LLR) mechanism is introduced to establish a dynamic feedback loop. This mechanism resolves structural conflicts by dynamically repairing local packing infeasibilities and translating unresolvable CoG violations into guided routing perturbations, thereby ensuring joint optimization rather than sequential evaluation.

The main contributions of this study are summarized as follows:

(1)

Mathematical modeling of coupled constraints: Unlike existing UAV routing models that abstract cargo as scalar demand and neglect spatial loading configurations, and unlike ground-based 3L-VRP formulations that incorporate three-dimensional packing geometry but do not model CoG stability, the 3DLC-UAVRP explicitly encodes the full transmission chain from customer visit sequence to LIFO loading order, from loading order to three-dimensional parcel coordinates, and from parcel coordinates to static pre-takeoff CoG within a single unified formulation. This encoding makes the static CoG feasibility condition a direct function of routing decisions rather than an external post hoc check. Consequently, CoG violations can be structurally attributed to specific route segments, which is the prerequisite that enables the CGPA mechanism to translate loading infeasibility into targeted routing perturbations rather than simply discarding infeasible solutions.

(2)

A guided collaborative algorithmic framework: The GLS-WSCPA framework is proposed to address the circular dependency between routing feasibility and loading evaluation that traditional decoupled methods cannot resolve. The core algorithmic innovation lies in the CGPA-LLR mechanism, which establishes a bidirectional feedback loop between combinatorial route search and physical loading feasibility: LLR attempts to restore CoG balance through localized loading adjustments without altering the route structure, while CGPA translates unresolvable loading infeasibility into tabu-guided routing perturbations that actively redirect the global search trajectory. This tight coupling ensures that physical stability constraints are enforced throughout the search process rather than verified post hoc.

(3)

Algorithmic performance and sensitivity evaluation: Comprehensive computational experiments are conducted to evaluate the performance and scalability of the proposed framework across multiple problem scales. Wilcoxon signed-rank tests confirm that the performance advantages of GLS-WSCPA over competing methods reach statistical significance at n ≥ 50, with p-values below 0.001 at larger scales. Ablation studies with statistical validation quantify the complementary contributions of CGPA and LLR under varying constraint intensities. Sensitivity analysis further quantitatively reveals the non-linear impact of CoG stability margins and payload constraints on routing structures and fleet configurations, providing a reference benchmark for stability-constrained combinatorial optimization.

The remainder of this paper is organized as follows. Section 2 reviews related research. Section 3 formulates the 3DLC-UAVRP model. Section 4 presents the GLS-WSCPA algorithmic framework. Section 5 reports and analyzes experimental results. Finally, Section 6 concludes the paper and outlines directions for future research.

2. Related Works

2.1. Advances in UAV Delivery Routing Optimization

Early research on UAV delivery primarily focused on single-destination routing and task allocation. Under this operational mode, UAVs typically execute point-to-point missions from a distribution center to a single customer [16]. Routing optimization in such settings mainly aims to minimize flight time [17], travel distance [18], or operational cost [19], often under simplified constraints and without considering complex operational factors.

With the expansion of UAV applications—particularly in urban logistics, emergency response, and express delivery services—research attention has gradually shifted toward multi-destination delivery scenarios. In these settings, UAVs must schedule and plan routes across multiple customer nodes within a single sortie, significantly increasing problem complexity [20]. To address multi-point delivery tasks, researchers have adopted classical combinatorial optimization models such as the traveling salesman or problem (TSP) and the capacitated vehicle routing problem (CVRP) [21,22,23].

For example, ref. [8] introduced a pure-play drone-based (PD) delivery model in which UAVs are capable of carrying multiple parcels per flight, collecting goods from multiple depots, and serving multiple customers. This operational mode substantially improves delivery efficiency compared with traditional single-parcel dispatch models. Building on this framework, ref. [24] formulated a mixed-integer linear programming model incorporating UAV capacity and endurance constraints and proposed heuristic algorithms for multi-UAV PD systems. Further extensions incorporate additional operational constraints, such as customer time windows and payload-dependent energy consumption rates, thereby enhancing realism in system modeling [25].

Overall, UAV routing research has evolved from geometric path optimization for single flights toward integrated planning models capable of handling multi-parcel, multi-task scenarios. Increasingly, realistic operational constraints are embedded into routing formulations, reflecting a transition from idealized modeling assumptions to more complex and system-oriented representations.

2.2. Joint Route–Loading Optimization

As delivery tasks become more complex, the interdependence between routing decisions and cargo loading configurations has gained growing attention. Traditional routing models typically abstract transported goods as demand quantities at customer nodes, considering only capacity or weight constraints [10]. In contrast, loading problems are often solved independently, failing to capture the practical coupling between route sequences and spatial packing feasibility.

Recent studies have begun to explore integrated route–loading optimization frameworks. These approaches often extend the capacitated vehicle routing problem by incorporating three-dimensional loading constraints—such as cargo volume, stacking order, and spatial feasibility—into routing models [11,12]. Compared with traditional capacity-based formulations, such integrated models more effectively capture the influence of loading configurations on transportation efficiency and safety, offering more realistic modeling paradigms for complex, multi-task delivery scenarios.

Similar concepts have been widely investigated in air and maritime transportation. In aircraft cargo loading problems, center-of-gravity positioning and load distribution are treated as critical factors affecting flight stability and safety. Optimization models focus on distributing cargo within the hold to maintain acceptable center-of-gravity ranges across different flight stages, thereby preventing instability caused by load imbalance [13]. In container ship stowage problems, center-of-gravity balance and load distribution similarly serve as key constraints to ensure vessel stability and structural safety during long voyages [14,15].

These studies collectively highlight the importance of center-of-gravity control in transportation systems. However, most existing work focuses on large-scale platforms such as aircraft or ships, where structural redundancy and relatively spacious loading compartments mitigate the risks associated with load imbalance. In contrast, UAVs operate within highly constrained spatial and payload limits, rendering flight stability particularly sensitive to loading configurations. Consequently, center-of-gravity constraints play a more critical role in UAV delivery systems.

2.3. Evolution of Solution Strategies

In early studies of loading optimization and integrated routing–loading problems, exact algorithms dominated the methodological landscape. Common approaches include mixed-integer programming [26] and branch-and-bound techniques [27]. These methods can provide globally optimal solutions for small-scale instances and serve as important theoretical benchmarks.

However, route–loading joint optimization problems are generally NP-hard. As problem size increases and constraint structures become more complex, the search space grows exponentially, rendering exact methods computationally prohibitive for medium- and large-scale instances [28].

Consequently, research has increasingly shifted toward heuristic and metaheuristic approaches that provide high-quality approximate solutions within reasonable computational time. Although these methods sacrifice guarantees of global optimality, they demonstrate superior scalability and practical applicability [29]. At the same time, to effectively address complex constraint structures, many scholars adjust algorithms based on specific problems while considering the constraints. To this end, common adaptation mechanisms include dynamic strategy adaptation [30], adaptive operator selection [31], and adaptation of penalty function methods [32], all of which help enhance the algorithm’s ability to avoid local optima.

To address these limitations, hybrid and hierarchical frameworks have been developed. Hybrid methods integrate complementary algorithms to balance global exploration and local exploitation, thereby improving both solution quality and computational efficiency [33,34]. Hierarchical approaches decompose the integrated problem into subproblems—such as routing and loading components—solve them using specialized algorithms, and subsequently coordinate them through integration mechanisms [27,35].

These methodological developments demonstrate a clear evolution from single-algorithm strategies toward integrated, framework-based solution paradigms, reflecting the growing need for adaptable and scalable optimization techniques in complex delivery systems.

3. Problem Description and Modeling

3.1. Problem Description

The Three-Dimensional Loading-Constrained UAV Routing Problem (3DLC-UAVRP) is formally defined as follows. A distribution center operates a homogeneous fleet of multi-rotor UAVs to serve a set of geographically dispersed customers. All order information, including parcel dimensions, weights, and delivery locations, is known in advance. A complete delivery plan consists of a set of UAV routes together with their corresponding three-dimensional loading configurations.

During mission execution, each UAV departs from the distribution center, follows its predetermined route, and returns to the distribution center after completing all deliveries on that route. Prior to departure, parcels are loaded into the UAV’s cargo bay according to a carefully designed loading scheme. This loading scheme must simultaneously satisfy the three-dimensional spatial constraints of the cargo compartment. Moreover, the loading layout must maintain strict sequential consistency with the delivery route. Specifically, at each customer location, the parcel(s) designated for that customer must be directly accessible from the corresponding side compartment without the need to reposition or unload any parcels intended for subsequent customers. As illustrated in Figure 1, each delivery route forms a sequence of customer visits. In the figure, parcels of different colors represent the delivery demands of different customers.

3.2. Mathematical Model

In the 3DLC-UAVRP, the transportation network is modeled as a complete directed graph $U=\left(T,E\right)$, where $T=T_0\cup T_n$ is the set of all nodes and $E=\left\{\left(i,i^{\prime }\right)\right|i,i^{\prime }\in T,i\ne i^{\prime }\}$ is the set of arcs. The node set consists of a single depot node $T_0$ and the customer node set $T_n=\left\{1,2,\dots ,n\right\}$. The principal notation used throughout the model is summarized in

Table 1. Notation and parameter definitions.

Type	Notation	Definitions
Parameters	$c_{i{i}^{\prime }}$	Distance from node $i$ to node $i^{\prime }$
	$G$	Maximum payload capacity of the UAV
	$I_{ik}$	The $k$-th cargo of customer $i$
	$d_{ik}$	Weight of the $k$-th cargo $I_{ik}$
	$l_{ik},w_{ik},h_{ik}$	Length, width, and height of the cargo $I_{ik}$
	$v_{ik}$	Volume of the cargo $I_{ik}$
	$I_l,I_r$	Collection of customer nodes whose cargoes are placed in the left ($I_l$) or right ($I_r$) loading zone
	$\left(a_1,a_2\right)$	Safe interval of the center of gravity along the $X$-axis
	$\left(b_1,b_2\right)$	Safe interval of the center of gravity along the $Y$-axis
	$\left(c_1,c_2\right)$	Safe interval of the center of gravity along the $Z$-axis
	$M$	A sufficiently large positive number
	$g_{safe}$	Threshold for the center of gravity deviation
Decision making variables	${\alpha }_{i{i}^{\prime }}^p$	$\left\{\begin{array}{l}1, Indicat that UAV p travels from node i to node i^{\prime }\hfill \\ 0, Otherwise\hfill \end{array}\right.$
	$b_i^p$	$\left\{\begin{array}{l}1, Indicat that customer i is served by UAV p\hfill \\ 0, Otherwise\hfill \end{array}\right.$
	${\delta }_{ik}$	$\left\{\begin{array}{l}1, Indicat that cargo I_{ik} is placed in the right zone I_r\hfill \\ 0, Indicat that cargo I_{ik} is placed in the left zone I_l\hfill \end{array}\right.$
	$S_i\in {\mathbb{Z}}^{+}$	Visitation sequence index of customer $i$ in the route (1, 2, …)
	${\lambda }_{ik,i^{\prime }{k}^{\prime }}^{dir}$	$\left\{\begin{array}{l}1, Cargo I_{ik} is placed to the dir of cargo I_{i^{\prime }{k}^{\prime }}\hfill \\ 0, Otherwise\hfill \end{array}\right.,dir\in \left\{Left,Right,Front,Back,Under,Top\right\}$
	${\xi }_{ik,i^{\prime }{k}^{\prime }}$	$\left\{\begin{array}{l}\begin{array}{c}1,I_{ik} is placed to the dir of cargo I_{i^{\prime }{k}^{\prime }}Indicate that parcels I_{ik},I_{i^{\prime }{k}^{\prime }} \\ have overlapping projections on the Y-Z plane\end{array}\hfill \\ 0, Otherwise\hfill \end{array}\right.$
	${\xi }_{i{i}^{\prime }}$	$\left\{\begin{array}{l}1,S_i<S_{i^{\prime }}\hfill \\ 0, Otherwise\hfill \end{array}\right.$

.

To ensure flight stability during multi-customer delivery missions, the UAV adopts a dual-compartment modular cargo structure. A longitudinal partition beam is installed along the central axis of the cargo bay, physically dividing the effective payload space into two symmetric compartments: the left loading compartment and the right loading compartment. A three-dimensional Cartesian coordinate system is established based on this structural design. The $X$-axis, $Y$-axis and $Z$-axis correspond to the cargo bay width $W$, length $L$, and height $H$, respectively. Consequently, the entire loading space is partitioned into two symmetric regions: the left loading region $I_L$ and the right loading region $I_R$, as illustrated in Figure 2.

The loading logic is defined as follows. In the left compartment, parcels are loaded starting from the front–bottom–left corner, denoted as point $\left(x_{ik}-w_{ik},y_{ik},z_{ik}-h_{ik}\right)$, and are arranged sequentially along the negative $X$-direction and positive $Y$-direction. The origin of the coordinate system $\left(0,0,0\right)$ is located at the center of the front edge of the cargo floor. In the right compartment, parcels are loaded with respect to reference point $\left(0,L,0\right)$, beginning at the rear–bottom–left corner $\left(x_{ik},y_{ik}-l_{ik},z_{ik}-h_{ik}\right)$, and arranged sequentially along the positive X-direction and negative Y-direction. This structured loading rule ensures spatial consistency with the delivery sequence and facilitates direct unloading at customer locations.

To formulate the model, the following assumptions are adopted:

(1)

Each UAV has sufficient endurance to complete its assigned delivery route.

(2)

Each UAV can complete its mission and return to the depot within the prescribed time horizon.

(3)

Each customer’s demand must be fully satisfied in a single visit; split deliveries are not allowed.

(4)

All parcels are placed orthogonally, with their edges aligned parallel to the cargo bay boundaries.

(5)

No parcel may be suspended; each parcel must be fully supported from below.

(6)

The CoG of the UAV coincides with the geometric center of the empty cargo bay. The model imposes CoG constraints only on the mass distribution of loaded parcels, and the structural weight of the UAV body is not explicitly modeled.

(7)

CoG window constraints are enforced only at the fully loaded (pre-takeoff) state. A conservative CoG deviation threshold is adopted to maintain safety margins against CoG shifts caused by parcel unloading during delivery operations.

The mathematical formulation of the 3DLC-UAVRP is presented as follows. Constraints (1)–(2) ensure that each customer is visited exactly once by a single UAV and that flow conservation is maintained at every visited node; that is, if a UAV arrives at a node, it must subsequently depart from that node. Constraints (3)–(6) enforce route feasibility for each UAV. Specifically, each UAV must depart from the depot, sequentially visit the customers assigned to its route, and ultimately return to the depot, thereby forming a closed delivery tour. Constraint (7) limits the total weight of parcels loaded on a UAV along any single delivery route, ensuring that the cumulative payload does not exceed the UAV’s maximum carrying capacity. Constraints (8) enforce spatial feasibility by ensuring that each parcel is fully contained within the geometric boundaries of the cargo compartment. Constraint (9) specifies the compartment assignment rule, requiring that each parcel be placed entirely within either the left loading region or the right loading region, without spanning both compartments. Constraint (10) enforces non-overlapping conditions, ensuring that any two parcels assigned to the same UAV do not occupy intersecting spatial regions within the cargo compartment. Constraint (11) establishes the delivery-sequence accessibility requirement. Specifically, if customer $i$ is visited before customer $j$, the parcel assigned to customer $i$ must be directly retrievable through the corresponding side door without being obstructed by the parcel designated for customer $j$. Constraint (12) represents the loading stability constraint.

$${\displaystyle \sum _{p=1}^P{b}_i^p}=1, i\in T_n$$

(1)

$${\displaystyle {\displaystyle \sum _{i=0}^n}{\alpha }_{i{i}^{\prime }}^p}={\displaystyle {\displaystyle \sum _{i^{\prime }=0}^n}{\alpha }_{i^{\prime }i}^p}=b_{i^{\prime }}^p, \forall i,i^{\prime }\in T_n, p\in P$$

(2)

$${\displaystyle {\displaystyle \sum _{i^{\prime }=0}^n}{\alpha }_{0i}^p}={\displaystyle {\displaystyle \sum _{i^{\prime }=0}^n}{\alpha }_{i0}^p\le 1}, \forall p\in P$$

(3)

$$S_i+1\le S_{i^{\prime }}+M\left(1-{\alpha }_{i{i}^{\prime }}^p\right),\forall i,i^{\prime }\in T_n,p\in P$$

(4)

$$1\le S_i\le n,\forall i\in T_n$$

(5)

$$S_0=0$$

(6)

$${\displaystyle {\displaystyle \sum }_{i=1}^n}{\displaystyle {\displaystyle \sum }_{k=1}^{m_i}}{d}_{ik}\cdot b_i^p\le G, \forall p\in P$$

(7)

$$\begin{array}{c}-{\displaystyle \frac{W}{2}}\le x_{ik},x_{ik}+w_{ik}\le {\displaystyle \frac{W}{2}},\forall i,k\\ 0\le y_{ik},y_{ik}+l_{ik}\le L,\forall i,k\\ 0\le z_{ik},z_{ik}+h_{ik}\le H,\forall i,k\end{array}$$

(8)

$$\begin{array}{l}{x}_{ik}+w_{ik}\le 0+M\cdot {\delta }_{ik},\forall i,k\in I_l\\ x_{ik}\ge 0-M\cdot \left(1-{\delta }_{ik}\right),\forall i,k\in I_r\end{array}$$

(9)

$$\begin{array}{c}{x}_{ik}+w_{ik}\le x_{i^{\prime }{k}^{\prime }}+M\left(1-{\lambda }_{ik,i^{\prime }{k}^{\prime }}^{Left}\right)\\ x_{i^{\prime }{k}^{\prime }}+w_{i^{\prime }{k}^{\prime }}\le x_{ik}+M\left(1-{\lambda }_{ik,i^{\prime }{k}^{\prime }}^{Right}\right)\\ y_{ik}+l_{ik}\le y_{i^{\prime }{k}^{\prime }}+M\left(1-{\lambda }_{ik,i^{\prime }{k}^{\prime }}^{Back}\right)\\ y_{i^{\prime }{k}^{\prime }}+l_{i^{\prime }{k}^{\prime }}\le y_{ik}+M\left(1-{\lambda }_{ik,i^{\prime }{k}^{\prime }}^{Front}\right)\\ z_{ik}+h_{ik}\le z_{i^{\prime }{k}^{\prime }}+M\left(1-{\lambda }_{ik,i^{\prime }{k}^{\prime }}^{Under}\right)\\ z_{i^{\prime }{k}^{\prime }}+h_{i^{\prime }{k}^{\prime }}\le z_{ik}+M\left(1-{\lambda }_{ik,i^{\prime }{k}^{\prime }}^{Top}\right)\\ {\lambda }_{ik,i^{\prime }{k}^{\prime }}^{Left}+{\lambda }_{ik,i^{\prime }{k}^{\prime }}^{Right}+{\lambda }_{ik,i^{\prime }{k}^{\prime }}^{Back}+{\lambda }_{ik,i^{\prime }{k}^{\prime }}^{Front}+{\lambda }_{ik,i^{\prime }{k}^{\prime }}^{Under}+{\lambda }_{ik,i^{\prime }{k}^{\prime }}^{Top}\ge 1\end{array}$$

(10)

$$\begin{array}{c}{x}_{ik}-x_{i^{\prime }{k}^{\prime }}\le M\cdot (1-{\xi }_{ik,i^{\prime }{k}^{\prime }})+M\cdot {\delta }_{ik}+M\cdot {\delta }_{i^{\prime }{k}^{\prime }}+M\cdot (1-{\xi }_{ik})\\ x_{ik}-x_{i^{\prime }{k}^{\prime }}\le M\cdot (1-{\xi }_{ik,i^{\prime }{k}^{\prime }})+M\cdot (1-{\delta }_{ik})+M\cdot (1-{\delta }_{i^{\prime }{k}^{\prime }})+M\cdot (1-{\xi }_{ik})\end{array}$$

(11)

$$\begin{array}{c}{W}_p={\displaystyle \sum _{i=1}^n{\displaystyle \sum _{k=1}^{m_i}{d}_{ik}}}{b}_i^p,\hspace{1em}\forall p\in P\\ a_1{W}_p\le {\displaystyle \sum _{i=1}^n{\displaystyle \sum _{k=1}^{m_i}{d}_{ik}}}\left(x_{ik}+{\displaystyle \frac{1}{2}}{w}_{ik}\right)b_i^p\le a_2{W}_p,\hspace{1em}\forall p\in P\\ b_1{W}_p\le {\displaystyle \sum _{i=1}^n{\displaystyle \sum _{k=1}^{m_i}{d}_{ik}}}\left(y_{ik}+{\displaystyle \frac{1}{2}}{l}_{ik}\right)b_i^p\le b_2{W}_p,\hspace{1em}\forall p\in P\\ c_1{W}_p\le {\displaystyle \sum _{i=1}^n{\displaystyle \sum _{k=1}^{m_i}{d}_{ik}}}\left(z_{ik}+{\displaystyle \frac{1}{2}}{h}_{ik}\right)b_i^p\le c_2{W}_p,\hspace{1em}\forall p\in P\end{array}$$

(12)

Based on the principle of minimizing the total routing distance, the objective function is formulated as Equation (13).

$$\min Z={\displaystyle \sum _{p=1}^P{\displaystyle \sum _{i^{\prime }=0}^n{\displaystyle \sum _{i=0}^n{c}_{i{i}^{\prime }}}}}\cdot {\alpha }_{i{i}^{\prime }}^p$$

(13)

4. Guided Hybrid Heuristic Solution Approach

The 3DLC-UAVRP requires the simultaneous optimization of routing sequences and three-dimensional loading configurations under coupled operational and CoG stability constraints, a combination that exposes three fundamental limitations in existing solution paradigms.

The dominant “route-first, load-second” evaluation strategy decouples routing from loading, causing high rejection rates when distance-optimal routes prove physically infeasible under CoG constraints. More fundamentally, existing loading heuristics are designed to verify spatial feasibility after a route is fixed rather than to actively balance mass distribution during packing construction, so infeasible solutions must be discarded entirely rather than repaired. Most critically, no existing framework channels loading infeasibility information back into the routing search. When a CoG violation is detected, the search restarts without any knowledge of which route segment caused it, preventing the algorithm from learning to avoid recurrent infeasible patterns and severely limiting scalability as problem size grows.

To address these limitations, this study proposes GLS-WSCPA. The framework assigns global route exploration to an Improved White Shark Optimization (IWSO) and spatially balanced loading arrangement to a Human-like Divide-and-Conquer Packing Strategy (HLDCPS). A Center-of-Gravity-Guided Path Adjustment and Local Loading Repair (CGPA–LLR) mechanism then coordinates the two components by translating unresolvable CoG violations into guided routing perturbations, so that physical stability is enforced continuously throughout the search rather than checked only at its end. The overall framework is illustrated in Figure 3.

4.1. Improved White Shark Optimization

The White Shark Optimization (WSO) algorithm [36] draws inspiration from the unique olfactory and auditory tracking behaviors of great white sharks, establishing a global exploration strategy based on movement towards prey and a local exploitation mechanism centered on optimal position tracking. These components correspond to the exploration and exploitation phases of the optimization process, respectively. Furthermore, WSO integrates fish school behavior to maintain population diversity, which optimizes the population update process and prevents the algorithm from falling into local optima.

To address the limitations of the original algorithm in discrete domains, this study develops an IWSO algorithm. By refining the search effectiveness and solution quality, IWSO achieves a superior balance between global search and local refinement. The workflow of the proposed IWSO algorithm is illustrated in Figure 4.

4.1.1. Encoding and Decoding Strategy

To bridge the gap between the continuous evolutionary search mechanism of WSO and the discrete nature of routing decision variables, a tailored encoding–decoding strategy is designed.

(1)

Encoding

A random-key real-number encoding scheme is adopted to resolve the structural inconsistency between the continuous search space of WSO and the discrete permutation-based routing problem.

Let $n$ denote the total number of customer nodes and $N_{pop}$ denote the population size. The position vector of the s-th search individual (where $s\in \left\{1,2,3,\dots ,N_{pop}\right\}$) is defined as an $n$-dimensional continuous real-valued vector $W_s$:

$$W_s=\left\{w_1^s,w_2^s,.,w_i^s,.,w_n^s\right\}$$

(14)

Under the adopted encoding scheme, the i-th component $w_i^s$ of the position vector $W_s$ represents the real-valued ranking key associated with customer node $i$. It should be noted that this value serves only as a continuous priority indicator. The algorithm determines the visiting sequence by comparing the relative magnitudes of these keys, thereby mapping the continuous position vector into a discrete customer visit order.

(2)

Decoding

In this study, the Smallest Position Value (SPV) rule is employed as the core decoding operator. The SPV rule transforms the continuous real-valued position vector generated during the evolutionary process into a physically feasible discrete delivery route, thereby establishing a correspondence between the search space of the metaheuristic and the decision variables of the routing problem. For any given search individual $s$ with position vector $W_s$, the decoding procedure proceeds as follows. An illustrative example of the SPV-based decoding process is provided in Figure 5.

Step 1: Sequence reconstruction

Sort the real-valued components of $w_i^s$ in ascending order. Define the index mapping rule such that if $w_i^s<w_{i^{\prime }}^s$ for $i<i^{\prime }$, then customer $i$ precedes customer $i^{\prime }$ in the generated visiting sequence.

Step 2: Closed route construction

Extract the ordered customer index sequence obtained from Step 1. Then append the depot node (denoted as 0) at both the beginning and the end of the sequence to form a complete closed UAV delivery route: ${\pi }_p=\left[0,\underset{n}{\underbrace{\dots ,i,i^{\prime },\dots }},0\right]$.

4.1.2. Fitness Function Design

Fitness evaluation is a crucial step in assessing the quality of each search individual $W_s$. The algorithm directly employs the total routing distance, as defined in Equation (13), as the criterion for evaluating the decoded delivery solution.

4.1.3. Population Initialization

The quality of population initialization directly influences the search diversity and convergence performance of the algorithm. According to the adopted encoding strategy, the initial population consists of $N_{pop}$ search individuals and can be represented as an $N_{pop}\times n$ matrix, where each row corresponds to an n-dimensional continuous position vector:

$$N_{pop}=\left[\begin{array}{ccc}{w}_1^1& \dots & w_n^1\\ ⋮& \ddots & ⋮\\ w_1^{N_{pop}}& \dots & w_n^{N_{pop}}\end{array}\right]$$

(15)

After determining the population structure, the values of each dimension must be initialized. The original WSO algorithm relies on a uniform random distribution, which often leads to uneven spatial distribution, resulting in the clustering of routing solutions and reduced global exploration capability. To address this limitation and enhance initial population diversity, this study introduces a Sine-Partition-Modulation (SPM) chaotic mapping to replace the traditional random initialization. The SPM mapping exhibits strong randomness and ergodicity, generating more uniformly distributed chaotic sequences and thereby improving the spatial coverage of initial solutions. The SPM mapping is defined as:

$$\begin{array}{ll}mod\left({\displaystyle \frac{x_k}{\eta }}+usin\left(\pi x_k\right)+rand,1\right),& 0\le x_k<\eta \\ mod\left({\displaystyle \frac{x_k/\eta }{0.5-\eta }}+usin\left(\pi x_k\right)+rand,1\right),& \eta \le x_k<0.5\\ mod\left({\displaystyle \frac{\left(1-x_k\right)/\eta }{0.5-\eta }}+usin\left(\pi \left(1-x_k\right)\right)+rand,1\right),& 0.5\le x_k<1-\eta \\ mod\left({\displaystyle \frac{1-x_k}{\eta }}+usin\left(\pi \left(1-x_k\right)\right)+rand,1\right),& 1-\eta \le x_k<1\end{array}$$

(16)

where $x_k$ denotes the chaotic variable at iteration $k$, and $\eta$ and $u$ are mapping parameters.

The generated chaotic values are then linearly transformed to the feasible range of each dimension:

$$w_n^s=l_n+x_{k+1}\times \left(u_n-l_n\right)$$

(17)

where $u_n$ and $l_n$ denote the upper and lower bounds of the n-th search dimension, respectively, and $rand\in \left(0,1\right)$ is a uniformly distributed random number.

This strategy improves the diversity of the initial population and enhances the algorithm’s global search capability.

4.1.4. Velocity Update

The velocity vector $V_s=\left\{v_1^s,v_2^s,.,v_i^s,.,v_n^s\right\}$ describes the dynamic adjustment process of the ranking keys in the continuous search space. The magnitude and direction of velocity updates determine the relative ordering of elements in $W_s$, thereby influencing the customer visiting sequence obtained through SPV decoding. In the original WSO algorithm, differences among search states are not sufficiently distinguished, and adaptability during the exploration stage is limited. The velocity update rule is expressed as:

$$v_s\left(t+1\right)=\mu \cdot \left[v_s\left(t\right)+\underset{global guidance term}{\underset{¯}{p_1\left(W_{gbest}\left(t\right)-W_s\left(t\right)\right)c_1}}+\underset{individual memory}{\underset{¯}{p_2\left(W_{best}^s\left(t\right)-W_s\left(t\right)\right)c_2}}\right]$$

(18)

where $v_s\left(t\right)$ and $v_s\left(t+1\right)$ denote the velocity vectors of the s-th search individual at iterations $t$ and $t+1$, respectively. The coefficients $p_1$ and $p_2$ are control parameters governing the influence of global and individual search components. During the UAV routing optimization process, $p_1$ guides the search individual toward the current globally best routing solution, thereby accelerating convergence. In contrast, $p_2$ preserves the individual’s historically best ordering information, helping maintain population diversity and preventing premature convergence. The coefficients $c_1$ and $c_2$ are random numbers uniformly distributed in the interval [0, 1]. The vector $W_{gbest}\left(t\right)$ represents the globally best position vector obtained by any search individual up to iteration t, while $W_{best}^s(t)$ denotes the historical best position vector of the s-th individual. The vectors $W_s\left(t\right)$ and $W_s\left(t+1\right)$ represent the position vectors of the s-th search individual at iterations $t$ and $t+1$, respectively.

Essentially, Equation (18) calculates the numerical adjustment step for the continuous sorting keys of the customer nodes. It determines the update direction by linearly combining three components: maintaining the previous adjustment inertia, pulling towards the globally best-known solution, and pulling towards the individual’s historically best solution. However, applying this continuous, numerical addition-based update mechanism to a discrete routing problem (which strictly requires combinatorial sequence permutations) causes a severe structural mismatch, leading to inefficient exploration.

To enhance the performance of the original WSO update rule in Equation (18), this study introduces a nonlinear reconstruction strategy and proposes a composite improvement mechanism.

(1)

Adaptive perturbation amplitude control

In Equation (18), the inertia coefficient $\mu$ is fixed, which may provide insufficient exploration capability in the early stages of iteration. To address this limitation, a nonlinear perturbation amplitude control function $A\left(t\right)$, dependent on the iteration index, is introduced to replace the constant inertia coefficient. At early iterations, a relatively large $A\left(t\right)$ is assigned to enlarge the adjustment range of ranking keys, thereby enhancing exploration across diverse routing sequences. As the iteration progresses, $A\left(t\right)$ gradually decreases to preserve structural stability of promising routing solutions and strengthen exploitation. The adaptive perturbation function is defined as:

$$A\left(t\right)=A_0{e}^{-\beta \cdot {\left({\displaystyle \frac{t}{T_{max}}}\right)}^2}$$

(19)

where $A_0$ denotes the initial perturbation amplitude, determining the exploration intensity in early iterations; $\beta$ is the decay control factor; $t$ represents the current iteration index; and $T_{max}$ denotes the maximum number of iterations.

(2)

Composite guidance term construction

The global guidance term in the original velocity update is isolated and combined with the adaptive perturbation amplitude to construct a composite guidance increment:

$$\Delta V_s=A\left(t\right)\cdot p_1\left(W_{gbest}\left(t\right)-W_s\left(t\right)\right)\times c_1$$

(20)

This formulation allows the strength of global guidance to vary dynamically throughout the search process. To further enhance diversity and prevent stagnation, a periodic perturbation mechanism is introduced. When the iteration count satisfies a predefined triggering condition, a reinforced guidance phase is activated, allowing large-scale adjustment of ranking keys. During other iterations, the algorithm operates in a regular search mode, temporarily disabling global guidance and relying primarily on individual memory to perform local refinements. The periodic trigger function is defined as:

$$T_m\left(t\right)={\displaystyle \frac{t}{T_{max}/10}}$$

(21)

Based on the above mechanisms, the updated velocity equation becomes:

$$v_s\left(t+1\right)=\left\{\begin{array}{ll}{v}_s\left(t\right)+p_2\left(W_{best}^s\left(t\right)-W_s\left(t\right)\right)c_2+\Delta V_s,\hfill & \mathrm{mod}(T_m\left(t\right))=0\hfill \\ v_s\left(t\right)+p_2\left(W_{best}^s\left(t\right)-W_s\left(t\right)\right)c_2,\hfill & Otherwise\hfill \end{array}\right.$$

(22)

This nonlinear and periodically enhanced velocity update strategy improves exploration capability in early iterations, maintains population diversity, and strengthens convergence stability in later stages.

4.1.5. Hierarchical Position Update Mechanism

To further enhance convergence stability and search efficiency, a hierarchical position update mechanism is introduced. This mechanism first updates the overall population positions and subsequently applies differentiated update strategies according to fitness-based ranking.

(1)

Position update

Based on the velocity $v_s\left(t+1\right)$ computed in Section 4.1.4, the position of the s-th search individual is updated using the following piecewise rule:

$$W_s\left(t+1\right)=\left\{\begin{array}{l}{W}_s\left(t\right)\cdot \neg \oplus w_o+u\cdot a+l\cdot b,rand<mv\\ W_s\left(t\right)+{\displaystyle \frac{v_s\left(t+1\right)}{f}},rand\ge mv\end{array}\right.$$

(23)

where $\neg$ denotes a logical complement operator, $w_o=\oplus (a,b)$ represents a logical auxiliary vector, $u$ and $l$ denote the upper and lower bounds of the search space, respectively, $a$ and $b$ are binary vectors generated via boundary detection, $mv$ is a movement threshold parameter, and $f$ is a scaling factor. The original WSO algorithm, designed for continuous optimization, typically sets ($f=0.90$). However, UAV routing is a discrete combinatorial problem. Applying an excessively large scaling factor severely disrupts the relative magnitudes of the ranking keys. Under SPV decoding, this causes the routing sequences to lose structural stability, degrading the guided search into a random walk. Therefore, $f$ is recalibrated to 0.67 in this study, effectively balancing global exploration with the preservation of advantageous route structures. The parameter $mv$ v varies nonlinearly with the iteration index to balance global exploration and local exploitation. It is defined as:

$${\displaystyle \frac{1}{\left(a_0+e^{\left(T_{max}/2-t\right)/a_1}\right)}}$$

(24)

where $a_0$ and $a_1$ are control parameters governing the trade-off between exploration and exploitation. The values used in this study are 5 and 50, respectively.

(2)

Fitness-based reconstruction strategy

Although the above position update generates new continuous solutions, applying identical update rules to all individuals may reduce search efficiency. Therefore, a fitness-based hierarchical reconstruction strategy is introduced.

Step 1: Decoding

The updated position vectors $W_s\left(t+1\right)$ and $W_{gbest}\left(t+1\right)$ are decoded using the SPV rule to obtain the corresponding delivery routes ${\pi }_p$ and the global best route ${\pi }_p^{best}$.

Step 2: Fitness ranking and population layering

The population is sorted according to fitness values. Let the population size be $N_{pop}$, and define a layering threshold ratio $\partial$. The top $\partial \cdot N_{pop}$ individuals are classified as the elite layer $s^{elite}$, while the remaining $(1-\partial )\cdot N_{pop}$ individuals constitute the ordinary layer $s^{ordinary}$.

Step 3: Elite-layer operation

Individuals in the elite layer are assumed to reside near promising regions and therefore undergo local refinement rather than large perturbations. An exchange mutation operator is introduced to adjust customer visit sequences. Specifically, two distinct customer positions in the current route ${\pi }_p(s^{elite})$ are randomly selected and swapped, while the relative order of other customers remains unchanged, yielding a new candidate route ${\pi }_p^{new}(s^{elite})$. This operation enhances local exploitation while preserving structural stability (see Figure 6).

Step 4: Ordinary-layer operation

For individuals in the ordinary layer, a locking-reconstruction operator is adopted. The current route ${\pi }_p(s^{ordinary})$ is compared position-by-position with the global best route ${\pi }_p^{best}$. If a customer node matches the global best at the same position, that position is locked. For unmatched positions, the remaining unassigned customer nodes are randomly permuted and reinserted. This produces a reconstructed route ${\pi }_p^{new}(s^{ordinary})$ that preserves advantageous structural segments while introducing diversity (see Figure 7).

Step 5: Fitness evaluation

The fitness values of both elite and ordinary reconstructed solutions are evaluated. If the new solution improves upon the original route ${\pi }_p^{new}$, it replaces the previous solution; otherwise, the original solution is retained.

Step 6: Position mapping

To ensure continuity in the next iteration, the accepted discrete route ${\pi }_p^{new}$ is mapped back into the continuous search space and re-encoded into the corresponding position vector $W_s{\left(t+1\right)}^{new}$.

4.2. Human-like Divide-And-Conquer Packing Strategy

The HLDCPS integrates divide-and-conquer principles with heuristic loading strategies. Given a routing solution as input, the algorithm sequentially performs loading feasibility verification, loading order construction, and left–right compartment balancing under multi-dimensional constraints. The overall procedure is illustrated in Figure 8.

For a UAV $p$ assigned route ${\pi }_p$, loading feasibility must satisfy the following conditions:

$${\displaystyle \sum }_{i\in {\pi }_p}{\displaystyle {\displaystyle \sum }_{k=1}^{m_i}}{d}_{ik}\le G$$

(25)

If this constraint is violated, the current route is deemed infeasible and fed back to IWSO for re-optimization. Otherwise, the loading sequence of UAV $p$ is defined according to the visit order:

$${\mathcal{I}}_p^{seq}=\left\{\dots ,\underset{m_i}{\underbrace{I_{ik},I_{i(k-1)},\dots I_{i1}}},\dots \right\}$$

(26)

Following Section 3.1, the cumulative payload difference between left and right cabins is initialized to zero. Customers are assigned to either cabin to minimize weight imbalance, providing a partitioning basis for subsequent spatial packing. The initial reference points of the left and right cabins are set to $\left(0,0,0\right)$ and $\left(0,L,0\right)$, respectively. The X- and Z-axes are defined as primary reference directions.

For each item $I_{ik}$ with original dimensions $(l_{ik},w_{ik},h_{ik})$, a finite rotation set is defined:

$$R=\left\{\begin{array}{l}\left(w_{ik},l_{ik},h_{ik}\right),\left(h_{ik},l_{ik},w_{ik}\right),\left(l_{ik},w_{ik},h_{ik}\right),\\ \left(w_{ik},h_{ik},l_{ik}\right),\left(l_{ik},h_{ik},w_{ik}\right),\left(h_{ik},w_{ik},l_{ik}\right)\end{array}\right\}$$

(27)

Each element represents a feasible orthogonal orientation in the Cartesian coordinate system (see Figure 9).

A dynamically updated candidate placement set is generated during packing. For candidate coordinate $\left(x_i^s,y_i^s,z_i^s\right)$ and rotated dimensions $\left(w_{ik}^{\prime },l_{ik}^{\prime },h_{ik}^{\prime }\right)$, boundary feasibility requires:

$$\left\{\begin{array}{c}-x_i^s-w_{ik}^{\prime }\ge W_x\\ y_i^s-l_{ik}^{\prime }\le L_y\\ z_i^s+h_{ik}^{\prime }\le H_z\end{array}\right.$$

(28)

where $W_x,L_y,H_z$ denote dynamic boundary reference limits. After boundary screening, non-overlapping verification is performed. Among feasible placements, the algorithm prioritizes: (1) minimal placement height, (2) then minimal horizontal gap. The process continues iteratively until all items are packed, producing the final loading plan ${\mathcal{I}}_p^{load}$.

4.3. Center-of-Gravity-Guided Path Adjustment and Local Loading Repair

To address infeasibility arising from the joint effects of three-dimensional packing and CoG constraints, two collaborative mechanisms are formally embedded into the metaheuristic search process. The pseudocode for the LLR and CGPA mechanisms is summarized in Algorithms 1 and 2, respectively. When the CoG deviation $\Delta g$ exceeds the safety threshold $g_{safe}$, the LLR mechanism is first activated to restore feasibility through localized physical adjustments without altering the route structure. If LLR fails, the CGPA mechanism is subsequently triggered, which identifies the most destabilizing route segment, incorporates it into a tabu set, and perturbs the corresponding ranking keys in the IWSO position vector to redirect the global search away from infeasible regions (see Figure 10).

4.3.1. Trigger Function

Given route ${\pi }_p$ and loading plan ${\mathcal{I}}_p^{load}$, the three-dimensional center of gravity is computed as:

$$\left\{\begin{array}{l}{X}_{cog}={\displaystyle \frac{1}{{\displaystyle {\displaystyle \sum }_{i=1}^n}{\displaystyle {\displaystyle \sum }_{k=1}^{m_i}}{d}_{ik}}}{\displaystyle {\displaystyle \sum }_{i=1}^n}{\displaystyle {\displaystyle \sum }_{k=1}^{m_i}}{d}_{ik}\left(x_{ik}+{\displaystyle \frac{1}{2}}{w}_{ik}\right)\hfill \\ Y_{cog}={\displaystyle \frac{1}{{\displaystyle {\displaystyle \sum }_{i=1}^n}{\displaystyle {\displaystyle \sum }_{k=1}^{m_i}}{d}_{ik}}}{\displaystyle {\displaystyle \sum }_{i=1}^n}{\displaystyle {\displaystyle \sum }_{k=1}^{m_i}}{d}_{ik}\left(y_{ik}+{\displaystyle \frac{1}{2}}{l}_{ik}\right)\hfill \\ Z_{cog}={\displaystyle \frac{1}{{\displaystyle {\displaystyle \sum }_{i=1}^n}{\displaystyle {\displaystyle \sum }_{k=1}^{m_i}}{d}_{ik}}}{\displaystyle {\displaystyle \sum }_{i=1}^n}{\displaystyle {\displaystyle \sum }_{k=1}^{m_i}}{d}_{ik}\left(z_{ik}+{\displaystyle \frac{1}{2}}{h}_{ik}\right)\hfill \end{array}\right.$$

(29)

To measure stability, the deviation from the ideal equilibrium point $\left(0,L/2,H/2\right)$ is defined as:

$$\Delta g=\sqrt{{(X_{cog})}^2+{(Y_{cog}-L/2)}^2+{(Z_{cog}-H/2)}^2}$$

(30)

If $\Delta g>g_{safe}$, the CGPA–LLR mechanism is activated.

4.3.2. Local Loading Repair

LLR attempts to restore balance without altering the route structure.

(1)

Posture mutation operator

The item contributing most to CG deviation is identified. Its orientation is adjusted to modify projected dimensions along the $X-$ or $Y$-axis, thereby correcting CG offset (see Figure 11).

(2)

Cross-cabin migration operator

Customers are treated as atomic units. The algorithm computes total customer weight and identifies candidates contributing most to left–right imbalance. Priority is given to moving entire customers to the lighter cabin. If spatial constraints prevent full migration, item-level exchanges between cabins are attempted (see Figure 12).

4.3.3. CGPA Mechanism

If LLR fails to satisfy the $\Delta g$ constraint, CGPA modifies the route structure. First, the algorithm identifies the contiguous customer segment ${\pi }_p^{infeasibl}$ that contributes most to $\Delta g$ imbalance. This segment is inserted into a tabu set $\mathcal{T}$:

$$\mathcal{T}\leftarrow \mathcal{T}\cup \left\{{\pi }_p^{infeasibl}\right\}$$

(31)

The construction of tabu set $\mathcal{T}$ in Equation (31) encodes a cross-domain mapping that distinguishes CGPA from conventional infeasibility handling. Standard metaheuristics treat CoG violations as binary outcomes and discard infeasible solutions without extracting structural information from them. CGPA instead treats $\Delta g$ as a diagnostic signal: the contiguous customer segment contributing most to the CoG imbalance is identified as the structural source of instability and incorporated into $\mathcal{T}$. The corresponding ranking keys in the IWSO position vector are then perturbed as described in Algorithm 2, so that spatial mass imbalance is translated into a discrete routing constraint and loading feasibility feedback actively shapes the combinatorial search trajectory rather than merely filtering its outputs.

During subsequent IWSO position updates, decoded routes containing tabu segments receive additional perturbation to break structural similarity and guide the search away from recurrent infeasible patterns. Thus, CGPA introduces memory-guided structural diversification into the metaheuristic process.

The pseudocode for LLR and CGPA is summarized in Algorithms 1 and 2.

Algorithm 1 LLR
Input: πp: the closed UAV delivery route of UAV p ${\mathcal{I}}_{\mathrm{p}}^{\mathrm{load}}$: the loading scheme of UAV p $g_{safe}$: the threshold of the centroid offset Output: ${\mathcal{I}}_{\mathrm{p}}^{\mathrm{load}}$ and $\Delta \mathrm{g}$ (if still infeasible, trigger CGPA)
1:	Compute Δg
2:	If $\Delta \mathrm{g}\le {\mathrm{g}}_{\mathrm{safe}}$ then
3:	return ${\mathcal{I}}_{\mathrm{p}}^{\mathrm{load}}$ and $\Delta \mathrm{g}$
4:	endif
5:	apply the posture mutation operator to ${\mathcal{I}}_{\mathrm{p}}^{\mathrm{load}}$
6:	recompute $\Delta \mathrm{g}$
7:	if $\Delta \mathrm{g}\le {\mathrm{g}}_{\mathrm{safe}}$ then
8:	return ${\mathcal{I}}_{\mathrm{p}}^{\mathrm{load}}$ and $\Delta \mathrm{g}$
9:	endif
10:	apply the cross-cabin migration operator to ${\mathcal{I}}_{\mathrm{p}}^{\mathrm{load}}$
11:	recompute $\Delta \mathrm{g}$
12:	if $\Delta \mathrm{g}>{\mathrm{g}}_{\mathrm{safe}}$ then
13:	trigger CGPA
14:	endif
15:	return ${\mathcal{I}}_{\mathrm{p}}^{\mathrm{load}}$ and $\Delta \mathrm{g}$

Algorithm 2 CGPA
Input: $W_s\left(t\right)$: the position vector of search individual s at iteration t $w_i^s$: the sorting key value of customer i in $W_s\left(t\right)$ $g_{safe}$: the threshold of the centroid offset $\mathcal{I}$: the global tabu set Output: $\mathcal{I}$
1:	If $\Delta g\le g_{safe}$ then
2:	return $\mathcal{I}$
3:	endif
4:	identify ${\pi }_p^{infeasibl}$
5:	$$\mathcal{T}\leftarrow \mathcal{T}\cup \left\{{\pi }_p^{infeasibl}\right\}$$
6:	decode $W_s\left(t\right)$ to obtain the route ${\pi }_p$
7:	if ${\pi }_p$ contains ${\pi }_p^{infeasibl}$ in $\mathcal{T}$
8:	randomly perturb the sorting key values $w_i^s$ corresponding to that segment to destroy the relative order
9:	endif
10:	return $\mathcal{T}$

4.4. Convergence Criterion and Computational Complexity Analysis

4.4.1. Convergence Criterion

The algorithm terminates when either of the following conditions is met: (1) the iteration count reaches the predefined maximum $T_{max}$; or (2) all UAV routes have obtained feasible loading plans satisfying $\Delta g\le g_{safe}$. Upon termination, the solution with the minimum total routing distance among all recorded feasible solutions is returned.

4.4.2. IWSO Routing Module

Let $n$ denote the number of customer nodes, $N_{pop}$ the population size, and $T_{max}$ the maximum iterations. Per iteration, SPV decoding sorts the three-dimensional position vector in $O\left(nlogn\right)$; fitness evaluation and both the elite-layer exchange mutation and ordinary-layer locking-reconstruction operators each run in $O\left(n\right)$. The total complexity of IWSO is therefore:

$$\mathcal{O}\left(T_{max}\cdot N_{pop}\cdot nlogn\right)$$

(32)

4.4.3. HLDCPS Loading Module

Let $M_p={\displaystyle \sum _{i\in {\pi }_p}}{m}_i$ denote the total number of parcels on route ${\pi }_p$. Loading sequence construction and left–right cabin balancing run in $O\left(M_p\right)$ and $O\left(\left|{\pi }_p\right|\right)$, respectively. For each parcel, six orthogonal orientations are enumerated; boundary feasibility is checked in $O\left(1\right)$ and non-overlap verification against all placed parcels in $O\left(M_p\right)$, giving $O\left(M_p\right)$ per parcel and $O\left(M_p^2\right)$ per route. Across a fleet of $P$ UAVs, the total complexity is:

$$\mathcal{O} \left({\displaystyle {\displaystyle \sum }_{p=1}^P}{M}_p^2\right)$$

(33)

4.4.4. CGPA–LLR

CoG deviation computation runs in $O\left(M_p\right)$. The posture mutation operator enumerates six orientations for the worst-offending parcel in $O\left(M_p\right)$; the cross-cabin migration operator scans $\left|{\pi }_p\right|$ customers and recomputes CoG in $O\left(\left|{\pi }_p\right|+M_p\right)$. CGPA segment identification enumerates all contiguous subsequences of the route in $O(|{\pi }_p{|}^2)$; tabu set update and key perturbation each run in $O\left(\left|{\pi }_p\right|\right)$. The per-call complexity of CGPA–LLR is thus:

$$\mathcal{O}(|{\pi }_p{|}^2+M_p)$$

(34)

4.4.5. Overall Complexity

Combining all modules, the total complexity of GLS-WSCPA is:

$$\mathcal{O} \left(T_{max}\cdot N_{pop}\cdot nlogn+T_{max}\cdot N_{pop}\cdot {\displaystyle {\displaystyle \sum }_{p=1}^P}(|{\pi }_p{|}^2+M_p^2)\right)$$

(35)

Under the typical assumption that $m_i$ is bounded by a constant and routes are balanced such that $\left|{\pi }_p\right|=O\left(n/P\right)$ and $M_p=O\left(\left|{\pi }_p\right|\right)$, this simplifies to:

$$\mathcal{O} \left(T_{max}\cdot N_{pop}\cdot {\displaystyle \frac{n^2}{P}}\right)$$

(36)

This polynomial growth with respect to problem scale is consistent with the complexity of comparable metaheuristic frameworks for three-dimensional loading vehicle routing problems reported in the literature, confirming the computational scalability of GLS-WSCPA.

5. Numerical Experiments

The proposed GLS-WSCPA framework was written in Python 3.10 and executed on a Windows 10 platform equipped with an NVIDIA RTX 3060 GPU (NVIDIA Corporation, Santa Clara, CA, USA). The parameter settings for the algorithm are summarized in

Table 2. Parameter settings for GLS-WSCPA.

Parameter	Description	Value
$W$, $L$, $H$	The cargo bay width, length, height	90 cm, 90 cm, 80 cm
$G$	Maximum payload capacity of the UAV	12 kg
$g_{safe}$	Threshold for the CoG deviation	10 cm
$N_{pop}$	Population size	30
$T_{max}$	Maximum number of iterations	500
$f$	Scaling factor	0.67
$A_0$	The initial perturbation amplitude	2.0
$a_0$, $a_1$	Control parameters	5, 50
$\partial$	Layering threshold ratio	0.3

.

5.1. Performance Comparison of IWSO

Before addressing the 3DLC-UAVRP, it is necessary to first evaluate the underlying optimization capability of the IWSO algorithm. In this experiment, the TSPLIB benchmark library, including representative instances such as eil51, berlin52, st70, and pr76, is used to compare IWSO with WSO, ABCSS [37], and DSMO [38].

ABCSS is an improved Artificial Bee Colony algorithm designed for the TSP. It introduces multiple update rules based on swap sequences and swap operations, combined with a roulette wheel selection mechanism and a K-opt perturbation strategy, thereby improving solution accuracy and stability. DSMO simulates the social behavior of spider monkeys, enabling information exchange among individuals through swap sequence and swap operator mechanisms, and employs a multi-population cooperative framework to enhance global search capability and effectively avoid local optima.

The parameter settings for all comparative algorithms are adopted from their respective original publications to ensure fairness. The computational results are summarized in

Table 3. Experimental results of IWSO and other algorithms.

No.	Instance	ABCSS	DSMO	WSO	IWSO
1	burma14	30.87	30.87	30.87	30.87
2	ulysses16	73.99	73.99	86.33	73.99
3	ulysses22	75.31	75.31	80.15	75.31
4	eil51	428.98	428.86	463.21	426
5	berlin52	7544.37	7544.37	8031.21	7542
6	st70	682.57	677.11	693.36	675
7	eil76	550.24	558.68	555.49	538
8	pr76	108,879.7	108,159.4	108,409	108,159
9	kroA100	21,299	21,298.21	21,459	21,282
10	kroB100	22,229.71	22,308	24,109	22,141
11	rd100	7944.32	8041.3	8062	7940
12	eil101	646.05	648.66	701.6	629
13	pr107	44,525.68	44,385.86	46,587.46	44,438
14	pr124	59,030.74	60,285.21	61,235.12	59,030
15	pr136	97,853.91	97,538.68	102,680	98,494
16	gr137	713.91	709.48	736.89	711.53
17	kroA150	26,981.98	27,591.44	28,693.73	26,524
18	kroB150	26,760.79	26,601.94	28,125.23	26,350.34
19	pr152	74,337.62	74,243.91	76,321.15	73,682
20	d198	16,270.22	15,978.13	16,350.56	16,061
21	kroA200	30,701.86	30,481.35	31,025	29,723
22	kroB200	31,508.85	30,716.5	32,545.7	30,451.3
23	gr202	507.27	501.83	512.36	491
24	tsp225	4140.24	4013.68	4537.24	4011
25	pr226	82,266	83,587.98	85,421.11	81,264
26	gr229	1713.54	1683.45	1733	1680.41
27	gil262	2526.99	2543.15	2853	2431
28	pr299	50,265.88	50,579.82	51,208.16	50,168.32
29	lin318	45,135.5	44,118.66	50,124	43,178
30	fl417	12,356.44	12,218.98	13,966	11,978.32

, where the best results obtained by IWSO are highlighted in bold. In addition, Figure 13 illustrates the optimal routing solutions generated by IWSO for four benchmark instances: berlin52, eil76, kroA100, and pr152.

As shown in Table 3, for most small- and medium-scale benchmark instances, IWSO achieves solution accuracy comparable to that of the other improved algorithms. However, for more complex instances such as kroA100 and pr152, IWSO demonstrates relatively better performance, producing competitive routing distances among the compared methods. In the largest-scale instance, fl417, IWSO achieves a 1.97% reduction in route length compared to DSMO in terms of route length reduction and achieves a 14.23% improvement over the baseline WSO. These results indicate that as the problem scale increases, IWSO shows relatively stable performance and helps alleviate premature convergence to local optima. Overall, the findings suggest the effectiveness and potential scalability of the proposed improvement strategies, particularly in large and complex routing scenarios.

5.2. Feasibility Verification of HLDCPS

To validate the physical adaptability and robustness of the proposed loading algorithm, simulation experiments are conducted based on a dataset constructed from real-world packing practices of JD Logistics. The proposed HLDCPS is compared against a baseline method referred to as the Sequence-Constrained Loading Strategy (SCLS). The SCLS strictly enforces the LIFO unloading accessibility constraint and organizes loading according to the service sequence. This ensures that each customer’s items can be directly retrieved at the corresponding stop without relocating other goods, thereby minimizing rearrangement operations during unloading [39]. The comparative results are presented as follows.

As illustrated in Figure 14, the two loading strategies exhibit fundamentally different behaviors when addressing the conflict between sequence constraints and weight balance. The SCLS strictly follows the predefined loading sequence and prioritizes placing heavier items at the bottom of the cabin to ensure basic vertical stability. However, when multiple heavy items appear consecutively in the loading sequence, the rigid sequential constraint forces them to accumulate along the same spatial direction. This inevitably results in an imbalanced horizontal mass distribution, as shown in Figure 14a. In contrast, while maintaining sequence accessibility constraints, HLDCPS employs a divide-and-conquer mechanism to decouple the loading space. Specifically, heavy items are alternately distributed between the left and right cabins, thereby achieving effective horizontal mass balance without violating unloading feasibility, as illustrated in Figure 14b. These results demonstrate that HLDCPS helps alleviate the structural conflict between operational accessibility and weight equilibrium through a more flexible spatial coordination strategy.

5.3. Algorithmic Evaluation and Mechanism Analysis

This section systematically evaluates the GLS-WSCPA framework by constructing a set of mixed benchmark instances. The benchmark instances are built with reference to literature [40], integrating multi-source benchmark data and adapting it to the drone delivery scenario as follows: Solomon’s VRPTW benchmark instances C101 (clustered distribution) and R101 (random distribution) are selected to define the spatial distribution of customer nodes, while cargo information is fused from the CVRP instances proposed by Gendreau et al. [41]. Given that the original data were designed for land-based container transportation, the dimensions and weight parameters of the heterogeneous cargos are proportionally scaled to satisfy the payload and cargo bay capacity constraints of drones. This process constructs a high-dimensional constrained test environment suitable for micro-cargo compartment scenarios.

5.3.1. Comparison with Other Methods

To quantitatively evaluate the performance superiority of the GLS-WSCPA framework in solving the 3DLC-UAVRP, this section conducts a comparative study against three representative hybrid optimization algorithms on the Solomon–Gendreau R101 hybrid benchmark instance. The comparison benchmarks include CP-EA [42], RSO-IGA [43], and ALNS-DBLF [44]. The comparison results are summarized in

Table 4. Comparative results on the Solomon–Gendreau R101 hybrid benchmark instance.

		Routing Distance			Wilcoxon Signed-Rank Tests
Scales	Method	Best	Mean	Std	vs. GLS-WSCPA
n = 25	GLS-WSCPA	309.42	311.79	1.15	-
	CP-EA	305.81	306.57	0.38	<0.682
	RSO-IGA	310.45	314.12	2.12	<0.05
	ALNS-DBLF	313.20	316.47	1.95	<0.01
n = 50	GLS-WSCPA	489.12	492.58	1.42	-
	CP-EA	524.35	534.28	7.15	<0.05
	RSO-IGA	516.82	524.30	5.34	<0.01
	ALNS-DBLF	552.40	561.37	4.82	<0.001
n = 75	GLS-WSCPA	768.45	772.26	2.05	-
	CP-EA	815.12	838.54	13.85	<0.001
	RSO-IGA	806.34	829.78	11.42	<0.001
	ALNS-DBLF	851.65	869.27	9.35	<0.001
n = 100	GLS-WSCPA	1391.54	1398.92	5.82	-
	CP-EA	1558.40	1602.60	14.12	<0.001
	RSO-IGA	1485.12	1544.63	9.45	<0.001
	ALNS-DBLF	1668.75	1695.85	16.20	<0.001

, while the corresponding route structures for the case of n = 50 are illustrated in Figure 15. Here, n represents the number of customer nodes.

Across the tested problem scales, GLS-WSCPA generally achieves better solutions than the other compared algorithms. Wilcoxon signed-rank tests confirm that these improvements reach statistical significance at n ≥ 50 for all competitors, and at n = 75 and n = 100 all p-values fall below 0.001, indicating highly consistent advantages at larger scales. The performance gap tends to widen as instance size grows: at n = 100, GLS-WSCPA achieves a best solution of 1391.54, approximately 16.7% lower than ALNS-DBLF (1668.75) and 11.3% lower than RSO-IGA (1485.12).

At n = 25, CP-EA obtains a marginally better best solution (305.81 vs. 309.42), and the difference against GLS-WSCPA is not statistically significant (p = 0.682), suggesting that at small scale, CP-EA’s constraint programming backbone provides comparable feasibility guarantees. However, this advantage disappears at n ≥ 50, where CP-EA’s mean value deteriorates substantially (534.28 vs. 492.58), consistent with the known scalability limitations of exact-based methods.

In terms of solution stability, GLS-WSCPA maintains the lowest standard deviation across all scales (e.g., 1.15 at n = 25, 5.82 at n = 100), while CP-EA and ALNS-DBLF exhibit notably higher variance on medium- to large-scale instances, reflecting increased solution instability as problem complexity grows.

5.3.2. Ablation Experiments

After validating the overall effectiveness of the proposed framework, an ablation study is conducted on the Solomon–Gendreau C101 hybrid benchmark instance to further examine the individual contributions of the CGPA and LLR components within the CGPA–LLR mechanism and to evaluate their collaborative effects.

To perform a controlled comparison, four algorithmic variants are designed:

(1)

Baseline: The CGPA–LLR mechanism is completely removed. If the center-of-gravity constraint is violated, the solution is declared infeasible and the search process restarts.

(2)

No CGPA: Only LLR is retained. When imbalance occurs, local physical repair operations—such as posture mutation and cross-compartment migration—are performed. If repair fails, the solution is discarded.

(3)

No LLR: Only CGPA is retained. When imbalance occurs, physical repair is skipped. Instead, conflicting route segments are identified and converted into tabu constraints to adjust the visit sequence.

(4)

GLS-WSCPA: The complete collaborative framework integrating both CGPA and LLR mechanisms.

Each scheme is independently executed 30 times for every problem scale, and average performance metrics are reported. The comparative results are summarized in

Table 5. Comparative results on the Solomon-Gendreau C101 hybrid benchmark instance.

Scales	Indicator	Baseline	No CGPA	No LLR	GLS-WSCPA
n = 25	Fleet size	3	3	3	3
	total routing distance	254.05 **	244.75 *	244.96 *	240.20
	CoG deviation	8.98	9.44	9.48	8.45
n = 50	Fleet size	4	4	4	4
	total routing distance	404.34 ***	369.82 **	370.28 **	361.81
	CoG deviation	8.82	9.33	9.77	9.14
n = 75	Fleet size	6	6	6	6
	total routing distance	742.04 ***	723.00 ***	668.73 *	659.27
	CoG deviation	9.15	9.24	9.31	9.06
n = 100	Fleet size	9	9	9	9
	total routing distance	1097.16 ***	997.11 ***	913.74 *	903.13
	CoG deviation	9.30	9.55	9.68	9.52

* denotes results of one-tailed Wilcoxon signed-rank tests vs. GLS-WSCPA (30 paired runs): * p < 0.05, ** p < 0.01, *** p < 0.001.

.

The results in Table 5 indicate that removing the CGPA–LLR mechanism leads to statistically significant deterioration in total routing distance across all problem scales. This suggests that the baseline strategy, which relies solely on re-searching when infeasibility occurs, lacks effective mechanisms for repairing infeasible solutions, and consequently tends to select more conservative but path-redundant feasible solutions.

The relative effectiveness of individual modules varies with problem scale. For the medium-scale instance (n = 50), the variant retaining only the LLR module achieves slightly shorter routing distances than the variant retaining only CGPA (369.82 vs. 370.28, p < 0.01). This indicates that under moderate constraint intensity, resolving stability conflicts through local physical loading adjustments alone can effectively restore feasibility without substantially altering route structures.

However, for large-scale instances (n = 100), the performance trend reverses. The No LLR variant outperforms No CGPA, achieving an 8.36% reduction in total routing distance (913.74 vs. 997.11, p < 0.001). This demonstrates that as problem complexity and constraint conflicts increase, structured optimization of the customer visiting sequence contributes more than localized physical adjustments under the tested large-scale settings.

When LLR and CGPA are integrated within the full GLS-WSCPA framework, the total routing distance for n = 100 is reduced by 17.68% compared with the Baseline strategy (p < 0.001). Although the center-of-gravity deviation is slightly higher than that of the Baseline, it consistently remains within the predefined safety threshold, indicating that the integrated framework better utilizes the feasible region within safety boundaries to achieve lower routing costs.

As shown in Figure 16, the Baseline method produces routes with notable spatial dispersion, where certain sorties extend over large service areas with visible route crossings between trips. When CGPA is removed, the overall route structure remains similar to the Baseline, with long-distance sorties to the right-side node cluster still present and limited improvement in spatial boundary definition. When LLR is removed, the long-distance extensions are partially mitigated, but the service areas of individual sorties remain spatially discontinuous in some regions. In contrast, GLS-WSCPA generates the most spatially compact route structure among the four schemes: long-distance extensions to the right-side cluster are substantially reduced, the boundaries among service sectors are more clearly delineated, and inter-route crossings are minimized.

5.3.3. Sensitivity Analysis

To comprehensively evaluate the robustness and generalizability of the proposed optimization framework, a sensitivity analysis was conducted based on the instances described in Section 5.3.2. Multiple intervals were defined for both the allowable CoG deviation and the maximum payload coefficient. The model performance was systematically examined under different parameter settings, and the variation in total routing distance was compared across scenarios. In the tables, “–” indicates that no feasible solution was obtained within 30 independent runs.

The results in

Table 6. Total route distance under different load factors.

Scale	The Payload Coefficient	Total Routing Distance	Scale	The Payload Coefficient	Total Routing Distance
25	0.8	278.18	75	0.8	687.84
25	1	241.12	75	1	660.13
25	1.2	186.07	75	1.2	590.91
25	1.4	166.67	75	1.4	548.41
25	1.6	166.24	75	1.6	495.99
50	0.8	408.19	100	0.8	1037.64
50	1	360.19	100	1	902.65
50	1.2	344.36	100	1.2	730.39
50	1.4	335.66	100	1.4	608.29
50	1.6	320.45	100	1.6	582.38

show that as the payload coefficient increases from 0.8 to 1.4, the total routing distance exhibits an overall decreasing trend. For the large-scale instance (n = 100), when payload capacity increases, the total routing distance decreases from 1037.64 to 568.29, representing a 43.87% reduction. This trend is directly related to the payload expansion mechanism: higher payload capacity allows each UAV to carry more parcels per sortie, thereby reducing the required number of flights and potentially improving overall routing efficiency.

Table 7. Total path distance under different centroid offset coefficients.

Scale	CoG Deviation Coefficients	Total Routing Distance	Scale	CoG Deviation Coefficients	Total Routing Distance
25	0.8	355.10	75	0.8	-
25	1	241.12	75	1	660.13
25	1.2	231.71	75	1.2	651.67
25	1.4	225.49	75	1.4	645.80
25	1.6	214.16	75	1.6	645.35
50	0.8	435.96	100	0.8	-
50	1	360.19	100	1	902.65
50	1.2	359.09	100	1.2	779.29
50	1.4	353.47	100	1.4	767.69
50	1.6	339.26	100	1.6	766.02

further indicates that under large-scale scenarios (n = 75 and n = 100), excessively strict CoG deviation coefficients (e.g., 0.8) prevent the algorithm from effectively handling heterogeneous loading configurations, resulting in infeasible solutions. This suggests that pursuing near-perfect balance may render practical delivery tasks inoperable. As the CoG deviation coefficient is relaxed from 1.0 to 1.4, the total routing distance decreases consistently across all instances. For example, when n = 100 and the coefficient increases to 1.2, the routing distance is reduced by 13.67%. However, when the coefficient exceeds 1.4, the marginal improvement in routing performance becomes negligible, indicating that beyond this threshold, the CoG constraint may no longer be the primary limiting factor under the tested settings.

5.4. Case Analysis

This case study is constructed using operational data from a JD Logistics fulfillment center located in the northeastern area of Jinan, China. The dataset consists of one distribution center and 20 customer nodes. The scenario is designed to evaluate the applicability of the proposed method in a typical urban last-mile delivery environment, characterized by dispersed demand patterns and heterogeneous payloads, thereby providing a representative test setting.

For performance evaluation, a conventional “route-first, loading-decoupled” baseline strategy is adopted for comparison. Under this approach, delivery routes are first determined using the IWSO algorithm. Subsequently, loading is executed using the SCLS, and payload and CoG constraints are verified post hoc. If the addition of a customer to a route results in a violation of physical constraints, the route is truncated at that customer node, and an additional UAV is activated. This process continues iteratively until all customers are assigned to feasible delivery routes. The comparative results are summarized in

Table 8. Fleet size and total routing distance under different strategies.

Strategy	Fleet Size	Total Routing Distance	SD	CV
Traditional	5	143.00	4.27	2.99%
GLS-WSCPA	4	133.20	1.22	0.92%

Note: Results are reported over 30 independent runs. A Wilcoxon signed-rank test (two-tailed) confirms a statistically significant difference between the two strategies ($W=0$, $p<0.001$, rank-biserial correlation $r=-1.000$).

, where CV = standard deviation/mean, which is used to measure the stability of the algorithm across different scenarios. A smaller CV indicates better robustness. The distribution of total routing distances across 30 independent runs is further illustrated in Figure 17, while the corresponding drone delivery route structures are presented in Figure 18.

As shown in Table 8, the conventional baseline strategy is constrained by its fixed loading rules. When the center-of-gravity deviation exceeds the allowable threshold, the original loading configuration cannot be repaired, and feasibility can only be restored by activating additional flight sorties. In contrast, the proposed GLS-WSCPA framework dynamically coordinates routing and loading decisions, thereby helping to mitigate potential loading conflicts before they lead to infeasibility. As a result, the required fleet size for serving the entire service region is reduced from five UAV sorties to four, while the total routing distance decreases by 6.85%. These findings demonstrate that collaborative optimization of routing and loading shows potential to improve operational efficiency and resource utilization, as demonstrated in this case study under stability constraints. Furthermore, statistical analysis over 30 independent runs confirms that GLS-WSCPA achieves substantially lower standard deviation (1.22 vs. 4.27) and coefficient of variation (0.92% vs. 2.99%) compared to the traditional strategy, indicating superior solution stability. A Wilcoxon signed-rank test (two-tailed) further validates that this performance advantage is statistically significant ($W=0$, $p<0.001$, rank-biserial correlation $r=-1.000$), confirming that the improvement is consistent and not attributable to random variation.

6. Conclusions

This study formulated the Three-Dimensional Loading-Constrained UAV Routing Problem (3DLC-UAVRP) and proposed GLS-WSCPA, a guided collaborative optimization framework that jointly addresses routing sequence optimization and three-dimensional loading feasibility under CoG stability constraints.

At the modeling level, the 3DLC-UAVRP encodes the transmission chain from customer visit sequence through LIFO loading order to three-dimensional parcel coordinates and static pre-takeoff CoG within a single formulation, making CoG feasibility a direct function of routing decisions rather than an external post hoc check.

At the algorithmic level, IWSO adapts the continuous White Shark Optimization to the discrete routing domain by introducing SPM chaotic initialization and a fitness-based hierarchical update strategy, which improves solution diversity and convergence stability relative to the original algorithm. HLDCPS addresses the conflict between LIFO unloading accessibility and left–right mass balance by distributing cargo across compartments during packing construction rather than treating these two requirements sequentially. These two components are coordinated through the CGPA–LLR mechanism: when a CoG violation cannot be repaired locally by LLR through parcel reorientation and cross-compartment relocation, CGPA identifies the responsible route segment and introduces it as a tabu constraint, redirecting the routing search away from infeasible regions. This feedback channel from loading evaluation back into route generation allows the framework to handle the coupling between routing decisions and loading outcomes within a single search process.

Experimental results demonstrate the effectiveness of the proposed framework. GLS-WSCPA generally achieves better results than the compared algorithms across the tested problem scales, with performance advantages becoming more pronounced as instance size increases. Ablation studies confirm the complementary roles of CGPA and LLR, and sensitivity analysis suggests that relaxing payload and CoG constraints tends to reduce routing distance, while maintaining acceptable stability margins. Case analysis further shows that GLS-WSCPA achieves a reduction in fleet size by 20% and in total delivery distance by 6.85% in the tested case study, with significantly lower solution variance.

Several limitations point to directions for future work. CoG constraints are currently enforced only in the pre-takeoff state; incorporating dynamic mass shift during parcel unloading and in-flight disturbances such as wind loading would bring the model closer to operational conditions. Extending the framework to stochastic demand environments and developing tighter theoretical lower bounds for the 3DLC-UAVRP also remain open for further investigation.