MEMOBWO: A Novel Multi-Objective Optimization Algorithm for UAV Path Planning in Complex Urban Environments

Abstract

Path planning for Unmanned Aerial Vehicles (UAV) in complex urban environments poses significant challenges for autonomous systems. This paper proposes a Multi-strategy Enhanced Multi-Objective Beluga Whale Optimization algorithm, termed MEMOBWO, to address these problems. The proposed MEMOBWO adopts a multi-objective optimization framework to overcome the limitations of traditional single-objective approaches while simultaneously enhancing exploration and exploitation capabilities through three complementary strategies. Firstly, a Chaotic Quasi-Opposition-Based Learning (CQOBL) strategy is introduced to enhance initial population diversity and quality. Secondly, a Hybrid Adaptive Position Update (HAPU) strategy is designed to dynamically balance global exploration and local exploitation. Finally, a Multi-Objective Thinking Innovation (MOTI) strategy is proposed as a targeted repair operator to overcome specific performance deficiencies of whale agents in weaker objectives. To evaluate its performance, the MEMOBWO was comprehensively tested through 20 standard multi-objective benchmark functions, as well as three-dimensional (3D) UAV path planning experiments in simulated urban environments with varying obstacle configurations, and was compared against a series of classical and recently proposed multi-objective optimization algorithms. Moreover, the overall performance of the algorithms was assessed using Hypervolume (HV) and Inverted Generational Distance (IGD) metrics and further tested using the Friedman test and Wilcoxon rank-sum test. Experimental results demonstrated that MEMOBWO achieved competitive performance across benchmark functions, and showed favorable overall performance against comparison algorithms in path planning tasks, attaining the lowest average Friedman rank as 1.14 and HV improvements of 15.24% to 30.86%. This study provides a promising optimization framework for multi-objective UAV path planning problems in urban environments, thereby lowering the tracking burden of downstream UAV flight-control and trajectory tracking.

1. Introduction

Unmanned Aerial Vehicles (UAVs) are aircraft operated via autonomous flight systems [1,2]. Currently, technologies for UAV are widely used in fields such as military reconnaissance [3], aerial photography [4], material transport [5], agricultural spraying [6], and more recently, in Internet of Things networks for energy-efficient data collection [7], etc. As one of the core issues of UAV technologies, the UAV path planning problems [8,9] have always been extensively studied. From the perspective of UAV trajectory tracking and flight-control execution, the quality of the planned reference path directly affects the downstream control burden. Paths with abrupt heading changes, excessive climbing-angle variations, or high curvature may increase tracking difficulty and impose larger demands on the flight-control system.

Discussions of path planning problems tend to be regarded as optimization problems. After the objective functions are defined, typically multiple, the selection of optimal paths requires the simultaneous minimization of all the objective functions. Since the minimum points of each function do not coincide, some are even conflicting, it means that a reduction in one single function may lead to an increase in another. To solve this issue, most existing studies choose to combine the functions into a single objective function by using a weighted sum approach [10], in which different objective functions are calculated with corresponding weights. Alternatively, through the main goal method, the constraint method is transformed into selecting the most important sub-goal from a single goal as the optimization goal, and the rest of the sub-goals as constraints. Each sub-goal is constrained by setting an upper bound. Although the methods mentioned above are straightforward to implement, it is still difficult to determine appropriate weights for each function in complex optimization problems while maintaining the optimality of the solution. Therefore, this paper employs the multi-objective optimization [11] approach to solve the UAV path planning problems.

As UAV technologies evolve towards more intelligent and interconnected paradigms, such as Digital Twin-assisted autonomous navigation [12] and IoT-enabled cooperative sensing [13], the operational environments for UAVs are becoming increasingly dynamic and complex. Depending on the differences of spatial dimension, the path planning problems can be categorized into two-dimensional (2D) and three-dimensional (3D) path planning [14]. The dimensionality of the environments significantly influences path planning approaches. In traditional 2D space, the simplified environmental modeling cannot intuitively reflect the complex terrain encountered by the UAVs during the urban missions, often leading to inaccurate optimization results. Three-dimensional spatial models can incorporate real terrain data to establish terrain models for the UAV flight, which are used to test the optimization performances of path planning algorithms, thereby enhancing the path planning capabilities of UAV and improving its efficiency in exploration and transport tasks [15]. Moreover, traditional path planning algorithms, including Artificial Potential Field Method [16], Rapidly-exploring Random Tree (RRT) [17], A* [18] and Dijkstra [19], are suitable for solving relatively simple path planning problems, such as operating obstacle avoidance on 2D maps. However, due to the difficulties handling dynamic obstacles, susceptibility to falling into local optima and high computational complexity, these methods are often difficult to apply in problems with complex 3D environments or diverse obstacle types. In contrast, meta-heuristic algorithms [20] are intelligent optimization methods based on the cooperative mechanisms of biological populations. These algorithms treat the feasible paths as candidate solutions and improve them via swarm intelligence methods [21], thus possessing advantages of high efficiency, flexibility and robustness when solving complex problems. Consequently, they are widely used for solving complex math problems [22]. Typical meta-heuristic algorithms include Genetic Algorithm (GA) [23], Particle Swarm Optimization (PSO) [24], Simulated Annealing (SA) [25] and Ant Colony Optimization (ACO) [26], etc. These meta-heuristic algorithms provide new approaches for solving UAV path planning problems [27], especially in the rather complex 3D urban environments.

In recent years, with the gradual development of related research, researchers worldwide have conducted further exploration into UAV path planning, proposing many improved path planning algorithms. Nature-based optimization techniques have been widely applied in relevant research. Among them, the Beluga Whale Optimization (BWO) algorithm [28] is a relatively novel swarm-based meta-heuristic algorithm inspired from the behaviors of beluga whales, which was proposed by Changting Zhong, et al. in 2022. It features a simple structure, ease of implementation and insensitivity to initial values, making it widely applicable to various kinds of optimization problems like feature selection [29], energy capacity allocation [30] and transportation scheduling [31] after its publication. At the same time, several variants of the algorithm have been introduced by relevant researchers. For example, SC Horng et al. have introduced an improved Beluga Whale Optimization algorithm for solving the simulation optimization problems with stochastic constraints [32]. In 2024, a multi-strategies improved Beluga Whale Optimization algorithm was proposed by Zhaoyong Fan et al. to optimize feature selection process [33]. Moreover, multiple Beluga Whale Optimization algorithms improved by hybrid strategies were introduced to solve engineering optimization problems [34,35]. However, few studies applied the BWO to UAV path planning problems, particularly in 3D urban environments. This research gap motivates our work to adapt the MEMOBWO for multi-objective path planning in 3D urban environments.

This paper proposes a novel algorithm called Multi-strategy Enhanced Multi-Objective Beluga Whale Optimization (MEMOBWO). The standard BWO is enhanced by three complementary improvement strategies. Chaotic Quasi-Opposition-Based Learning (CQOBL) is used to improve the quality and diversity of the initial population. Hybrid Adaptive Position Update (HAPU) dynamically balances global exploration and local exploitation through nonlinear energy factor. Multi-Objective Thinking Innovation (MOTI) is used as a targeted repair operator to overcome the performance defects of the solution on the weaker objective dimension. Our contributions in this work include: (i) the establishment of kinematic model of UAV, the objective function system of optimization model and the simulated complex 3D urban environments, (ii) proposal of three collaborative improvement strategies to systematically improve the performance of the algorithm in convergence, diversity and solution quality, covering the deficiency of standard BWO when dealing with more complex multi-dimensional optimization problems, (iii) verification of the effectiveness of MEMOBWO in dealing with multiple conflicting targets in simulated 3D urban environments.

To evaluate the performance of MEMOBWO in addressing UAV path planning problems, extensive experiments were conducted in simulated three-dimensional urban environments with varying obstacle configurations. The algorithm was tested against a series of multi-objective optimization algorithms, including MORBMO [36], NMOPSO [37], MOHHO, NSGA-II and standard MOBWO. NSGA-II is a classical optimization algorithm while the others are all recently published algorithms. The experimental results demonstrated that the MEMOBWO consistently generated flight paths with better performance across multiple conflicting objectives, and exhibited enhanced convergence characteristics toward the Pareto front while maintaining greater solution diversity. This underscores the effectiveness of MEMOBWO in balancing the exploration and exploitation behaviors, validating its advantages on offline path planning.

The remainder of this paper is organized as follows: Section 2 focuses on three-dimensional environmental modeling and objective function design for the path planning problem, Section 3 proposes the MEMOBWO algorithm based on the standard Beluga Whale Optimization algorithm and analyzes its main innovative components, and Section 4 involves comparative experiments conducted on a simulation platform using the proposed algorithm to verify its effectiveness, Section 5 summarizes the paper and outlines potential areas for further improvement in future research.

2. Problem Formulation

The UAV path planning problem is described as follows: under the physical and kinematic constraints of UAV, as well as the certain terrain obstacles and non-flying zones of the environments, a series of optimal flight paths of the UAV from a start point to a destination are planned. Efficient solutions to this question by optimization algorithms require three key formulations, including simulated environment modeling, kinematic modeling and the defining of objective functions.

2.1. Simulated Environment Modeling

Assuming the map constraints are known, one of the challenges in three-dimensional environment modeling is acquiring ground elevation parameters. This can be addressed by creating a 3D matrix to include the terrain size and resolution, where the value of each element indicates the elevation above ground at its corresponding position. The constraints in UAV path planning problems can be specifically categorized into physical constraints and environmental constraints. The first category includes constraints on the geometric properties of the generated paths, such as heading angle, climbing angle and curvature. The second category encompasses terrain obstacles and electromagnetic interference in urban environments. Currently, the main safety threats to flight in urban environments come from tall buildings, trees and street lights, as well as electromagnetic field interference generated by signal towers or power plants.

To represent typical urban flight constraints in a computationally tractable form, building-like obstacles are modeled as cuboids, small urban structures such as trees, street lamps, and poles are modeled as cylinders, and electromagnetic interference (EMI) risk regions are modeled as hemispherical soft-threat regions. These geometric primitives are used as planning-level obstacle and risk proxies for controllable and repeatable algorithm evaluation.

2.2. UAV Kinematic Model

On the premise that a certain safe distance is left between the UAV and the environmental obstacles, and the obstacle avoidance distance is much larger than the body size, the structure of UAV can be regarded as a particle, indicating that most of the influence of the physical conditions is ignored. However, due to the limited inertia and motor thrust of the real UAV, the generated paths still necessitates constructing mathematical models that accurately reflect the motion characteristics of UAV and fully accounting for its inherent physical performance limitations.

A point-mass kinematic model is adopted at the path geometric planning level. Its core state variables are the position $P={\left(x,y,z\right)}^T$ of UAV in three-dimensional space and its velocity v. The motion is determined by the heading angle $\psi$ and climbing angle $\theta$, with the fundamental kinematic equations given by:

$$\begin{array}{cc}\hfill \dot{x}& =vcos\theta cos\psi \hfill \\ \hfill \dot{y}& =vcos\theta sin\psi \hfill \\ \hfill \dot{z}& =vsin\theta \hfill \end{array}$$

(1)

where v denotes the velocity scalar along the forward axis of the UAV body. Equation (1) is defined to describe the geometric relationship between the position change rate, velocity and orientation angle of the UAV. The climbing angle $\theta$, heading-angle change $\mathsf{\Delta }\psi$, climbing-angle change $\mathsf{\Delta }\theta$ and curvature of the generated path are constrained as:

$$\left\{\begin{array}{c}\left|\theta \right|\le {\theta }_{max}\hfill \\ \left|\mathsf{\Delta }\psi \right|\le \mathsf{\Delta }{\psi }_{max}\hfill \\ \left|\mathsf{\Delta }\theta \right|\le \mathsf{\Delta }{\theta }_{max}\hfill \\ \kappa \le {\kappa }_{max}\hfill \end{array}\right.$$

(2)

After defining the kinematic parameters of the UAV, the minimum turning radius $R_{min}$ of UAV is derived from the maximum roll angle ${\varphi }_{max}$. Accordingly, the curvature $\kappa \left(s\right)$ at any point on the paths must satisfy the constraint of $R_{min}$ to ensure that the UAV has the ability to perform the turning maneuver required by the path.

$$\begin{array}{cc}\hfill & R_{min}={\displaystyle \frac{v^2}{g·tan{\varphi }_{max}}}\hfill \\ \hfill & \kappa \left(s\right)\le {\displaystyle \frac{1}{R_{min}}}\hfill \end{array}$$

(3)

where g = 9.8 m/s². denotes the gravitational acceleration, variable s should be valued between 0 and the total length of the flight path.

Additionally, the initial paths produced by discrete algorithms are polylines, which contain multiple sharp and angular turns that would be mechanically impossible for the UAV to execute at a certain speed. The MEMOBWO algorithm utilizes the continuous Bezier smoothing technique to improve geometric continuity of generated paths and reduce abrupt local direction changes. The satisfaction of the constraints above is quantitatively evaluated before and after smoothing in the experimental section.

2.3. Objective Functions Definition

To evaluate the performance of the path planning algorithm, considering the simulated environments in 3D space and the constraints mentioned in the previous sections, the following four objective functions are designed: path spatial length, path vertical maneuver cost, path smoothness and path threat degree. Let the path P be represented by a sequence of N waypoints, the objective vector is defined as $\mathit{F}={\left[f_1,f_2,f_3,f_4\right]}^{\mathrm{T}}$. Each component is detailed below. All the objective functions and constraints are designed to guide the algorithm to generate collision-free and geometrically executable paths.

2.3.1. Path Spatial Length Function

Assuming the UAV flies at a constant speed during the flight, generated paths with shorter length implies reduced flight time and energy consumption. Therefore, the spatial length of a feasible path is one of the crucial factors for assessing the algorithm’s performance.

The path spatial length function aims to minimize the total Euclidean distance of the flight path to ensure the UAV reaches the target in a possibly minimum time, to increase its operation efficiency. Let N be the number of waypoints and $P_i$ be the coordinate of the ith waypoint. The cost function is defined as:

$$f_1={\displaystyle \sum _{i=1}^{N-1}}{\parallel P_{i+1}-P_i\parallel }_2$$

(4)

where $\parallel P_{i+1}-P_i{\parallel }_2$ denotes the Euclidean norm of two contiguous waypoints in the generated path. Minimizing $f_1$ ensures the shortest geometrical route between the start and goal configurations. By minimizing the $f_1$ function, the algorithm can gradually find a shorter path from the starting point to the end point, reducing the task consumption time.

2.3.2. Path Vertical Maneuver Cost Function

The power consumption of the UAV is non-linearly related to changes in height, where ascending actions require higher motor torque to overcome gravity compared to level flight. During UAV flight, frequent altitude variations may increase the energy demand and impose additional burden on vertical maneuvering.

Let $h_i=z_{i+1}-z_i$ represent the altitude difference between two consecutive waypoints, the objective function models the asymmetrical energy cost of vertical maneuvers as:

$$f_2={\displaystyle \sum _{i=1}^{N-1}}\epsilon \left(\mathsf{\Delta }{h}_i\right)$$

(5)

The path vertical maneuver cost function is formulated to penalize climbing or descending:

$$\epsilon \left(\mathsf{\Delta }{h}_i\right)=\left\{\begin{array}{cc}{\mu }_{\mathrm{climb}}·\mathsf{\Delta }{h}_i,\hfill & \mathrm{if}\mathsf{\Delta }{h}_i>0\hfill \\ {\mu }_{\mathrm{desc}}·\left|\mathsf{\Delta }{h}_i\right|,\hfill & \mathrm{Otherwise}\hfill \end{array}\right.$$

(6)

where ${\mu }_{\mathrm{climb}}$ and ${\mu }_{\mathrm{desc}}$ are weighting coefficients. The parameters are set as ${\mu }_{\mathrm{climb}}>{\mu }_{\mathrm{desc}}$, typically ratio 3:1, like ${\mu }_{\mathrm{climb}}$ = 0.3 while ${\mu }_{\mathrm{desc}}$ = 0.1, to impose a higher penalty on climbing than on descending due to the higher energy consumption of climbing maneuvers.

The explanation of $f_2$ function is as Figure 1. Minimizing $f_2$ naturally encourages the algorithm to find paths similar to the contour type that could avoid unnecessary or sudden altitude changes in the flight.

2.3.3. Path Smoothness Function

During the flight of UAV, excessive turns and pitch maneuvers, as well as excessively large turning angles, can affect travel speed and increase energy consumption and the risk of damage. Therefore, a feasible path should minimize the number of sharp turns and reduce the maximum turning angle. The path smoothness objective function is expressed as:

$$f_3={\alpha }_1{\displaystyle \sum _{i=1}^{N-2}}{\sigma }_{\mathrm{yaw}}^2\left(i\right)+{\alpha }_2{\displaystyle \sum _{i=1}^{N-2}}{\sigma }_{\mathrm{pitch}}^2\left(i\right)$$

(7)

where ${\sigma }_{\mathrm{yaw}}$ and ${\sigma }_{\mathrm{pitch}}$ represent the horizontal heading angle and vertical climbing angle of the UAV at the ith waypoint, respectively, and ${\alpha }_1$ and ${\alpha }_2$ are the weight coefficients of turn and pitch actions of UAV, typically set as ${\alpha }_1$ = ${\alpha }_2$ = 0.5.

${\sigma }_{\mathrm{yaw}}$ is defined as:

$${\sigma }_{\mathrm{yaw}}=arccos\left({\displaystyle \frac{V_i^{x}y·V_{i+1}^{x}y}{\parallel V_i^{x}y\parallel ·\parallel V_{i+1}^{x}y\parallel }}\right)$$

(8)

where $V_i^{x}y$ is the projection vector of each flight segment on the horizontal plane $Oxy$, $\parallel ·\parallel$ represents the modulus length of the vector. To ensure numerical stability, $\left|\right|V_i^{x}y\left|\right|$ is defined to be greater than 0.

${\sigma }_{\mathrm{pitch}}$ is defined as:

$${\sigma }_{\mathrm{pitch}}={\displaystyle \frac{arctan\mathsf{\Delta }z}{\parallel {\left(\mathsf{\Delta }x\right)}^2+{\left(\mathsf{\Delta }y\right)}^2{\parallel }_2}}$$

(9)

where $\mathsf{\Delta }x$, $\mathsf{\Delta }y$ and $\mathsf{\Delta }z$ represent the changes of 3D coordinates of each flight segment in the path. This angle is the absolute value of the difference between the elevation angles of the two preceding and following flight segments, which is used to measure the changes of altitude in the vertical direction during the flight. By minimizing $f_3$, the flight path generated by the algorithm will be smoother, and the burden of UAV performing turning and pitching actions will be reduced, generating smooth flight paths that are more suitable as reference paths for subsequent trajectory tracking.

2.3.4. Path Threat Degree Function

A feasible path must guarantee avoidance of physical collision while simultaneously mitigating the risk of control link loss due to the electromagnetic interference (EMI). In this work, the urban environment contains physical obstacles and spectral threats. Let $C_{phys}$ be the physical collision risk and $C_{spec}$ be the interference risk, the composite safety objective is expressed as:

$$f_4={\displaystyle \frac{1}{N}}{\displaystyle \sum _{k=1}^N}\left(C_{phys}\left(p_k\right)+C_{spec}\left(p_k\right)\right)$$

(10)

where $p_k$ denotes the UAV position at sample k, and $E_n$ represents the nth EMI source with center $C_n$ and interference radius $R_n$.

The physical collision cost $C_{phys}$ is calculated based on $d_v$, $S_{uav}$ and $S_{safe}$, which means the distance to the nearest structure, the diameter of UAV and the safety distance between it and the obstacles, respectively. $C_{phys}\left(p_k\right)$ is then defined as:

$$C_{phys}\left(p_k\right)=\left\{\begin{array}{cc}\infty ,\hfill & d_v\le S_{uav}\hfill \\ 1-{\displaystyle \frac{d_v-S_{uav}}{S_{safe}}},\hfill & S_{uav}<d_v<S_{uav}+S_{safe}\hfill \\ 0,\hfill & d_v\ge S_{uav}+S_{safe}\hfill \end{array}\right.$$

(11)

A hemisphere interference is defined to simulate the potential field threat. Let $D_n=\parallel p_k-C_n\parallel$ be the Euclidean distance to the EMI center, the spectral cost $C_{spec}$ is defined as:

$$C_{spec}\left(p_k\right)=\left\{\begin{array}{cc}\mu ·exp\left(\alpha ·{\displaystyle \frac{1.5R_n-D_n}{R_n}}\right),\hfill & \mathrm{if}{D}_n<1.5R_n\hfill \\ 0,\hfill & \mathrm{Otherwise}\hfill \end{array}\right.$$

(12)

$C_{spec}\left(p_k\right)$ implies that the UAV perceives EMI sources as soft repulsive fields that extend 1.5 times beyond their physical radius $R_n$, hence forcing the path planner to bend the path away from high-interference zones to preserve communication integrity. The explanation of the EMI interference zone is shown in Figure 2. Minimizing $f_4$ encourages the algorithm to generate flight paths with higher safety in complex urban environment, thus ensuring the reliability and the operational safety of the UAV.

3. The Proposed Approach

3.1. Standard BWO

The Beluga Whale Optimization algorithm simulates the swimming, foraging, and the ’Whale Fall’ behaviors of a specific kind of animals called beluga whales. Belugas are highly social animals, usually gathering in groups of 2 to 25 members. When summer comes, many creatures gather in some estuaries, making the whales gather to forage. Belugas with the coordination group attack and feed the fish by guiding them into shallow water. In addition, some whales may die and fall into the deep ocean during the migration or escaping from predators, which is also known as the ’Whale Fall’ behaviors, providing plenty of food for creatures without sunlight or oxygen. The BWO algorithm process contains three certain phases, including exploration phase, exploitation phase and ’Whale Fall’ phase. The exploration phase of BWO is established by considering the swimming behavior of beluga whales. Instead of a search agent moving randomly on its own, it updates its position based on the position of a randomly selected partner whale from the population. Therefore, the positions for beluga whales are updated as:

$$X_{i,j}^{T+1}=\left\{\begin{array}{cc}{X}_{i,P_j}^T+(X_{r,P_j}^T-X_{l,P_j}^T)·(1+r_1)·sin\left(2\pi r_2\right),\hfill & j=\mathrm{even}\hfill \\ X_{i,P_j}^T+(X_{r,P_j}^T-X_{l,P_j}^T)·(1+r_1)·cos\left(2\pi r_2\right),\hfill & j=\mathrm{odd}\hfill \end{array}\right.$$

(13)

where T is the current iteration, $X_{i,j}^{T+1}$ is the new position for the ith beluga whale on the jth dimension, $P_j$ ($j=1,2,\dots ,d$) is a random number selected from d-dimension, $X_{i,P_j}^T$ is the position of the ith beluga whale on $P_j$ dimension, $X_{r,P_j}^T$ and $X_{l,P_j}^T$ are the current positions for rth and lth beluga whale, which are both randomly chosen, $r_1$ and $r_2$ are random numbers, $sin\left(2\pi r_2\right)$ and $cos\left(2\pi r_2\right)$ mean fins of the mirrored beluga whales are toward the surface. According to the dimension chosen by odd and even number, the updated position reflects the synchronous or mirror behaviors of pair swimming or diving.

The exploitation phase of BWO is inspired by the preying behavior of beluga whales. They prey by sharing the information of positions for each other, considering the best candidate and others. The Levy flight strategy is introduced in the exploitative phase of BWO to enhance the convergence. The mathematical model of the exploitation phase is expressed as:

$$X_i^{T+1}=r_3{X}_{best}^T-r_4{X}_i^T+C_1{L}_F(X_r^T-X_i^T)$$

(14)

where T is the current iteration, $X_i^T$ and $X_r^T$ are current position for the ith beluga whale and a random beluga whale, $X_i^{T+1}$ is the position of new position of the ith beluga whale, $X_{best}^T$ is the best position among beluga whales, $r_3$ and $r_4$ are random numbers between $(0,1)$, $C_1=2r_d\left(1-{\displaystyle \frac{t}{T_{max}}}\right)$ is the random jump strength that measuring the intensity of Levy flight. LF is the Levy flight function, calculated as:

$$L_F=0.05·{\displaystyle \frac{u·\sigma }{{\left|v\right|}^{\frac{1}{\beta }}}}$$

(15)

$$\sigma ={\left({\displaystyle \frac{\mathsf{\Gamma }(1+\beta )·sin(\frac{\pi }{2}·\beta )}{\mathsf{\Gamma }\left(\frac{1+\beta }{2}\right)·\beta ·2^{\frac{\beta -1}{2}}}}\right)}^{{\displaystyle \frac{1}{\beta }}}$$

(16)

where u and v are normally distributed random numbers, $\beta$ is the default constant equal to $1.5$.

The beluga whales either migrate elsewhere or die. In order to keep the population size constant, the positions of beluga whales and step size of ’whale fall’ phase are used to establish the updated position. The formula is as:

$$X_l^{T+1}=r_5{X}_l^T-r_6{X}_r^T+r_7{X}_{step}$$

(17)

where $r_5,r_6,r_7\in (0,1)$ are random numbers, $X_{step}$ is the step size of ’whale fall’, which is established as:

$$X_{step}=(u_b-l_b)exp(-C_2{\displaystyle \frac{T}{T_{max}}})$$

(18)

where $C_2=2W_f·n$ is the step factor which is related to the probability of ’whale fall’ and population size, $u_b$ and $l_b$ are upper and lower boundary of variables, respectively. It can be seen that the step size is affected by the boundaries of design variables, iteration and maximum iterative number. The probability of ’whale fall’ $W_f$ is calculated as:

$$W_f=0.1-0.05·{\displaystyle \frac{T}{T_{max}}}$$

(19)

3.2. The Proposed MEMOBWO

The original BWO algorithm, due to its own limitations, will face problems of insufficient diversity of the solution set and be easy to fall into local optima when solving some more complex optimization problems. In order to apply BWO algorithm to multi-objective optimization problems and make it more adaptable in path planning problems, a series of corresponding improvement approaches is proposed in this section.

3.2.1. Chaotic Quasi-Opposition-Based Learning

The diversity and quality of the initial population are important for constructing a well-distributed Pareto archive in the early stage of the search.Traditional initialization methods [38] based on random distribution often lead to an uneven exploration of the search space, potentially trapping the algorithm in local optima. To overcome these limitations, this paper proposes a Chaotic Quasi-Opposition-Based Learning (CQOBL) strategy. This strategy introduces Quasi-Opposition-Based Learning [39] (QOBL) to evaluate both the current solution and its opposite solution simultaneously. It also introduces a Piecewise chaotic sequence to control the generation of quasi-opposite candidates between the center of the search space and the opposite point. After generating both the random population and the chaotic quasi-opposition population, non-dominated sorting and objective evaluation are used to select higher-quality individuals for the initial population. This design provides a more diverse sampling pattern while keeping the generated candidates within the feasible variable bounds.

Let $X_{ij}$ be the jth dimension of the ith individual in the population, bounded by $[l{b}_j,u{b}_j]$. The standard opposition solution ${\tilde{X}}_{ij}$ is defined as:

$${\tilde{X}}_{i,j}=l{b}_j+u{b}_j-X_{i,j}$$

(20)

The center of the search space, $M_j$, is calculated as $M_j=(l{b}_j+u{b}_j)/2$. The proposed CQOBL strategy generates a candidate solution $X_{\{i,j\}}^{\mathrm{new}}$ dynamically within the interval $[M_j,{\tilde{X}}_{\{i,j\}}]$, governed by a chaotic sequence. The formulation is defined as:

$$X_{i,j}^{\mathrm{new}}=M_j+{\gamma }_k·({\tilde{X}}_{i,j}-M_j)$$

(21)

where ${\gamma }_k$, valued between $(0,1)$, represents the chaotic value generated by the Piecewise chaotic map at iteration k. The Piecewise map is selected for its superior ergodicity and uniform distribution properties compared to the Logistic or Tent maps. The mathematical expression of the Piecewise map is defined as:

$${\gamma }_{k+1}=\left\{\begin{array}{cc}{\displaystyle \frac{{\gamma }_k}{P}},\hfill & 0\le {\gamma }_k<P\hfill \\ {\displaystyle \frac{{\gamma }_k-P}{0.5-P}},\hfill & P\le {\gamma }_k<0.5\hfill \\ {\displaystyle \frac{1-{\gamma }_k-P}{0.5-P}},\hfill & 0.5\le {\gamma }_k<1-P\hfill \\ {\displaystyle \frac{1-{\gamma }_k}{P}},\hfill & 1-P\le {\gamma }_k<1\hfill \end{array}\right.$$

(22)

By utilizing ${\gamma }_k$, the algorithm can perform a fine-grained, non-linear scan of the potential optimal region between the geometric center and the opposition point. The operation principle of CQOBL is shown in Figure 3. This mechanism enhances the algorithm’s ability to jump out of local optima and improves the diversity of the initial population and subsequent generations.

3.2.2. Hybrid Adaptive Position Update

Existing meta-heuristic algorithms tend to introduce Lévy flight to enhance global exploration or spiral search to strengthen local exploitation. In contrast, HAPU is designed to improve the position update process of MEMOBWO by coordinating long-range exploration and local exploitation in a stage-adaptive manner. It combines three components in a unified update mechanism, including a nonlinear energy factor, an adaptive switching probability and a hybrid stochastic exploration operator.

The behavioral patterns of the beluga population are influenced by their energy states. To simulate this biological characteristic and to quantitatively describe the temporal evolution of the search process, a nonlinear adaptive energy factor $B_t$ is introduced to the algorithm. Unlike linear attenuation strategies, $B_t$ reflects energy fluctuations and nonlinear decay in complex environments. The formulation of $B_t$ is defined as:

$$B_t=B_0{\left(1-{\displaystyle \frac{t}{T_{max}}}\right)}^{\alpha }cos(2\pi ·\mathrm{rand})$$

(23)

where $B_0$ represents the initial energy state, typically set as constant 2, t and $T_{max}$ denote the current iteration index and the maximum number of iterations, respectively, and $\alpha$ is a nonlinear exponent controlling the decay rate.

The introduction of $B_t$ provides the algorithm with a global evolutionary mark, whose oscillatory decay trend characterizes the overall transition of the population from broad exploration to fine-grained exploitation. Under the energy evolution governed by $B_t$, an adaptive switching probability $P_a\left(t\right)$ is introduced as the core control parameter to explicitly guide the behavioral decisions of individuals at different evolutionary stages. To achieve a smooth transition from exploration-dominated to exploitation-dominated behavior, the switching threshold $P_a\left(t\right)$ is designed as a cosine-based monotonically decreasing function of the iteration index t defined as:

$$P_a\left(t\right)=\mu +(1-\mu )·cos\left({\displaystyle \frac{\pi ·t}{2·T_{max}}}\right)$$

(24)

where $\mu$ is a constant coefficient that controls the lower bound of the switching probability. With the introduction of $B_t$ and $P_a\left(t\right)$, the mechanism of dynamic cooperation between Lévy flight and spiral search is defined as:

$$X_{t+1}=\left\{\begin{array}{cc}L\left(X_t\right),\hfill & \mathrm{if}{B}_t<P_a\left(t\right)\hfill \\ S\left(X_t\right),\hfill & \mathrm{Otherwise}\hfill \end{array}\right.$$

(25)

where $L\left(X_t\right)$ and $S\left(X_t\right)$ denote the Lévy flight operator and the spiral search operator, respectively. During the early stages of iteration, $P_a\left(t\right)$ remains at a relatively high level, leading to a high probability of triggering the Lévy flight operator $L\left(X_t\right)$. Leveraging its heavy-tailed distribution, Lévy flight drives individuals to perform long-distance jumps, thereby maximizing the coverage of the search space. As the iteration proceeds, $P_a\left(t\right)$ gradually decreases, causing the algorithm to favor the spiral search operator $S\left(X_t\right)$, which guides individuals to approach the current best solution via logarithmic spiral trajectories for fine-grained exploitation.

The mathematical definition of cooperative operators is as follows. Focusing on global exploration, the position update rule is defined as:

$$L_c(\lambda ,\gamma ,t)=\left[1-\omega \left(t\right)\right]·L_F+\omega \left(t\right)·\mathrm{Cauchy}(0,\gamma )$$

(26)

where $\mathrm{Cauchy}(0,\gamma )$ represents the standard Cauchy distribution with location parameter zero and scale parameter $\gamma$. The time-varying mixing weight $\omega \left(t\right)$ is given by:

$$\omega \left(t\right)=1-exp\left({\displaystyle \frac{-\kappa ·t}{T_{max}}}\right)$$

(27)

where $\kappa$ is usually set to 3 to govern the transition speed of the weight. The scale parameter $\gamma$ is scaled according to the problem dimension as $\gamma =0.1·{\displaystyle \frac{u_b-l_b}{\sqrt{d}}}$, where $u_b$ and $l_b$ are the upper and lower bounds of the search space and d is the dimensionality. The innovation lies in combining the heavy-tail property of the Lévy distribution, which facilitates large-step global exploration, with the broad-tail characteristic of the Cauchy distribution, which promotes fine-grained local search. Through $\omega \left(t\right)$, the HAPU strategy forms a multi-level adaptive optimization framework.

Focusing on deep exploitation, this rule is defined as:

$$S\left(X_t\right)=D^{\prime }·e^{bl}·cos\left(2\pi t\right)+X_{\mathrm{best}}$$

(28)

where $D^{\prime }$ denotes the distance between the individual and the current best solution, b is a constant defining the shape of the logarithmic spiral, and $l\in [-1, 1]$ is a random number. This equation enables search agents to scan the target region locally with a smoothly contracting path.

The proposed HAPU strategy provides a staged and probabilistic update mechanism that adjusts both the search mode and perturbation structure during the iteration process. By overcoming the limitations of single-mechanism strategies in terms of search breadth and depth, it enhances the global optimization efficiency of the algorithm.

3.2.3. Multi-Objective Thinking Innovation Strategy

The search operators in optimization algorithms usually apply uniform evolutionary pressure on all decision variables, which lacks the flexibility to repair specific performance deficiencies. Inspired by the concept of “Thinking and Innovation” [40], the Multi-Objective Thinking Innovation (MOTI) strategy uses different elite guidance for different individuals according to their weakest objectives. The repair direction for the population is no longer uniform, but is adaptively determined by the objective-level deficiency of each individual.

Given the disparate units and magnitudes of the objective functions, direct comparisons of objective values is infeasible. At each generation t, the objective values of the population are dynamically normalized. Let $f_m\left(x_i\right)$ denote the mth objective value of the ith whale, the normalized value ${\widehat{f}}_m\left(x_i\right)$ is calculated as:

$${\widehat{f}}_m\left(x_i\right)={\displaystyle \frac{f_m\left(x_i\right)-f_m^{min}}{f_m^{max}-f_m^{min}+\epsilon }}$$

(29)

where $f_m^{max}$ and $f_m^{min}$ denote the maximum and minimum values of the population on the mth objective dimension, respectively, and $\epsilon$ is a small constant introduced to avoid division by zero. Through this linear mapping, all objective functions are projected onto a dimensionless interval $[0, 1]$, thereby eliminating the scale discrepancies among different physical quantities of the objective functions. This normalization provides a unified measurement basis for subsequent performance evaluation.

After the normalization of dynamic objectives, the algorithm identifies the performance deficiency of each individual based on the normalized objective matrix. For minimization problems, a larger normalized objective value indicates poorer performance on that dimension. Accordingly, the weakest objective index $k_i$ of individual $X_i$ is determined as:

$$k_i=arg\underset{m\in \{1,\dots ,M\}}{max}\left\{{\widehat{f}}_m\left(X_i\right)\right\}$$

(30)

This step identifies the primary performance deficiency of the current solution, providing explicit directional guidance for subsequent targeted search.

When the weakest dimension $k_i$ is determined, the algorithm performs targeted refinement. Specifically, the individual $X_i$ learns from the elite solution $X_{\mathrm{best},k}$, which exhibits the best performance on objective $k_i$ within the current population. The position update rule is defined as:

$$X_i^{\mathrm{new}}=X_i+\phi ·(X_{\mathrm{best},k}-X_i)+\gamma ·(R_1-R_2)$$

(31)

where $X_{\mathrm{best},k}$ denotes the individual with the minimum fitness value on the objective, $\phi \in [0, 1]$ is a random learning rate that controls the degree of attraction toward the elite solution, and $\gamma$ is a disturbance coefficient. Term $(R_1-R_2)$ represents a random difference vector, which is introduced to preserve population diversity and prevent premature convergence.

The flow chart of the MOTI strategy is shown as Figure 4. This proposed strategy acts as a supplementary local refinement mechanism that helps individuals improve their most deficient objective while maintaining the multi-objective search framework.

3.3. Time Complexity Analysis

The time complexity of the MEMOBWO algorithm is mainly determined by the population size N, number of objectives M, the maximum number of iterations T, the problem dimension D and the objective evaluation cost of one candidate path $C_f$. The original BWO algorithm has a complexity of $O(T\times N\times D)$.

The proposed MEMOBWO introduces three improvement strategies. Their complexities are analyzed by examining the mathematical formulas involved:

CQOBL: This strategy is applied only once during initialization. It generates a chaotic sequence and computes a quasi-oppositional solution for each individual. Each operation involves basic arithmetic and logic per dimension, resulting in a complexity of $O(N\times D)$. Since it is not repeated iteratively, its contribution does not multiply by T.

HAPU: For each iteration, this strategy updates every individual’s position using the energy factor $B_t$, the adaptive switching probability $P_a\left(t\right)$, and either the Lévy flight or spiral search operator. All calculations are performed per dimension per individual, with a complexity of $O(N\times D)$ per iteration. Over T iterations, the total complexity is $O(T\times N\times D)$.

MOTI: In each iteration, this strategy normalizes the objective values (Equation (29), $O(N\times M)$, where M is a small constant), identifies the weakest objective dimension (Equation (30), $O(N\times M)$), and performs a targeted position update (Equation (31), $O(N\times D)$). All operations are linear in N and D, leading to a per-iteration complexity of $O(N\times D)$. Thus, the total cost is $O(T\times N\times D)$.

The objective evaluation of all individuals requires $O(N\times C_f)$ per iteration. For UAV path planning, $C_f$ includes the computation of path spatial length, vertical maneuver cost, smoothness, and safety-related threat cost, and is therefore reported separately. Archive management involves dominance comparison and crowding-distance-based pruning. During archive updating, the dominance comparison cost is $O\left(M{(A+N)}^2\right)$, and the crowding-distance pruning cost is approximately $O(M\times AlogA)$. Thus, the per-iteration complexity can be expressed as:

$$O_{\mathrm{iter}}=O(N\times D)+O(N\times M)+O(N\times C_f)+O\left(M{(A+N)}^2\right)+O(M\times AlogA)$$

(32)

Therefore, the overall time complexity of MEMOBWO is described as:

$$O\left(\mathrm{MEMOBWO}\right)=O(N\times D)+T{O}_{\mathrm{iter}}$$

(33)

The analysis confirms that the proposed improvement strategies of MEMOBWO remain linear with respect to N and D, while the complete multi-objective implementation also includes objective evaluation and archive management costs. Since M and A are fixed in the experiments, and each run of path planning experiments can be completed within 1 min, the computational cost remains acceptable for offline mission planning scenarios.

3.4. Implementation of MEMOBWO

The flowchart of MEMOBWO is described as Figure 5. Furthermore, Figure 6 shows a schematic diagram of the MEMOBWO algorithm system model, where Figure 5 constitutes the core layer of the algorithm.

The detailed path planning steps of the proposed MEMOBWO algorithm are operated as Algorithm 1:

Algorithm 1 Implementation of MEMOBWO

1: Initialize parameters: population size N, maximum iterations $T_{max}$, archive size $A_{size}$   2: Begin   3: Initialize chaotic sequence $\gamma$ using Piecewise map   4: Initialize population $P_{\mathrm{rand}}$ with random positions within $[l_b,u_b]$   5: Generate quasi-opposition population $P_{\mathrm{quasi}}$ using CQOBL strategy by Equation (21).   6: Select top N individuals from $\{P_{\mathrm{rand}}\cup P_{\mathrm{quasi}}\}$ to form initial population $X_0$   7: Evaluate objective functions $F\left(X_0\right)={[f_1,f_2,f_3,f_4]}^T$   8: Initialize archive A with non-dominated solutions from $X_0$   9: Set iteration counter $t=0$ 10: while $t<T_{max}$ do 11:       Update nonlinear energy factor $B_t$ using Equation (23) 12:       Update adaptive switching probability $P_a\left(t\right)$ using Equation (24) 13:       for each whale $X_i$ in population do 14:     if $B_t<P_a\left(t\right)$ then 15:        Update position using Lévy Flight operator $L(X,t)$ by Equation (26) 16:     else 17:        Update position using Spiral Search operator $S(X,t)$ by Equation (28) 18:     end if 19:     Amend violated boundaries 20:       end for 21:       Calculate normalized objective values $f_m\left(X\right)$ by Equation (29) 22:       for each whale $X_i$ in population do 23:     Identify weakest dimension index $k,i=argmax\left\{f_m\left(X_i\right)\right\}$ 24:     Identify dimension elite $X_{\mathrm{best},k}$ 25:     Update position using MOTI by Equation (31) 26:       end for 27:       Evaluate fitness of updated population $X_{t+1}$ 28:       Update archive A by adding non-dominated solutions from $X_{t+1}$ 29:       Remove dominated solutions from A 30:       if $\left|A\right|>A_{size}$ then 31:     Apply crowding distance sorting 32:     Prune excess solutions to maintain diversity 33:       end if 34:       $t=t+1$ 35: end while 36: Return Pareto optimal solution set A 37: End

4. Experiments and Comparisons

Relevant simulated urban environments, as well as a series of benchmark functions, are prepared on the certain data simulation platform in order to test the feasibility and optimization performance of MEMOBWO algorithm in face of UAV path planning problem in complex urban environment.

4.1. Performance Metrics

4.1.1. Simulation Scenarios Setup

The data simulation platform is MATLAB R2023b. Due to the restriction that the flight height of UAV should not be higher than 120 m in the certain countries, the simulated city flight area established on the simulation platform is set as a three-dimensional space with dimensions of 1000 m in length, 1000 m in width and 120 m in height. The main threat sources are cubic building obstacles in the city, cylindrical obstacles such as trees and light poles, and electromagnetic interference from signal towers in the city. The discrete flight paths are generated by randomly defining a number of way points in the three-dimensional space, and then combined with the starting point and ending point. All the discrete paths are smoothed by Bezier interpolation method to obtain the final flight path. In the three-dimensional simulated urban environment, the MEMOBWO algorithm is used to search and optimize the navigation points in the path, so as to obtain better path results. Each simulated environment is randomly changed to verify the generality of the algorithm in urban environment: scenario 1 is the case of building obstacles only, scenario 2 to add the elongated cylindrical obstacles such as telephone poles, trees and building roofs, and scenario 3 the urban environment with electromagnetic interference areas, in addition to the obstacles above. The algorithm parameters in the experiments were configured as population size N = 100, maximum iterations $T_{max}$ = 500, archive size $A_{size}$ = 50.

4.1.2. Normalized Objective Functions

It is necessary to note that during the iteration process of MEMOBWO algorithm, the algorithm generates a comprehensive Pareto front to preserve solution diversity and accommodate decision-makers with different preferences in practical use. The weighted-sum method was employed solely for the objective comparison of the final selected solution across different algorithms in our simulation studies. This approach provides a single, scalar performance metric to clearly illustrate the comprehensive optimization ability of each algorithm. The evaluation function is defined as:

$$F_{\mathrm{nor}}={\displaystyle \sum _{i=1}^4}{w}_i{f}_i$$

(34)

where $f_i$ represents the four objective functions mentioned in Section 2.3, from $f_1$ to $f_4$ are path spatial length, path vertical maneuver cost, path smoothness and path threat degree. $w_i$ is the weighted value of each objective function, hence the sum of all $w_i$ in the formula should be 1. In our experiments, the weighted values of all the problems and scenarios were set as $w_1=w_2=w_3=w_4=0.25$ based on fairness of experiment and preferences of users in practical problems. In addition to the weighting coefficients in different cases to be balanced, all objective functions need to be normalized to facilitate data comparisons.

4.1.3. Hypervolume Metric

Hypervolume (HV) metric is one of the core indicators to evaluate the quality of the solution set of multi-objective optimization algorithms. It is able to measure the convergence and diversity of the solution set at the same time, and is frequently used to evaluate the comprehensive operation ability of the algorithm. For minimization problems, a larger HV value indicates better convergence and diversity of the solution set. The formula of HV calculation is defined as:

$$HV\left(P\right)=\mathrm{Vol}\left(\bigcup _{x\in P}[f_1\left(x\right),r_1]\times \cdots \times [f_m\left(x\right),r_m]\right)$$

(35)

where P is the set of non-dominated solutions, r is the reference point selected for HV index calculation. All objective values are normalized to $[0, 1]$ before HV calculation. The reference point is defined based on the worst objective values among all algorithms and remains identical for all experiments, defined as $r=(1.1f_1^{max},\dots ,1.1f_m^{max})$, which is set slightly worse than the worst objective values obtained among all algorithms.

4.1.4. Inverted Generational Distance Metric

Inverted Generational Distance (IGD) is a convergence metric that assesses how well the approximations align with the Pareto Front achieved by the algorithms. The metric is calculated as follows:

$$IGD={\displaystyle \frac{\sqrt{{\sum }_{i=1}^n{d}_i^2}}{n}}$$

(36)

where n represents the total quantity of elements in the True Pareto Front, and $d_i$ denotes the Euclidean distance between the ith solution in Pareto Front and its closest corresponding point within the True Pareto Front. A zero value of this metric signifies that all generated solutions exactly coincide with the True Pareto Front.

4.1.5. Friedman Test

To statistically validate the performance differences among the algorithms, the Friedman test [41] was employed. This non-parametric test is appropriate for comparing multiple algorithms across different test scenarios. For each scenario, the comparison algorithms were ranked based on their final HV values from multiple independent runs. The Friedman test statistic is calculated as:

$${\chi }_F^2={\displaystyle \frac{12N}{k(k+1)}}{\displaystyle \sum _{j=1}^k}\left[R_j^2-{\displaystyle \frac{k{(k+1)}^2}{4}}\right]$$

(37)

where k is the number of comparison algorithms, N is the number of simulation scenarios, and $R_j$ represents the average rank of the jth algorithm. This test reveals statistically significant differences among the comparison algorithms. The algorithm with lower rank shows better performance.

4.1.6. Wilcoxon Rank Sum Test

As a non-parametric test method, the core advantage of Wilcoxon rank sum test [42] is that the sample data does not depend on specific distributions, which makes it particularly suitable for dealing with the non-normality and heteroscedasticity problems that may exist in the performance metrics of the algorithms.

4.2. Performance on Multi-Objective Benchmark Suites

Experiments of benchmark suites are tested on the proposed MEMOBWO algorithm to evaluate its performance. The rationale for conducting experiments across such a diverse range of benchmark problem sets is to establish a testing framework that ensures fairness, consistency, and neutrality in the evaluation of the algorithms. Twenty standard multi-objective test problems were selected from three benchmark suites. The performance of MEMOBWO is evaluated on the benchmark problems. For all the experiments, 100 search agents were employed, with a maximum number of iterations set as 500. The statistical results are shown in

Table 1. Statistical results for the IGD metric on ZDT, DTLZ and WFG benchmark functions generated by the comparison algorithms.

Function	Index	Algorithms
		NSGA2	MEMOBWO	MOHHO	NMOPSO	MOBWO	MORBMO
ZDT1	Mean	5.68 × 10−3	1.93 × 10−3	2.98 × 10−2	7.91 × 10−2	2.01 × 10−3	2.07 × 10−3
	Std	8.19 × 10−3	1.31 × 10−4	1.81 × 10−2	5.24 × 10−2	1.42 × 10−4	1.74 × 10−3
ZDT2	Mean	1.79 × 10−1	1.92 × 10−3	4.50 × 10−2	2.87 × 10−1	1.93 × 10−3	1.14 × 10−1
	Std	1.02 × 10−1	8.37 × 10−5	7.06 × 10−2	4.67 × 10−2	1.17 × 10−4	1.21 × 10−1
ZDT3	Mean	1.14 × 10−2	1.43 × 10−3	2.64 × 10−2	7.48 × 10−2	1.46 × 10−3	1.59 × 10−3
	Std	1.15 × 10−2	6.03 × 10−5	1.88 × 10−2	5.82 × 10−2	9.50 × 10−5	9.29 × 10−5
ZDT4	Mean	2.13 × 100	1.91 × 100	1.91 × 100	2.55 × 100	1.50 × 100	1.18 × 100
	Std	4.52 × 10−16	4.47 × 10−3	4.52 × 10−16	8.75 × 10−1	6.41 × 10−1	7.84 × 10−1
ZDT6	Mean	1.34 × 10−2	1.45 × 10−3	4.64 × 10−2	7.78 × 10−1	1.51 × 10−3	7.64 × 10−1
	Std	3.82 × 10−2	7.57 × 10−5	5.55 × 10−2	4.89 × 10−1	8.94 × 10−5	1.61 × 10−1
DTLZ1	Mean	5.79 × 10−1	1.89 × 10−2	2.20 × 100	7.58 × 100	3.75 × 100	5.17 × 100
	Std	7.26 × 10−1	6.68 × 10−4	1.46 × 100	4.83 × 100	1.02 × 100	1.04 × 100
DTLZ2	Mean	5.15 × 10−2	4.33 × 10−2	6.09 × 10−2	1.22 × 10−1	4.77 × 10−2	5.47 × 10−2
	Std	2.50 × 10−3	1.51 × 10−3	7.47 × 10−3	4.89 × 10−2	3.79 × 10−3	4.32 × 10−3
DTLZ4	Mean	5.34 × 10−2	6.48 × 10−2	9.00 × 10−2	5.69 × 10−1	5.02 × 10−2	1.37 × 10−1
	Std	1.67 × 10−2	1.18 × 10−1	6.57 × 10−2	1.11 × 10−1	5.35 × 10−3	6.04 × 10−2
DTLZ5	Mean	4.72 × 10−3	3.23 × 10−3	8.67 × 10−3	4.28 × 10−2	4.49 × 10−3	1.07 × 10−2
	Std	9.09 × 10−4	1.84 × 10−4	5.78 × 10−3	2.60 × 10−2	6.92 × 10−4	2.46 × 10−3
DTLZ6	Mean	3.11 × 10−3	3.01 × 10−3	3.46 × 10−2	4.22 × 10−1	3.10 × 10−3	6.46 × 10−1
	Std	1.43 × 10−4	1.16 × 10−4	3.07 × 10−2	4.22 × 10−1	1.47 × 10−4	1.61 × 10−1
DTLZ7	Mean	1.91 × 10−1	1.73 × 10−1	2.49 × 10−1	1.29 × 10−1	4.08 × 10−2	5.31 × 10−2
	Std	1.37 × 10−1	1.27 × 10−1	6.68 × 10−2	1.29 × 10−1	1.07 × 10−2	3.25 × 10−2
WFG1	Mean	1.75 × 10−1	3.73 × 10−2	1.83 × 10−2	1.33 × 10−1	4.46 × 10−1	2.35 × 10−1
	Std	3.69 × 10−2	3.03 × 10−3	1.89 × 10−2	5.66 × 10−2	3.94 × 10−4	1.05 × 10−2
WFG2	Mean	1.09 × 10−2	3.30 × 10−3	2.74 × 10−2	3.28 × 10−2	3.92 × 10−3	1.37 × 10−2
	Std	2.14 × 10−3	2.17 × 10−4	9.46 × 10−3	1.12 × 10−2	2.96 × 10−4	9.91 × 10−4
WFG3	Mean	1.30 × 10−1	1.25 × 10−1	1.41 × 10−1	1.29 × 10−1	1.25 × 10−1	1.34 × 10−1
	Std	3.03 × 10−3	4.95 × 10−5	6.72 × 10−3	8.92 × 10−3	2.22 × 10−4	1.13 × 10−3
WFG4	Mean	1.64 × 10−2	5.07 × 10−3	2.21 × 10−2	1.85 × 10−2	9.68 × 10−3	2.20 × 10−2
	Std	2.97 × 10−3	2.89 × 10−4	4.56 × 10−3	4.81 × 10−3	1.35 × 10−3	1.59 × 10−3
WFG5	Mean	2.43 × 10−2	2.23 × 10−2	2.63 × 10−2	3.36 × 10−2	2.32 × 10−2	2.59 × 10−2
	Std	1.57 × 10−3	8.63 × 10−5	2.69 × 10−3	1.42 × 10−2	1.47 × 10−3	1.65 × 10−3
WFG6	Mean	2.42 × 10−2	1.92 × 10−2	6.77 × 10−2	6.03 × 10−2	9.53 × 10−2	1.95 × 10−2
	Std	2.11 × 10−2	8.27 × 10−3	3.78 × 10−2	2.86 × 10−2	9.10 × 10−4	2.52 × 10−3
WFG7	Mean	1.37 × 10−2	5.16 × 10−3	2.71 × 10−2	4.28 × 10−2	5.32 × 10−3	1.71 × 10−2
	Std	4.00 × 10−3	2.42 × 10−4	8.23 × 10−3	2.66 × 10−2	2.92 × 10−4	1.97 × 10−3
WFG8	Mean	5.68 × 10−2	4.37 × 10−2	1.52 × 10−2	2.16 × 10−2	5.02 × 10−2	6.28 × 10−2
	Std	3.81 × 10−3	7.95 × 10−4	3.30 × 10−3	2.16 × 10−2	8.03 × 10−4	3.01 × 10−3
WFG9	Mean	6.68 × 10−3	5.93 × 10−3	2.62 × 10−2	3.98 × 10−2	7.27 × 10−3	1.67 × 10−2
	Std	1.62 × 10−3	5.35 × 10−4	2.30 × 10−3	4.07 × 10−2	3.43 × 10−4	2.32 × 10−3

and

Table 2. Statistical results for the HV metric on ZDT, DTLZ and WFG benchmark functions.

Function	Index	Algorithms
		NSGA2	MEMOBWO	MOHHO	NMOPSO	MOBWO	MORBMO
ZDT1	Mean	4.18 × 10−1	4.21 × 10−1	2.95 × 10−1	3.58 × 10−1	4.21 × 10−1	4.21 × 10−1
	Std	8.83 × 10−3	4.21 × 10−3	6.91 × 10−2	3.77 × 10−2	4.79 × 10−2	7.74 × 10−3
ZDT2	Mean	4.41 × 10−1	3.86 × 10−1	3.72 × 10−1	3.05 × 10−1	3.86 × 10−1	3.86 × 10−1
	Std	1.08 × 10−2	3.87 × 10−2	5.45 × 10−2	0.00 × 100	3.45 × 10−2	3.87 × 10−2
ZDT3	Mean	3.71 × 10−1	3.77 × 10−1	2.76 × 10−1	2.98 × 10−1	3.80 × 10−1	3.80 × 10−1
	Std	2.13 × 10−2	7.66 × 10−2	2.82 × 10−1	8.24 × 10−2	1.83 × 10−2	1.83 × 10−2
ZDT4	Mean	8.05 × 10−2	1.95 × 10−4	1.69 × 10−3	1.52 × 10−4	1.52 × 10−4	1.52 × 10−4
	Std	9.01 × 10−2	8.73 × 10−4	0.00 × 100	8.73 × 10−4	8.73 × 10−4	8.73 × 10−4
ZDT6	Mean	2.88 × 10−1	2.71 × 10−1	2.59 × 10−1	2.55 × 10−1	2.59 × 10−1	2.59 × 10−1
	Std	5.74 × 10−2	2.71 × 10−2	8.04 × 10−2	3.43 × 10−2	8.04 × 10−2	8.04 × 10−2
DTLZ1	Mean	5.59 × 10−1	9.96 × 10−1	2.38 × 10−3	0.00 × 100	0.00 × 100	0.00 × 100
	Std	2.80 × 10−2	7.94 × 10−3	3.52 × 10−3	0.00 × 100	0.00 × 100	0.00 × 100
DTLZ2	Mean	9.16 × 10−1	9.17 × 10−1	9.13 × 10−1	8.81 × 10−1	9.18 × 10−1	9.18 × 10−1
	Std	1.16 × 10−2	1.77 × 10−2	1.29 × 10−2	1.77 × 10−2	3.11 × 10−2	3.11 × 10−2
DTLZ4	Mean	9.20 × 10−1	9.17 × 10−1	9.23 × 10−1	3.24 × 10−1	9.17 × 10−1	9.17 × 10−1
	Std	1.99 × 10−2	3.29 × 10−2	1.03 × 10−2	9.59 × 10−1	3.29 × 10−2	3.29 × 10−2
DTLZ5	Mean	9.20 × 10−1	8.88 × 10−1	8.86 × 10−1	8.70 × 10−1	8.87 × 10−1	8.82 × 10−1
	Std	2.10 × 10−2	8.88 × 10−1	1.15 × 10−2	2.95 × 10−2	8.88 × 10−1	8.88 × 10−1
DTLZ6	Mean	8.76 × 10−1	8.58 × 10−1	8.88 × 10−1	5.53 × 10−1	8.58 × 10−1	8.69 × 10−1
	Std	2.10 × 10−2	8.25 × 10−1	1.15 × 10−2	2.95 × 10−2	8.37 × 10−1	8.88 × 10−1
DTLZ7	Mean	2.74 × 10−1	2.24 × 10−1	2.18 × 10−1	1.07 × 10−1	2.24 × 10−1	2.24 × 10−1
	Std	5.75 × 10−2	2.24 × 10−1	2.18 × 10−1	1.07 × 10−1	2.24 × 10−1	2.24 × 10−1
WFG1	Mean	6.50 × 10−1	6.37 × 10−1	5.86 × 10−1	7.05 × 10−1	6.37 × 10−1	6.37 × 10−1
	Std	8.79 × 10−2	6.37 × 10−1	5.86 × 10−1	7.05 × 10−1	6.37 × 10−1	6.37 × 10−1
WFG2	Mean	5.41 × 10−1	5.46 × 10−1	5.47 × 10−1	7.42 × 10−1	5.46 × 10−1	5.46 × 10−1
	Std	9.61 × 10−3	5.46 × 10−1	5.47 × 10−1	7.42 × 10−1	5.46 × 10−1	5.46 × 10−1
WFG3	Mean	5.76 × 10−1	5.77 × 10−1	5.83 × 10−1	7.05 × 10−1	5.77 × 10−1	5.77 × 10−1
	Std	2.99 × 10−3	5.77 × 10−1	5.83 × 10−1	7.05 × 10−1	5.77 × 10−1	5.77 × 10−1
WFG4	Mean	2.54 × 10−1	2.64 × 10−1	2.45 × 10−1	2.51 × 10−1	2.64 × 10−1	2.64 × 10−1
	Std	4.10 × 10−3	2.64 × 10−1	2.45 × 10−1	2.51 × 10−1	2.64 × 10−1	2.64 × 10−1
WFG5	Mean	2.56 × 10−1	2.56 × 10−1	2.71 × 10−1	2.74 × 10−1	2.56 × 10−1	2.56 × 10−1
	Std	1.07 × 10−2	2.56 × 10−1	2.71 × 10−1	2.74 × 10−1	2.56 × 10−1	2.56 × 10−1
WFG6	Mean	2.67 × 10−1	2.57 × 10−1	2.45 × 10−1	2.24 × 10−1	2.57 × 10−1	2.57 × 10−1
	Std	1.75 × 10−2	2.57 × 10−1	2.45 × 10−1	2.24 × 10−1	2.57 × 10−1	2.57 × 10−1
WFG7	Mean	2.56 × 10−1	2.63 × 10−1	2.61 × 10−1	2.24 × 10−1	2.63 × 10−1	2.63 × 10−1
	Std	3.79 × 10−3	2.63 × 10−1	2.61 × 10−1	2.24 × 10−1	2.63 × 10−1	2.63 × 10−1
WFG8	Mean	2.46 × 10−1	2.62 × 10−1	2.63 × 10−1	2.51 × 10−1	2.62 × 10−1	2.62 × 10−1
	Std	4.82 × 10−3	2.62 × 10−1	2.63 × 10−1	2.51 × 10−1	2.62 × 10−1	2.62 × 10−1
WFG9	Mean	2.60 × 10−1	2.63 × 10−1	2.55 × 10−1	2.51 × 10−1	2.63 × 10−1	2.63 × 10−1
	Std	6.03 × 10−3	2.63 × 10−1	2.55 × 10−1	2.51 × 10−1	2.63 × 10−1	2.63 × 10−1

.

As shown in the table, MEMOBWO demonstrates competitive performance across the majority of benchmark functions. Notably, on DTLZ2, DTLZ5 and DTLZ6, the IGD values of MEMOBWO closely approach those of the best-performing algorithms, while on DTLZ1, MEMOBWO achieves the highest HV value. On functions with more complex landscapes such as ZDT4 and DTLZ7, MEMOBWO exhibits comparatively higher IGD values, indicating that its domain-adaptive design is effectively leveraged in the multi-objective path planning context.

To validate the statistical significance of the observed differences, the Friedman test and Wilcoxon rank sum test were conducted. For the HV metric in

Table 3. Friedman Average Rank and Wilcoxon rank sum test comparisons of HV metric, where ≈ means no significant difference, + means better and $+\ast$ means significantly better, / means not applicable.

Algorithms	Average Rank	Significance
NMOPSO	2.42	≈
MEMOBWO	2.80	/
MORBMO	3.48	+
MOHHO	3.61	+
MOBWO	3.91	$+\ast$
NSGA2	4.50	$+\ast$

, MEMOBWO achieves statistically better improvements, while showing no significant difference when compared with NMOPSO algorithm. For the IGD metric in

Table 4. Friedman Average Rank and Wilcoxon rank sum test comparisons of IGD metric, where ≈ means no significant difference, + means better and $+\ast$ means significantly better, / means not applicable.

Algorithms	Average Rank	Significance
MORBMO	1.66	≈
MEMOBWO	2.41	/
NMOPSO	3.26	+
NSGA2	4.10	+
MOHHO	4.28	+
MOBWO	5.20	$+\ast$

, MEMOBWO significantly outperformed the majority of the comparison algorithms. However, it did not quite come first in both of the tests.

It should be emphasized that the primary strength of MEMOBWO is demonstrated through the comprehensive path planning experiments in Section 4.5. The benchmark experiments serve as a supplementary reference to establish the general optimization capability of MEMOBWO, confirming its competitive performance across diverse problem types.

4.3. MEMOBWO Path Planning Results

Figure 7 showed the paths generated by the MEMOBWO algorithm in the simulated urban environment of different scenarios. The results showed that the selected paths had no collision with the obstacles in the environment, and successfully reached the target position.

The paths shown in Figure 7 were merely part of the numerous non-dominated solutions in the Pareto front obtained by MEMOBWO algorithm. Each non-dominated solution has relative superiority in particular objectives, depending on the application when solving practical problems, thus demonstrating the intelligent balance of the MEMOBWO’s preference between energy efficiency and risk avoidance.

4.4. Ablation Studies

Ablation studies were conducted to systematically verify the effectiveness of each specific strategies introduced in this paper. MEMOBWO₁, MEMOBWO₂ and MEMOBWO₃ represent that the CQOBL, HAPU and MOTI strategies, respectively, are independently selected and used in the standard MOBWO algorithm.

Figure 8 illustrated the optimal paths generated by the algorithms above in the simulated environments. It can be observed that the paths generated by the standard MOBWO algorithm passes dangerously close to the obstacle boundary, resulting in a high safety cost. In contrast, the MEMOBWO path proactively adjusted its heading angle well before approaching the threats. It maintains a safe clearance distance while minimizing the total spatial length. Figure 9 showed the Pareto front generated by the algorithms. Figure 10 and Figure 11 reveals the convergence trend of objective functions and Hypervolume values during the iteration process of each algorithm, respectively. The MEMOBWO and MEMOBWO₁ began with a significantly lower fitness value compared to standard MOBWO and other algorithms, verifying that the CQOBL strategy successfully distributes the initial population more evenly across the search space. Moreover, the MEMOBWO₃ tended to have better outcome Hypervolume and objective functions values compared to other algorithms except the MEMOBWO algorithm. This proved the specific effects of the MOTI strategy. Although the MEMOBWO₂ lacked advantages in the initialization and outcome phase, its non-dominated solution set still performed better diversity and homogeneity, which was shown in Figure 9.

Table 5. Path planning results of MEMOBWO and its ablation versions.

Scenarios & Objectives		Algorithms
		MOBWO	MEMOBWO1	MEMOBWO2	MEMOBWO3	MEMOBWO
1	$f_1$	1233.88	1286.85	1214.83	1166.47	1171.81
	$f_2$	15.73	26.92	24.54	30.34	19.10
	$f_3$	2.01 × 10−3	3.35 × 10−3	2.16 × 10−3	2.01 × 10−3	2.01 × 10−3
	$f_4$	1.30 × 10−3	1.68 × 10−3	1.82 × 10−3	1.68 × 10−3	1.68 × 10−3
2	$f_1$	1226.36	1207.37	1195.38	1178.36	1172.34
	$f_2$	13.05	21.94	24.59	15.84	8.96
	$f_3$	7.03 × 10−3	4.26 × 10−3	2.54 × 10−3	3.01 × 10−3	2.42 × 10−3
	$f_4$	1.71 × 10−3	1.68 × 10−3	1.71 × 10−3	1.69 × 10−3	1.68 × 10−3
3	$f_1$	1224.38	1190.60	1196.37	1215.62	1185.24
	$f_2$	18.23	22.31	39.59	21.84	9.95
	$f_3$	9.40 × 10−3	3.84 × 10−3	3.73 × 10−3	4.68 × 10−3	4.25 × 10−3
	$f_4$	1.72 × 10−3	1.80 × 10−3	1.68 × 10−3	1.76 × 10−3	1.68 × 10−3

detailed the optimal solutions of objective values in each scenario.

The ablation experiments confirmed that the MEMOBWO has advantages over the standard MOBWO, as well as the single strategy versions of itself.

4.5. Comparison with Other Optimization Algorithms

The operating results of MEMOBWO are compared with other classical swarm intelligence algorithms. The comparison algorithms include MORBMO algorithm, standard MOBWO algorithm, MOHHO algorithm, NSGA-II algorithm and NMOPSO algorithm. To ensure the fairness and reproducibility of the comparative experiments, all algorithms were executed under the same population size, maximum number of iterations, archive size, and number of independent runs. The parameter settings were shown in

Table 6. Parameter settings of the compared algorithms. The common experimental settings, including population size N, maximum iteration number $T_{max}$, and archive size A, were kept identical for all algorithms in the same experiment and are therefore not repeatedly listed in the table. Here, D denotes the decision-variable dimension.

Algorithms	Specific Settings
NSGA-II	$p_c=0.9$, $p_m=1/D$, ${\eta }_c=20$, ${\eta }_m=20$
NMOPSO	$w_0=1$, $w_{\mathrm{damp}}=0.98$, $c_1=1.5$, $c_2=1.5$, $V_{max}=0.2(ub-lb)$, $\mu =0.5$, $\delta =20$
MOHHO	$E=2E_0(1-t/T_{max})$, $E_0\in [-1, 1]$, Lévy flight parameter $\beta =1.5$, Lévy scale factor $=0.01$
MORBMO	$n_{\mathrm{grid}}=5$, $CF={(1-t/T_{max})}^{2t/T_{max}}$, subgroup sizes $n_1\in [2, 5]$, $n_2\in [10, N]$
MOBWO	$B_f=1-t/T_{max}$, $W_f=0.1-0.05(t/T_{max})$
MEMOBWO	$\alpha =1.5$, $B_0=2$, $\mu =0.1$, $\kappa =3$, $b=1$, $\phi =0.3$, ${\gamma }_{\mathrm{MOTI}}=0.1$, $P=0.4$

.

4.5.1. Path Constraints Violation Analysis

To quantitatively examine whether the obtained Pareto paths satisfy the kinematic constraints, all non-dominated solutions stored in the Pareto archive were evaluated before and after the order-preserving Bézier smoothing procedure. A path was regarded as constraint-violating if it violated at least one of the heading angle, climbing angle change or curvature constraints. The constraint violation rate (CVR) was calculated as:

$$CVR={\displaystyle \frac{N_{\mathrm{vio}}}{N_{\mathrm{arc}}}}\times 100\%$$

(38)

where $N_{\mathrm{arc}}$ denotes the number of non-dominated paths in the final archive, and $N_{\mathrm{vio}}$ denotes the number of archived paths that violate at least one constraint. In addition, a normalized constraint violation severity (CVS), defined as the sum of the normalized excess values of the four considered constraints, was used to measure the degree of violation.

For the final Pareto archive containing 50 non-dominated paths, all raw discrete waypoint paths violated at least one constraint before smoothing, resulting in an $CV{R}_{\mathrm{pre}}$ of 100.00%. The violations were mainly caused by excessive heading-angle changes between adjacent waypoint segments, indicating that the raw waypoint sequences may still contain abrupt local directional changes. After Bézier smoothing, the constraint violation rate decreased to 4%. The mean CVS was reduced significantly after smoothing, while the maximum CVS reduced into the reasonable range.

A further inspection of individual constraint types shows that all 50 pre-smoothed archive paths violated the heading-change constraint, whereas only 2 smoothed paths still exhibited minor heading-change violations. No climbing angle, climbing-angle change or curvature violations were observed after smoothing. The average smoothing time was $7.9\times {10}^{-5}$ s per archived path. Therefore, the adopted constraint-aware evaluation and Bézier smoothing procedure substantially reduces local geometric constraint violations with negligible additional computational cost.

4.5.2. Path Planning Results in the 2D Test Scenario

To further evaluate the general applicability of MEMOBWO across different planning complexities, supplementary experiments were conducted in 2D environments. Accordingly, the objective functions were reduced to three dimensions: path spatial length ($f_1$, considering only X and Y coordinates), path smoothness ($f_3$, considering only horizontal turning angles), and path threat degree ($f_4$, keeping the same logic). The path vertical maneuver cost objective function ($f_2$) was omitted as altitude remains constant. All comparison algorithms were tested under identical parameter settings of population size N = 100 and maximum iterations $T_{max}$ = 500. Figure 12 illustrates the planned paths generated by each algorithm in this 2D environment. It can be observed that MEMOBWO successfully navigated through the obstacle-dense region while maintaining a smooth trajectory, achieving a balanced compromise between the length and safety of paths.

Table 7. Comparison of algorithms on path planning Objectives. The weighted sum is used to compare the overall performance of the paths generated by each algorithm.

Algorithms	${\mathit{f}}_1$ (Length)	${\mathit{f}}_3$ (Smoothness)	${\mathit{f}}_4$ (Safety)	Weighted Sum
MEMOBWO	1231.67	0.0003	0.00	0.0109
MORBMO	1229.03	0.0004	0.00	0.0277
NMOPSO	1256.84	0.0003	0.07	0.4016
NSGA-II	1260.32	0.0015	0.00	0.4102
MOHHO	1364.70	0.0011	0.00	0.5564
MOBWO	1247.90	0.0013	0.05	0.5720

presents the quantitative comparison of the objective values and the weighted-sum score. As shown, MEMOBWO achieved the lowest overall weighted sum, outperforming all comparison algorithms. Specifically, MEMOBWO demonstrated notable advantages in path smoothness and safety, while maintaining competitive path length. In contrast, algorithms such as MOHHO and MOBWO obtained significantly higher weighted-sum scores, primarily due to their inferior performance in safety or smoothness dimensions.

4.5.3. Path Planning Results in 3D Scenarios

In the comparison experiments, the parameter can be adjusted according to the emphasis of the specific problem on different objective functions to further verify the performance of MEMOBWO algorithm.

Figure 13 illustrated that all the algorithms can generate complete non-collision paths. Figure 14 and Figure 15 showed the final Pareto non-dominated solution set and the specific optimization trend on each objective function obtained by each experiment algorithm, respectively. Figure 16 revealed the normalized objective functions values of the comparison algorithms. During the iteration process, while algorithms like NMOPSO achieve competitive results in path vertical maneuver cost, they still showed a sharp spike in the path spatial length, indicating the fact that they easily sacrifice a shorter distance to maintain a lower altitude. In contrast, MEMOBWO algorithm maintained a balanced profile with no extreme spikes, suggesting a more safe compromise solution. The MEMOBWO algorithm outperformed other algorithms in objectives of path spatial length and path threat degree. Algorithms like NMOPSO and NSGA-II are relatively good at the results of vertical maneuver cost. However, by flying in a low altitude, they are forced to take longer routes or make sharp turns to avoid building obstacles, failing significantly in the path spatial length or path threat degree. Contrarily, MEMOBWO algorithm accepted a slightly higher altitude in exchange for a better global compromise, which meant a series of straighter, smoother, and safer paths.

According to data revealed on

Table 8. Path planning results of the comparison algorithms in 3D environment scenarios 1, 2 and 3.

Scenarios & Objectives		Algorithms
		MEMOBWO	MORBMO	NMOPSO	MOHHO	MOBWO	NSGA2
1	$f_1$	1178.51	1231.68	1229.18	1217.95	1312.49	1198.41
	$f_2$	8.88	27.13	13.58	25.72	11.43	14.94
	$f_3$	2.70 × 10−3	2.14 × 10−3	7.70 × 10−3	7.60 × 10−3	5.64 × 10−3	3.37 × 10−3
	$f_4$	1.30 × 10−3	1.41 × 10−3	1.30 × 10−3	1.30 × 10−3	1.30 × 10−3	1.80 × 10−3
2	$f_1$	1169.42	1195.50	1179.62	1195.95	1185.56	1241.68
	$f_2$	9.97	24.22	22.61	9.30	8.09	9.56
	$f_3$	2.1 × 10−3	6.00 × 10−3	4.9 × 10−3	5.6 × 10−3	3.89 × 10−3	2.80 × 10−3
	$f_4$	1.68 × 10−3	1.84 × 10−3	1.80 × 10−3	1.83 × 10−3	1.84 × 10−3	1.82 × 10−3
3	$f_1$	1172.00	1290.93	1277.15	1247.96	1232.62	1308.17
	$f_2$	19.60	21.72	18.93	25.13	18.98	14.91
	$f_3$	3.27 × 10−3	5.11 × 10−3	7.15 × 10−3	7.50 × 10−3	7.24 × 10−3	6.06 × 10−3
	$f_4$	1.68 × 10−3	2.07 × 10−3	1.68 × 10−3	1.83 × 10−3	1.82 × 10−3	1.68 × 10−3

, the superior performance of MEMOBWO can be attributed to the adaptive exploration-exploitation balance mechanism, which enables better diversity preservation in the early stage and faster convergence in the later stage.

4.5.4. Hypervolume Metric Comparisons

In addition to the comparisons on the ability to deal with the path planning problems, this paper also set up the Hypervolume metric test functions to test the performance of the proposed MEMOBWO algorithm and compare it with other classical optimization algorithms. Figure 17 detailed the iterative convergence process of the Hypervolume metric of all the algorithms. It can be seen that although other algorithms converged rapidly in the initial phase, they plateau prematurely. The distinct characteristics of the MEMOBWO curve were the staircase-like rise in the iteration process and the slight oscillation during the initial iterations. Unlike the smooth, rapid saturation seen in the MORBMO curve, the MEMOBWO curve exhibited a step-wise improvement pattern and actively maintained population diversity to avoid the premature convergence observed in the MORBMO and NMOPSO curves. The MEMOBWO algorithm flattened out later than others but at a higher level, demonstrating that the adaptive exploration-exploitation balance allowed it to refine the solution set when others have stagnated.

The innovative strategies of the MEMOBWO required it to constantly jump out of the optimal solution, so it did not have advantages over other comparison algorithms in HV convergence speed, but exceeds all other experiment functions in the final score. As the algorithm progressed, the exploitation ability is strengthened, leading to a more stable final HV value. Minor fluctuations in the later stage indicated continuous diversity maintenance rather than premature convergence. This phenomenon can be attributed to the HAPU strategy. Moreover, MEMOBWO experienced a sudden jump in performance in the particular stage, allowing the algorithm to explore more promising regions of the search space while other algorithms stagnate in local optima. Ultimately, this led to a final Hypervolume value that is 15.24% to 30.86% higher than that of the competitors.

4.5.5. Statistical Stability Experiments

To comprehensively evaluate the statistical stability of each algorithm under random initialization, 30 independent runs and the mean and standard deviation of each objective function were carried out for all the comparison algorithms. The results were shown in

Table 9. Statistical stability results of MEMOBWO, its ablation variants, and the comparison algorithms over 30 independent runs.

Algorithms	Objectives	Ave	Std
MEMOBWO	$f_1$	1177.08	12.56
	$f_2$	7.45	2.79
	$f_3$	4.13 × 10−3	5.91 × 10−4
	$f_4$	1.68 × 10−3	<1 × 10−12
MOBWO	$f_1$	1241.05	34.71
	$f_2$	28.07	4.67
	$f_3$	6.44 × 10−3	1.56 × 10−3
	$f_4$	1.68 × 10−3	<1 × 10−12
MEMOBWO1	$f_1$	1459.84	20.08
	$f_2$	20.83	6.27 × 10−3
	$f_3$	9.12 × 10−3	1.75 × 10−3
	$f_4$	1.73 × 10−3	<1 × 10−12
MEMOBWO2	$f_1$	1193.05	18.78
	$f_2$	15.94	5.26
	$f_3$	5.18 × 10−3	8.70 × 10−4
	$f_4$	1.75 × 10−3	<1 × 10−12
MEMOBWO3	$f_1$	1166.04	30.92
	$f_2$	28.48	2.7 × 10−3
	$f_3$	8.75 × 10−3	2.22 × 10−3
	$f_4$	1.76 × 10−3	9.25 × 10−4
MORBMO	$f_1$	1200.71	20.12
	$f_2$	11.76	3.57
	$f_3$	5.40 × 10−3	6.08 × 10−4
	$f_4$	1.68 × 10−3	<1 × 10−12
NMOPSO	$f_1$	1185.95	21.23
	$f_2$	25.27	3.51
	$f_3$	3.94 × 10−3	8.08 × 10−4
	$f_4$	1.79 × 10−3	5.98 × 10−5
MOHHO	$f_1$	1437.28	594.86
	$f_2$	38.05	11.67
	$f_3$	3.41 × 10−2	7.38 × 10−2
	$f_4$	2.03 × 10−3	5.34 × 10−4
NSGA2	$f_1$	1191.57	29.51
	$f_2$	13.90	4.73
	$f_3$	3.68 × 10−3	1.05 × 10−3
	$f_4$	1.78 × 10−3	6.17 × 10−5

, where the mean value represented the optimization ability of the algorithm, and the standard deviation indicated the degree of dispersion of the results. From the statistical perspective, MEMOBWO algorithm showed smaller standard deviation. Although algorithms like NSGA-II had similar performance with MEMOBWO in individual experiments, they were still not dominant in comprehensive statistical evaluation.

The Friedman test was applied to assess the statistical significance of performance differences among the comparison algorithms. The test statistic ${\chi }_F^2$ approximately followed a ${\chi }^2$ distribution with 6 degrees of freedom, and the significance level was set to $\alpha =0.05$. Average ranks are presented in

Table 10. Friedman test results of comparison algorithms for the HV metric, where the lower rank represents the better overall performance.

Algorithms	Performance
	Average Rank
MEMOBWO	1.14
MORBMO	2.86
NMOPSO	3.29
NSGA-II	3.86
MOHHO	4.71
MOBWO	5.43

. The results indicated that MEMOBWO achieved the lowest rank, thus having the best performance.

To further investigate the performance differences, the Wilcoxon rank sum test was conducted on the HV metric obtained from 30 independent runs across all test scenarios. The results are presented in

Table 11. Wilcoxon rank sum test results for the HV metric, where ≈ means no significant difference, + means better and +* means significantly better, / means not applicable.

Pair (MEMOBWO vs. Algorithm)	p-Value	Significance
MEMOBWO	/	/
MORBMO	2.45 × 10−1	≈
NMOPSO	1.57 × 10−2	+ *
MOBWO	<1.00 × 10−3	+ *
MOHHO	6.40 × 10−3	+
NSGA2	<1.00 × 10−3	+ *

. As shown in the table, MEMOBWO achieves statistically significant improvements over the majority of comparison algorithms. The only exception is the comparison with MORBMO, where no statistically significant difference is observed. This can be attributed to the fact that both MEMOBWO and MORBMO employ adaptive balance mechanisms between exploration and exploitation, leading to comparable solution diversity in terms of the comprehensive HV metric.

4.5.6. Parameters Sensitivity Analysis

To examine the stability of MEMOBWO against parameter variations, a sensitivity analysis was conducted on five ZDT benchmark functions, including ZDT1, ZDT2, ZDT3, ZDT4, and ZDT6. Three representative parameters were considered: the nonlinear decay exponent $\alpha$ in HAPU, the learning coefficient $\phi$ in MOTI, and the disturbance coefficient $\gamma$ in MOTI. For each parameter setting, 30 independent runs were performed, while the remaining parameters were fixed to their default values.

For $\alpha \in \{0.5,1.0,1.5,2.0,2.5\}$, the average HV and IGD values over the ZDT functions remained almost unchanged. The average HV stayed around 0.3589, and the average IGD stayed around 0.0207. This indicates that MEMOBWO is relatively insensitive to moderate variations of the nonlinear decay exponent under the current benchmark setting.

For the MOTI-related parameters, $\phi \in \{0.1,0.2,0.3,0.4,0.5\}$ and $\gamma \in \{0,0.05,0.1,0.2,0.5\}$ were further tested. The results show that these two parameters have moderate effects on the search behavior, but no severe performance degradation was observed within the tested ranges. When $\phi$ varied from 0.1 to 0.5, the average HV remained within a narrow range. A similar pattern was observed when $\gamma$ varied from 0 to 0.5. These results suggest that the MOTI strategy maintains stable performance under moderate variations of its learning and disturbance coefficients.

The Piecewise-map parameter P in CQOBL and the auxiliary constants in HAPU, such as $B_0$, $\mu$, $\kappa$, and b, were fixed according to the implemented algorithm configuration. Since CQOBL mainly affects initialization and these constants serve as auxiliary control factors, this study focuses on the representative parameters that directly affect the exploration–exploitation transition and the objective-specific refinement process. Overall, the default settings $\alpha =1.5$, $\phi =0.3$, and $\gamma =0.1$ are adopted as a stable and balanced configuration rather than a universally optimal parameter combination.

5. Conclusions

This paper proposed a novel improved algorithm called MEMOBWO, which contains a series of optimization strategies to improve its performance on the problem of the UAV path planning in urban environment. The subsequent simulation results show that MEMOBWO can effectively improve the comprehensive performance of UAV path planning algorithm while maintaining stable performance across independent runs. It can play a certain auxiliary role in UAV flight control and trajectory tracking.

However, the current MEMOBWO algorithm still has room for further improvement in terms of solution accuracy and optimization speed. In future research, more sophisticated decision-making models will be explored to solve the existing and potential problems. Exploration of hybrid frameworks, such as integrating deep learning with meta-heuristic optimization, could be a promising research direction. Combination of these complementary paradigms could sufficiently leverage the strengths of both approaches, thereby accelerating the convergence speed of population-based algorithms like MEMOBWO.

Moreover, while the experimental results demonstrate the effectiveness of MEMOBWO in solving UAV path planning problems in complex urban environments, we acknowledge that the testing scope has certain limitations that the experimental validation in this study is limited to simulated urban environments with basic geometric primitives like cubes, cylinders and hemispheres. To further strengthen the practical relevance of the results, future work should validate the proposed MEMOBWO on more realistic 3D city models based on real-world environments, or directly tested in real-world experiment fields. Such validation would provide a more comprehensive assessment of the algorithm’s strengths and limitations under realistic deployment conditions.