A quantum-enhanced heuristic algorithm for optimizing aircraft landing problems in low-altitude intelligent transportation systems

Abstract

As the low-altitude economy expands, optimizing the landing schedules of manned aerial vehicles has emerged as a critical challenge. The aircraft landing time window problem is a complex, high-dimensional optimization task frequently hindered by local optima. This study presents an Quantum-Enhanced Whale Optimization Algorithm (QEWOA) that incorporates quantum computing to tackle these challenges. By utilizing a quantum random number generator, the algorithm improves the diversity of the initial population, preserving potential optimal solutions for a more effective global search. Additionally, the incorporation of an enhanced quantum tunneling mechanism enables the algorithm to escape local optima and perform more extensive searches. The combination of the Artificial Bee Colony algorithm and Whale Optimization further strengthens both global search and local optimization capabilities. Experimental results demonstrate that QEWOA significantly improves global search ability, optimization precision, and convergence speed, surpassing traditional methods in optimizing landing time windows for low-altitude manned aerial vehicles.

Introduction

As the low-altitude economy rapidly develops, the application prospects of low-altitude manned aircraft continue to look increasingly promising. By leveraging low-altitude airspace, sectors such as aircraft manufacturing, aviation services, air tourism, air logistics, aerial agriculture, and environmental protection are positioned as a promising avenue for future economic growth¹. A key driver of the low-altitude economy is the efficient transport capability of low-altitude manned aircraft, which can perform swift and flexible flights in urban areas and their surroundings, providing innovative solutions for urban transportation and air taxi services. However, this rapid development presents significant challenges in flight safety and route planning. In particular, ensuring efficient and safe landings in complex environments has become a critical research topic.

The landing problem for low-altitude manned aircraft, similar to traditional aircraft landing issues, requires effective and safe task execution in dynamic, multi-constrained environments. Typically modeled as the Aircraft Landing Problem (ALP), this challenge involves developing optimal landing sequences and schedules that meet safety requirements while optimizing operational efficiency and minimizing resource consumption. This problem is often addressed using single-machine or parallel-machine scheduling models². However, traditional optimization techniques, such as linear programming and dynamic programming, often prove inefficient and susceptible to local optima, making them unsuitable for real-time response requirements. Therefore, there is an urgent need for advanced optimization techniques.

In recent years, several novel metaheuristic algorithms have emerged, such as the Draco Lizard Optimizer³ and the Eurasian Lynx Optimizer⁴, both demonstrating promising performance. However, the Whale Optimization Algorithm (WOA), a population-based optimization method inspired by the hunting behavior of humpback whales, has garnered increasing attention in the optimization research community. Its advantages—including simple implementation, few control parameters, and strong global search capabilities—have positioned WOA as a prominent and widely studied algorithm in this field⁵. However, traditional WOA is hindered by issues such as susceptibility to local optima and slow convergence speeds, particularly when addressing high-dimensional and multimodal optimization problems. The advent of quantum computing technology offers innovative solutions to these challenges. Quantum algorithms have demonstrated significant advantages in discrete optimization, dynamic programming, boosting, and clustering, often outperforming state-of-the-art classical algorithms due to substantially better polynomial dependency on certain problem dimensions⁶. Features such as quantum superposition and tunneling effects can significantly enhance global search capabilities and accelerate search processes. Notably, quantum tunneling allows optimization algorithms to escape local minima by tunneling through energy barriers, thereby enhancing the efficiency of global searches⁷. Thus, integrating quantum computing with WOA holds significant potential to overcome the limitations of traditional approaches and address the challenges of optimizing low-altitude manned aircraft landings. In summary, this paper proposes a Quantum-Enhanced Whale Optimization Algorithm (QEWOA) that integrates quantum random number generation mechanisms, quantum tunneling effects, and the Artificial Bee Colony (ABC) algorithm to improve the algorithm’s global search capabilities and local optimization precision. The main contributions of this study are outlined as follows:

• a) A quantum random number generation mechanism is designed to enhance the population initialization of WOA, increasing population diversity and improving global search capabilities.

• b) A quantum tunneling-based jumping mechanism is adopted, which introduces quantum tunneling effects into WOA. This mechanism generates large-scale random steps to escape local optima and search for new global optima, significantly enhancing the algorithm’s global exploration ability.

• c) Integration with the ABC algorithm: To further improve QEWOA’s local search capabilities, this study integrates ABC with WOA to enhance QEWOA’s overall performance through precise local searches.

The structure of this paper is organized as follows: Sect. 2 reviews the related research on low-altitude ALP, focusing on the strengths and weaknesses of existing optimization methods and the potential applications of quantum computing. Section 3 introduces the overall framework and process of the proposed algorithm. Section 4 provides a detailed explanation of the design and implementation of QEWOA, including quantum random number generation, quantum tunneling mechanisms, and integration with ABC. Section 5 describes the experimental design and result analysis, validating the superior performance of QEWOA in solving low-altitude ALP through comparisons with traditional methods. Finally, Sect. 6 summarizes the findings and suggests directions for future research.

Literature review

Driven by the growth of the low-altitude economy, Low-Altitude Intelligent Transportation (LAIT) has emerged as an innovative transportation paradigm, transforming traditional systems through ongoing technological advancements. LAIT provides efficient and flexible solutions for urban transportation systems and shows considerable potential in fields such as emergency rescue and aerial ride-sharing. The core concept of LAIT is a three-dimensional transportation system that integrates both manned and unmanned aircraft operations within urban or intercity environments. Its rapid development has injected innovation and vitality into numerous industries⁸. Researchers have extensively explored LAIT’s system architecture, infrastructure, and key technologies, proposing intelligent management methodologies, such as a hierarchical architecture based on Cyber-Physical Systems (CPS)¹. These studies provide comprehensive lifecycle solutions for urban air transportation systems, offering clear guidance from design and construction to operation and management.

Nevertheless, the development of LAIT presents several potential challenges. Studies suggest that, without appropriate regulatory advancements and efficient traffic management systems, low-altitude areas may experience traffic congestion, reducing operational efficiency and increasing systemic risks⁹. This threatens the safety of people and property in both air and ground contexts, placing greater demands on critical aspects, such as ALP. As a key component of LAIT, ALP is an essential element of low-altitude intelligent transportation systems (LAITS), and its optimization remains a central focus of academic research.

In addressing ALP, various methods and models have been introduced by research teams. For example, some researchers have integrated optimization and simulation techniques to develop a sliding-window Opt-Sim closed-loop feedback framework designed to evaluate scheduling schemes under uncertain conditions¹⁰. The results indicate that this method significantly mitigates potential aircraft conflicts and is applicable to optimizing a range of scheduling problems. To address winter runway scheduling challenges, researchers introduced an integrated optimization model combining runway snow removal plans with aircraft scheduling, demonstrating computational efficiency and significant cost savings validated by real-world data¹¹. Furthermore, a two-stage stochastic programming model has been developed, where the initial approach fix (IAF) assignment serves as the first-stage decision, and fixed IAF allocation serves as the problem input¹². An analysis of real-world cases at Charles de Gaulle Airport in Paris demonstrated that integrating two-stage stochastic programming with IAF reassignment effectively addresses uncertainties and significantly enhances aircraft arrival management.

In recent studies, heuristic algorithms have experienced continuous iterative improvements. For example, researchers proposed a novel metaheuristic algorithm—the Artificial Mongoose Algorithm (AMA)—which simulates cooperative behavior within mongoose populations. By incorporating multi-strategy search and multi-phase elimination mechanisms, AMA effectively addresses complex optimization problems¹³. Another example is the Fishing Cat Optimizer (FCO), a metaheuristic algorithm inspired by the distinctive hunting behavior of fishing cats. It employs a four-phase search strategy to balance global exploration with local exploitation, thereby achieving high convergence performance and robustness in solving complex optimization problems¹⁴. The application of these emerging heuristic algorithms in ALP research has been steadily increasing. Studies suggest that heuristic algorithms outperform traditional methods in both solution speed and quality. For example, researchers employed ADS-B technology to develop a heuristic search method that significantly enhances flight efficiency and adaptability to dynamic demands by updating landing schedules in real-time¹⁵. Meanwhile, heuristic algorithms based on probabilistic models have demonstrated exceptional performance in addressing uncertainties¹⁶. By integrating time-decomposition sliding windows with simulated annealing algorithms, this approach achieves conflict detection and resolution at waypoints, thereby significantly improving schedule stability. Additionally, heuristic approaches that integrate delay prediction models and dynamic speed control have been shown to alleviate congestion in Terminal Maneuvering Areas (TMA), thereby reducing fuel consumption and minimizing environmental impact¹⁷ .

To obtain superior solutions, recent research has increasingly focused on integrating ALP with multi-strategy optimization algorithms. For example, researchers proposed a hybrid optimization algorithm that integrates multi-strategy Particle Swarm Optimization (PSO) and Ant Colony Optimization (ACO), improving taxiway planning efficiency and performance by dynamically adjusting search directions and pheromone strategies¹⁸. Other studies employed wind-network technologies to optimize aircraft scheduling models, revealing that incorporating wind speed networks substantially reduces uncertainties in waypoint arrival times¹⁹. Innovative optimization algorithms, such as TMAOpt, have been applied to terminal airspace scheduling problems, integrating linear programming with metaheuristic algorithms to efficiently generate feasible solutions while ensuring convergence²⁰ .

Building upon this foundation, researchers have further improved algorithmic performance by integrating quantum computing with heuristic algorithms. One study proposed a parallel hybrid algorithm based on quantum annealing and dual-elite spiral search (QA-DESS), effectively combining the efficiency of quantum annealing for integer optimization with the global search capability of DESS in continuous domains. This algorithm was applied to mixed-integer optimal control problems in engineering, outperforming benchmark algorithms across multiple standard functions and practical cases²¹. Another study applied a quantum annealing algorithm based on the quantum tunneling mechanism to the fault section location problem in power distribution networks. A quantum Hamiltonian function incorporating both potential and kinetic energy terms was constructed, and an improved quantum annealing algorithm (IQA) was proposed. The algorithm demonstrated superior performance compared to conventional intelligent methods in IEEE standard test systems²². Further research addressed the impact of tidal factors on the operational efficiency of small and medium-sized container ports by constructing a berth and quay crane assignment optimization model under tidal constraints (T-B&QC). A chaotic quantum whale optimization algorithm (CQWOA), integrating chaotic mapping with quantum theory, was proposed and empirically demonstrated to outperform traditional assignment models and algorithms in terms of coordinated scheduling and optimization performance²³. Additionally, the multi-commodity network flow problem was formulated as a mixed-integer programming (MIP) NP-hard problem and solved using quantum annealing. The results demonstrated that quantum annealing achieved superior solution quality and computational efficiency compared to traditional methods in large-scale transportation and logistics optimization scenarios²⁴. Moreover, in solving the two-dimensional spin-glass problem, a quantum annealing correction (QAC) mechanism was introduced and implemented on the D-Wave quantum annealer, enabling large-scale, high-precision optimization. This work demonstrated, for the first time, a quantum advantage in algorithmic scaling over classical parallel tempering with isoenergetic cluster moves (PT-ICM) for approximate optimization, marking a significant milestone in realizing algorithm-level quantum acceleration for low-energy state sampling problems²⁵ .

Among various multi-strategy optimization algorithm combinations, the WOA, an emerging heuristic method, has garnered significant attention for its flexibility and efficiency in addressing complex optimization challenges. It has been successfully applied to a wide range of optimization challenges across various domains. Initially, whales locate potential prey by randomly exploring the search space from their current positions. By gradually converging on probable targets and continuously evaluating the distance between their current positions and the targets, they determine the optimal target location. After determining the optimal target, the whale updates its position using Eq. (4) to approach the target with greater accuracy, thereby improving hunting efficiency and optimization precision. This process is outlined as follows²⁶ :

$$\:D=|C\cdot\:{X}_{best}-{X}_{i}(t\left)\right|$$

(1)

$$\:A=2*a\cdot\:{r}_{1}-a$$

(2)

$$\:C=2\cdot\:{r}_{2}$$

(3)

$$\:{X}_{i}\left(t+1\right)={X}_{Best}-A\cdot\:D$$

(4)

Here, $\:{X}_{Best}$​ denotes the best position identified thus far, while a decreases linearly from 2 to 0 over the course of the iterations. This facilitates the progressive narrowing of the encirclement. Random numbers $\:{r}_{1}$​ and $\:{r}_{2}$​ are sampled from the interval [0,1].

The encircling mechanism enhances the search process by updating the position of the humpback whale, as defined in Eq. (4). Simultaneously, Eq. (5) modifies the computation of the parameter D, while the shrinking behavior is preserved by reducing the value of a in Eq. (2). Whales progressively narrow their encirclement and follow a spiral path, as described in Eq. (6), to approach the optimal solution. The specific equations for position updates are provided below²⁷ :

$$\:{D}^{{\prime\:}}=|{X}_{best}-{X}_{i}\left(t\right)|$$

(5)

$$\:{X}_{i}\left(t+1\right)=D{\prime\:}\cdot\:{e}^{bl}\cdot\:\text{c}\text{o}\text{s}\left(2\pi\:l\right)+{X}_{best}$$

(6)

Furthermore, to more efficiently search for the global optimal solution, the algorithm randomly selects an individual as a reference for position updates, while still utilizing the variation in vector A to guide the search process. Unlike the exploitation phase, in this phase, Eqs. (7) and (8) require a randomly selected search agent, rather than the current best agent, to perform position updates. This mechanism can be described as follows²⁸ :

$$\:D=|C\cdot\:{X}_{rand}\left(t\right)-{X}_{i}\left(t\right)|$$

(7)

$$\:{X}_{i}(t+1)={X}_{rand}\left(t\right)-A\cdot\:D$$

(8)

Currently, research on the WOA primarily focuses on enhancing its search mechanism. Mohamed Abdel-Basset et al. extended WOA to multi-objective optimization by incorporating the Nelder-Mead simplex method, dynamically generating distance control factors, which significantly enhance the algorithm’s adaptability and flexibility²⁹. Additionally, Mohamed Abd Elaziz et al. proposed the Multi-Leader Whale Optimization Algorithm (MLWOA), which integrates memory mechanisms, multi-leader strategies, self-learning approaches, and the Lévy flight method. These enhancements improve the algorithm’s global exploration capability and mitigate premature convergence issues³⁰. Jiquan Wang et al. introduced an improved algorithm combining mutation and dissimilarity (CRWOA), which significantly enhances local search capabilities and solution distribution through adaptive parameter adjustments and improvements to the bubble-net attack mechanism³¹ .

However, several challenges remain in these approaches, particularly when addressing problems involving complex constraints. The limited local search capability of WOA and its lack of population diversity may cause the algorithm to become trapped in local optima in high-dimensional search spaces. To address this challenge, this study aims to further investigate WOA’s limitations in solving complex constrained optimization problems and proposes an improved hybrid heuristic algorithm that integrates ABC optimization and quantum mechanisms. The research objectives include enhancing WOA’s local search performance and population diversity, introducing quantum random numbers and quantum tunneling effects, optimizing time windows for ALP, reducing the initial convergence threshold, minimizing convergence variance, improving the final penalty value, and enhancing the algorithm’s global exploration capability and optimization efficiency.

Methodology

To address the optimization problem in the landing process of low-altitude manned aerial vehicles, this paper presents an optimization model aimed at minimizing the total landing penalty value. The model comprehensively defines the time window and separation constraints, along with their impact on the accuracy of optimal solutions for low-altitude aircraft landing scheduling, thereby forming a three-stage optimization framework. In the model design, a quantum random number generator is employed to optimize the initial population, thereby enhancing population diversity and mitigating the potential deficiencies in diversity associated with traditional random number generation methods. This approach maximizes the retention of potentially optimal solutions, thereby laying a solid foundation for subsequent optimization. During the search phase, a collaborative search strategy combining WOA and the ABC algorithm is designed to achieve a dynamic balance between global search capability and local exploitation, thereby improving solution quality and search efficiency. Finally, in the optimization phase, a quantum tunneling-based jumping mechanism is introduced. By designing an improved quantum tunneling strategy, the algorithm’s ability to escape local optima during local searches is enhanced, further improving global search performance. The experimental section systematically validates the algorithm’s performance, and the results demonstrate that under complex constraints, the proposed algorithm significantly outperforms traditional methods in terms of global search capability, convergence speed, and solution accuracy. This underscores its exceptional performance in solving low-altitude aircraft landing optimization problems. The overall structural framework is illustrated in Fig. 1.

This method is designed to overcome the performance bottlenecks of traditional optimization algorithms when handling complex constrained problems, offering a novel approach to optimizing aircraft landing schedules within intelligent transportation systems.

Fig. 1

Overall framework of QEWOA.

In this framework, the initialization of the original population has traditionally relied on pseudo-random numbers generated by program code. Such randomness is not truly random but rather reveals inherent “patterns.” These pseudo-random numbers function as finite state machines with a limited number of states, meaning they will eventually return to previous states and repeat the sequence after a finite number of steps³². This can lead to insufficient uniformity in population distribution, thereby affecting the comprehensive coverage of the search space. This limitation is particularly problematic for high-dimensional problems, as it may result in an initial population lacking diversity, thereby increasing the risk of falling into local optima. Furthermore, for larger population sizes, population initialization may be influenced by periodic fluctuations, further restricting the distribution range. To address these issues, this paper employs quantum random numbers to improve population diversity and distribution uniformity, thereby enhancing the algorithm’s global search capability.

Quantum measurement processes facilitate the generation of genuinely unpredictable and private random numbers. By projecting a pure quantum state and ensuring that the state is not an eigenstate of the measurement projector, the outcomes become unpredictable, enabling the creation of true random numbers³³. These quantum-generated initial solutions exhibit high diversity, more uniform distribution, and robust adaptability, making them especially suitable for complex network topologies³⁴. This study employs quantum random numbers provided by the Australian National University, which convert digitized photocurrent into a sequence of random numbers³⁵. The generated random numbers can be arbitrarily correlated with specific subsets of quantum fluctuations while maintaining high robustness, remaining largely unaffected by environmental variations. The core of this conversion process lies in the precise modeling and mathematical description of the measurement signals to ensure that random number generation captures the inherent uncertainty and randomness of quantum mechanics. The measurement signal is represented as follows:

$$\:{X}_{m}={X}_{v}+{X}_{e}$$

(9)

In this context, $\:{X}_{m}$ represents the total signal obtained from the measurement, $\:{X}_{v}$ refers to the signal component caused by quantum vacuum fluctuations, and $\:{X}_{e}$ denotes the electronic noise superimposed on the vacuum fluctuation, $\:{X}_{v}$. Since $\:{X}_{e}$ and $\:{X}_{v}$ can be modeled as independent Gaussian distributions with zero mean and variances $\:{V}_{e}$ and $\:{X}_{v}$, respectively, the measurement signal, $\:{X}_{m}$, will also follow a Gaussian distribution. The conditional probability can be expressed as:

$$\:P\left({X}_{m}|{X}_{v}\right)=\frac{1}{\sqrt{2\pi\:{V}_{e}}}{e}^{-\frac{{({X}_{m}-{X}_{v})}^{2}}{2{V}_{e}}}$$

(10)

The Gaussian-distributed photodetector current signal obtained from the measurement is then transformed into a uniform distribution to standardize the random number generation process:

$$\:{Y}_{m}=\frac{\left[1+\text{e}\text{r}\text{f}\left(\frac{{X}_{m}}{\sqrt{2{V}_{m}}}\right)\right]}{2}$$

(11)

Here, erf represents the error function, and $\:{V}_{m}$ denotes the noise variance associated with the measurement signal. The cumulative distribution function (CDF) of the Gaussian distribution is employed to remap the original signal to the interval [0, 1], yielding a uniformly distributed output signal, $\:{Y}_{m}$.

Finally, Eq. (12) is derived, thereby generating genuinely unpredictable random numbers.

$$\:{I}_{E}=1+{P}_{e,\:e}{log}_{2}\left({P}_{e,\:e}\right)+{(1-P}_{e,\:e}){log}_{2}\left(1-{P}_{e,\:e}\right)$$

(12)

To enhance the diversity and overall optimization capability of the random initial population, we integrate the ABC algorithm with the hunting behavior of humpback whales, redesigning the individual update mechanism within the WOA. By extending the individual behavior of humpback whales to simulate the communication and cooperation patterns of bee colonies, the algorithm can more effectively optimize group foraging behavior. Specifically, in WOA, the hunting behavior of each whale is treated as an independent optimization process. By introducing the cooperative behavior of bees, the global search ability and local development capacity of the population can be significantly enhanced.

ABC simulates the collective intelligence of a bee colony during foraging tasks, aiming to maximize nectar accumulation in the hive³⁶. During foraging, bees use a pheromone communication strategy to select higher fitness areas while exploring potential solution spaces, thereby improving overall search efficiency. Similarly, in the ALP, the algorithm searches the candidate solution space for the optimal landing times that meet the constraints and optimization objectives. ABC employs distinct roles, such as employed bees, onlooker bees, and scout bees, to balance global exploration with local exploitation. In the ALP optimization, this strategy is adapted into a collaborative mechanism involving employed whales, observer whales, and scout whales, enhancing search diversity and identifying better solutions in local regions. In terms of fitness evaluation, the fitness values in ABC typically reflect the distance and quality between the bee dance and the food source, guiding the group search process. In ALP, fitness evaluation is based on the feasibility of each landing time scheme and the degree of optimization of the objectives, ensuring that the solution meets both flight safety and airport operational efficiency requirements.

After transitioning from the initialization of a truly random population to the search phase, the algorithm may encounter the issue of becoming trapped in local optima. To address this, we employ the quantum tunneling effect and develop a jump mechanism. When a whale becomes trapped in a local optimum, it can escape by generating a large random step through quantum random numbers, facilitating tunneling behavior that bypasses the local optimum and identifies a new solution. The tunneling effect represents the algorithm’s capacity to escape local optima³⁷. It simulates the behavior of quantum particles traversing high-energy potential barriers, thereby enhancing the algorithm’s global search capability. By combining quantum random numbers with large step jumps, individuals can rapidly overcome local constraints and discover new solutions. The introduction of historical direction and adaptive step-size adjustment ensures that the algorithm maintains high-precision local development capability during the convergence phase. Finally, by adjusting sensitivity factors and jump control parameters based on the characteristics of the optimization problem, the mechanism can be fine-tuned for optimal application of the algorithm.

QEWOA

QEWOA combines ABC and WOA, utilizing quantum random numbers to enhance the diversity of the population. The quantum tunneling effect is introduced to balance global search with local optimization. The sections “Quantum Random Number Initialization of the Population” and “Quantum Tunneling-Based Jumping Mechanism” provide a detailed description of the components and specific implementation of qewoa.

Quantum random number initialization of the ropulation

Population initialization is a critical step in swarm intelligence optimization algorithms, as its diversity significantly impacts the algorithm’s global search capability. Traditional population initialization typically relies on random numbers generated by code to produce the initial solutions. However, these random numbers exhibit periodicity and statistical regularity, potentially limiting the randomness and coverage of the population distribution. Quantum random numbers, based on quantum mechanics principles, possess true randomness and unpredictability, effectively avoiding the limitations of pseudo-random numbers, enhancing population initialization diversity, and improving the algorithm’s global search capability.

The random numbers are generated using a single-mode laser as the light source, splitting the beam into two equal-intensity beams, which are detected by a pair of photodetectors in a balanced zero-differential configuration. When the average amplitude α of the laser field significantly exceeds the vacuum field fluctuations, the photocurrent subtracted from the detector pair becomes proportional to the quadrant amplitude $\:{X}_{v}$ of the vacuum field. Random numbers are then generated through radio-frequency demodulation of the photocurrent, followed by low-pass filtering, using only sideband frequencies that are significantly higher than the laser technology noise frequency^35,38 .

Consider the solution space of the ALP as D-dimensional, with N whales in the population, and the domain defined as [L, U], where $\:L=[{L}_{1},\:{L}_{2},\:\cdots\:{L}_{D}]$ represents the lower bounds of the variables, and $\:U=[{U}_{1},\:{U}_{2},\:\cdots\:{U}_{D}]$ represents the upper bounds. The goal of initializing the population X is to generate a solution set that is evenly distributed and sufficiently covers the entire solution space, ensuring that the population has adequate representativeness and exploration capacity across the solution space. This objective can be represented by Eq. (13):

$$X = \left[ {\begin{array}{*{20}c} {x_{{11}} } & \cdots & {x_{{1D}} } \\ \vdots & \ddots & \vdots \\ {x_{{N1}} } & \cdots & {x_{{ND}} } \\ \end{array} } \right]$$

(13)

In other words, generate a D-dimensional random number for each individual:

$$\:{x}_{ij}={L}_{j}+{r}_{j}\cdot\:({U}_{j}-{L}_{j})$$

(14)

In this context, $\:{r}_{j}$ denotes the quantum random number. The following figure compares the whale population initialization for the ALP using quantum random numbers with that using pseudo-random numbers generated by the program code, across populations of varying sizes.

For small-scale populations, a heatmap of the population distribution is used to compare and analyze the distribution characteristics of both methods. As shown in the heatmap of Fig. 2, the distribution coverage and randomness of the former (using quantum random numbers) significantly outperform those of the latter (using program-generated random numbers). This difference primarily arises from the optimization of the population initialization strategy, which allows the small-scale population to achieve a more even distribution within the solution space. In contrast, program-generated random numbers may exhibit some degree of pattern or bias due to the limitations of pseudo-random number generation algorithms. This comparison not only confirms the importance of population initialization methods in enhancing algorithm performance but also further substantiates the advantage of randomness in optimization algorithms under small-scale population conditions, thus establishing a solid foundation for subsequent search stages.

Fig. 2

Heatmap of small-scale population distribution.

For large-scale population initialization, similar trends can be observed in Fig. 3. The optimized population initialization strategy effectively addresses the issue of uneven coverage in the solution space, ensuring a more balanced and diverse distribution of solutions. This ensures sufficient exploration space for subsequent global search and local optimization.

Fig. 3

Heatmap of large-scale population distribution.

The Shannon entropy formula is applied as follows:

$$\:H\left(X\right)=-\sum\:_{i=1}^{n}p\left({x}_{i}\right){log}_{2}p\left({x}_{i}\right)$$

(15)

The calculated average entropy of the quantum random numbers is 4.19 bits, compared to 4.16 bits for the pseudo-random numbers, suggesting that the quantum random numbers demonstrate greater randomness.

Quantum tunneling-based jumping mechanism

To improve WOA’s ability to escape local optima, this paper introduces a jump mechanism based on the quantum tunneling effect. This mechanism utilizes quantum random numbers to generate large random steps, emulating the behavior of quantum particles traversing high-energy barriers, thereby enhancing the algorithm’s global search capability.

Let the current position of the whale be denoted by $\:{x}_{i}$,with the goal of escaping the local optimal solution, $\:{x}_{local}$. The rule for updating an individual’s position via the quantum tunneling mechanism is presented as follows:

$$\:{x}_{i}^{new}=\left\{\begin{array}{c}{x}_{i}+\varDelta\:x,\:\:\:\:\:\:\:if\:rand<{P}_{tunnel}\\\:{x}_{i},\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:otherwise\end{array}\right.$$

(16)

In this context, $\:{\Delta\:}x$ represents the jump step size, while $\:{P}_{tunnel}$ denotes the tunneling probability, which characterizes the probabilistic behavior of the individual during the tunneling process.

The jump step size, $\:{\Delta\:}x$,can be formulated as follows:

$$\:{\Delta\:}x=\eta\:\cdot\:{rand}_{q}\cdot\:\left({x}_{global}-{x}_{i}\right)+\gamma\:\cdot\:{rand}_{q}^{2}\cdot\:{v}_{i}$$

(17)

This mechanism ensures that the individual conducts an extensive search toward the global optimum, while incorporating historical search directions to enhance local solution space exploration. In this context, $\:\eta\:$ represents the global step size control factor, regulating the individual’s search range within the solution space, where $\:\eta\:>0$, with a value range of [0.1, 0.5]. $\:{rand}_{q}$ represents the quantum random number, uniformly distributed in the range of [-1, 1]. $\:\gamma\:$ represents the local disturbance factor, fine-tuning the step size, where $\:\gamma\:>0$, with a value range of [0.5, 1]. Finally, $\:{v}_{i}$ denotes the individual’s historical search direction, which is defined as follows:

$$\:{v}_{i}={x}_{i}-{x}_{i}^{prev}$$

(18)

The tunneling probability can be expressed as follows:

$$\:{P}_{tunnel}=\frac{1}{1+\text{e}\text{x}\text{p}(-k\cdot\:\frac{f\left({x}_{local}\right)-f\left({x}_{i}\right)}{{\Delta\:}{f}_{avg}})}$$

(19)

When the difference in objective function values is substantial, the tunneling probability decreases, thereby promoting the exploration of individuals near higher-quality solutions. Conversely, when the objective function values are similar, the tunneling probability increases, thereby aiding in the escape from local optima. Here, k represents the sensitivity factor, which controls the response of the tunneling probability to differences in the objective function, with a value range of^5,10. $\:{\Delta\:}{f}_{avg}$ represents the average difference in objective function values across the population, serving as a normalization factor. This enables the dynamic adjustment of the algorithm’s step size and tunneling probability, addressing problems of varying scales and complexities. It is defined as:

$$\:{\Delta\:}{f}_{avg}=\frac{1}{N}\sum\:_{j=1}^{N}|f\left({x}_{j}\right)-f\left({x}_{global}\right)|$$

(20)

Thus, by integrating Eqs. (16), (17), and (19), the whale’s position update can be formulated as follows:

$$x_{i}^{{new}} = \left\{ {\begin{array}{*{20}c} {x_{i} + \eta \cdot rand_{q} \cdot \left( {x_{{global}} - x_{i} } \right) + \gamma \cdot rand_{q}^{2} \cdot \left( {x_{i} - x_{i}^{{prev}} } \right),~~~~~~~if~rand < \frac{1}{{1 + \exp \left( { - k \cdot \frac{{f\left( {x_{{local}} } \right) - f\left( {x_{i} } \right)}}{{\Delta f_{{avg}} }}} \right)}}} \\ {x_{i} ,~~\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,~~~~~otherwise} \\ \end{array} } \right.$$

(21)

Equation (21) integrates quantum random numbers with large step jumps, enabling individuals to effectively overcome local constraints and discover novel solutions. The incorporation of historical directions and adaptive step size adjustments ensures that the algorithm retains high-precision local optimization capabilities, even in the final stages of convergence. The sensitivity factor k, along with the jump control parameters $\:\eta\:$ and $\:\gamma\:$, enhances the algorithm’s efficiency and practical applicability.

Improved basic process

Basic process of integrating WOA and ABC

In the proposed QEWOA algorithm, the integration of the WOA and ABC algorithms aims to balance global exploration with precise local exploitation, capitalizing on the strengths of both frameworks. The WOA component primarily governs the global search process through mechanisms inspired by the hunting behavior of humpback whales. This includes the encircling mechanism and the spiral bubble-net strategy, mathematically represented by Eqs. (1)–(4) and (6). In particular, Eq. (4) updates an individual’s position based on the difference between the current and best-known positions, modulated by the control parameter a, which decreases linearly over iterations to transition from exploration to exploitation. Meanwhile, Eq. (6) models a logarithmic spiral path that mimics the natural hunting patterns of whales, enhancing the algorithm’s ability to explore the vicinity of the best solution more effectively.

To improve local search capabilities, the ABC algorithm is embedded within the WOA framework. In this hybrid approach, each whale is conceptualized as both an optimizer and a bee-like agent capable of dynamic role adaptation. The employed bee behavior is reflected when whales exploit the current best solution through direct attraction, as described by Eq. (4), similar to food source exploitation in ABC. The onlooker phase in ABC, which selects promising solutions based on fitness probabilities, is incorporated through the probabilistic selection mechanism in Eq. (25), enabling whales to focus their search stochastically on more promising regions. The scout bee behavior is emulated when a whale stagnates beyond a defined threshold; in this case, the individual is reinitialized using quantum random numbers, effectively resetting its position to maintain population diversity, as guided by Eq. (14).

This integration enables each whale to flexibly alternate between roles analogous to employed, onlooker, and scout bees, depending on their performance and context. By combining the exploratory strength of WOA with the adaptive local refinement of ABC, QEWOA effectively mitigates premature convergence, maintains solution diversity, and ensures detailed optimization near promising areas. This synergy is crucial for solving the complex, high-dimensional ALP, where both broad search and precise adjustment are necessary to navigate constrained, multimodal landscapes.

QEWOA basic process

In QEWOA, the population is first initialized, and as described by Eq. (14), we derive:

$$\:{x}_{ij}^{\left(0\right)}={L}_{j}+{r}_{j}\cdot\:\left({U}_{j}-{L}_{j}\right),\:\:\:\:\:j=1,\:2,\:\cdots\:,\:D$$

(22)

This process, leveraging quantum random numbers, ensures an even distribution of the population within the solution space. Subsequently, the fitness value $\:f\left({x}_{i}\right)$ of each individual is computed to evaluate its quality:

$$\:f\left({x}_{i}\right)=\sum\:_{d=1}^{D}{f}_{d}\left({x}_{id}\right),\:\:\:\:\:{x}_{id}\in\:\mathbb{R}$$

(23)

Here, $\:{f}_{d}\left({x}_{id}\right)$ denotes the objective function value in the d-th dimension. Subsequently, the employed whale refines the current solution position through a local search, and the updated position is determined using Eq. (4):

$$\:{x}_{i}^{new}={x}_{i}+A\cdot\:D$$

(24)

Here, D represents the distance between the current individual and the target individual, while A is a parameter controlling the migration of the individual towards the target, as described by Eqs. (1) and (2). The observer whales subsequently select based on the fitness of the employed whale, where the selection probability is proportional to the employed whale’s fitness, as follows:

$$\:P=\frac{f\left({x}_{i}\right)}{{\sum\:}_{i=1}^{N}f\left({x}_{i}\right)}$$

(25)

Here, $\:f\left({x}_{i}\right)$ represents the fitness of the i-th individual. The observer whales select a target from the employed whales and subsequently update their position. The scout whales, conversely, prevent getting trapped in local optima by randomly selecting a new position within the solution space. The update of their position is described by Eq. (22):

$$\:{x}_{i}^{new}={L}_{j}+{r}_{j}\cdot\:\left({U}_{j}-{L}_{j}\right)$$

(26)

Thus, the position update formulas for employed and observer whales, based on fitness, are as follows:

$$\:{x}_{i}^{new}={x}_{i}+\alpha\:\cdot\:({x}_{best}-{x}_{i})$$

(27)

If an individual’s fitness is low, quantum tunneling may be triggered. Upon the occurrence of quantum tunneling, the individual’s position is updated according to the following formula:

$$\:{x}_{i}^{new}={x}_{i}+{P}_{tunnel}\cdot\:({x}_{jump}-{x}_{i})$$

(28)

In summary, each solution, corresponding to a whale individual, represents a sequence of landing times for a set of aircraft, encoded as the vector $\:[{t}_{1},\:{t}_{2},\:\dots\:,\:{t}_{N}]$ The initial population is generated using quantum random numbers, significantly enhancing both the diversity of the population and the scope of potential solutions. Next, the fitness value of each solution is calculated using the objective function, with a primary focus on the total delay time and associated penalty terms. Throughout the search process, the WOA employs a bubble-net attack strategy to conduct a global search within the solution space, exploring potential solutions. To further refine the search, the ABC algorithm is introduced for local search, optimizing solutions within local regions. When the algorithm becomes trapped in a local optimum, the quantum tunneling effect’s jumping mechanism is triggered to update the current solution’s position. This process continues iteratively until the termination conditions—such as reaching the maximum number of iterations or meeting the convergence criteria—are satisfied. Through this series of steps, QEWOA effectively optimizes the ALP and identifies high-quality solutions that meet multiple constraint conditions.

Convergence is one of the key characteristics that determine the performance of metaheuristic algorithms³⁹. QEWOA integrates the collaborative mechanism of the ABC algorithm with key concepts from quantum computing, aiming to address the challenges of local optima and slow convergence rates commonly encountered in traditional metaheuristic algorithms. Initially, the algorithm employs Eq. (14) for quantum random number-based population initialization. Compared to pseudo-random numbers, QRNs exhibit superior uniformity and traversal, ensuring that the initial population, X, is more uniformly distributed and exhibits greater diversity in the D-dimensional solution space. This theoretically enhances the likelihood of covering the region containing the global optimum, providing a more advantageous starting point for the search and reducing the risk of premature convergence to local optima due to biases in the initial population. Furthermore, during the iterative search process, the algorithm utilizes Eq. (4) to surround the prey, Eq. (6) for the spiral bubble-net attack mechanism, and the role division and information-sharing strategies of ABC. The size of parameter A in Eq. (2) and the random selection of the reference individual, $\:{X}_{rand}$, along with other mechanisms, regulate the balance between exploration and exploitation. The division of roles among leader whales, observer whales, and scout whales, along with the probabilistic selection of f(X) in Eq. (25), further optimizes the allocation of search resources and effectively directs the population towards regions with higher fitness.

The quantum tunneling jump mechanism equips the algorithm with the ability to escape local optima, which is crucial for ensuring global convergence. When an individual $\:{x}_{i}$‘s search stagnates or exhibits little improvement in fitness, the algorithm triggers the quantum tunneling jump in Eq. (21) based on the probability $\:{P}_{tunnel}$ from Eq. (19). This probability, $\:{P}_{tunnel}$, is related to the difference between the current fitness $\:f\left({x}_{i}\right)$ and the local optimum fitness $\:f\left({x}_{local}\right)$; when they are close, the jump probability increases, and vice versa. The jump step length, $\:{\Delta\:}x$, in Eq. (17) is designed by integrating global guidance towards the current global optimum solution $\:{x}_{global}$, local information from the individual’s historical search direction $\:{v}_{i}$ in Eq. (18), and random perturbations introduced by quantum random numbers $\:{rand}_{q}$ and $\:{rand}_{n}$. This jump mechanism enables individuals to “tunnel” through energy barriers in the fitness landscape with a non-zero probability, escaping the attraction of local optima and exploring broader, previously unexplored regions of the solution space. According to the convergence theory of random algorithms (e.g., Markov Chain-based analytical frameworks), if an optimization algorithm ensures the preservation of elite solutions (as QEWOA does by continuously updating the global optimum solution $\:{X}^{*}$) and can transition with a non-zero probability from any solution state to a neighboring region containing the global optimum (the QT jump mechanism significantly enhances this global reachability), the algorithm will probabilistically converge to the global optimum solution after sufficient iterations.

The population size, N, and the maximum number of iterations, $\:{T}_{max}$, as fundamental control parameters, directly influence the algorithm’s exploration capability and convergence depth. A small population size may result in insufficient coverage of the search space, rendering the algorithm susceptible to trapping in local optima. In contrast, appropriately increasing the population size enhances the global search capability but also escalates computational resource consumption. Moreover, the maximum number of iterations determines the depth of the optimization process. If set too low, the algorithm may terminate prematurely before adequately exploring the optimal regions; if set too high, while convergence precision can be improved, computational time will increase substantially. In addition to these fundamental parameters, the dynamic control parameters incorporated into QEWOA also play a critical role in optimization performance. The global step size factor, $\:\eta\:$, and the local perturbation factor, $\:\gamma\:$, jointly regulate the scale and direction of individual search behaviors. During the early stages, a larger $\:\eta\:$ value facilitates wide-ranging jumps across the solution space, promoting exploration, while in the later stages of convergence, a smaller $\:\eta\:$ value refines the search within local regions, further enhancing solution quality. Similarly, the local perturbation factor, $\:\gamma\:$, primarily enhances the algorithm’s fine-tuning capability during the final stages of the search.

The overall time complexity of QEWOA is $\:O\left({T}_{max}ND\right)$, where $\:{T}_{max}$ represents the maximum number of iterations, N denotes the population size, and D signifies the dimensionality of the solution space. In other words, the algorithm demonstrates a linear relationship with respect to population size, problem dimensionality, and number of iterations, making it well-suited for high-dimensional optimization problems in ALP.

In comparison to other advanced heuristic and metaheuristic algorithms, QEWOA offers significant advantages. By integrating ABC, quantum random numbers, and the quantum tunneling effect’s jumping mechanism, QEWOA achieves a balanced integration of global search and local optimization, significantly enhancing both convergence speed and solution quality. Furthermore, QEWOA excels in handling dynamic changes and uncertainty, effectively addressing the complex challenges encountered in real-world applications. The algorithm’s flowchart (shown in Fig. 4) and pseudocode are presented below.

Fig. 4

Flowchart of the algorithm.

Algorithm 1

QEWOA Pseudocode.

Experiment

This study conducts experiments using the aircraft landing dataset provided by the OR-Library, aiming to optimize the landing scheduling plan through QEWOA, minimize penalty values, and accelerate the convergence process. The primary objective of the experiment is to evaluate the effectiveness of QEWOA in solving the ALP. Throughout the experiment, we will model the ALP using QEWOA and solve it with the dataset provided by the OR-Library. By adjusting various parameters and running different configurations, we will perform a comprehensive evaluation of the algorithm’s performance. The experiment will investigate the effects of varying iteration counts, population sizes, and other parameter settings on the final results, thereby analyzing the algorithm’s convergence, stability, and exploration capabilities within the solution space. This experiment aims to explore the potential application of WOA in solving the ALP, provide more effective scheduling solutions, and offer new insights into landing scheduling issues in low-altitude economies within LAITS. Furthermore, the study will investigate the application and development of optimization algorithms for complex practical problems.

Dataset

The dataset files include the number of planes (p) and the freezing time. For each aircraft i (where $\:i=1,\:\dots\:,\:p$), the dataset provides the appearance time, earliest landing time, target landing time, latest landing time, landing penalty per unit time, and the penalty cost for each unit of landing delay. Furthermore, for each aircraft j (where $\:j=1,\:\dots\:,\:p$), the dataset specifies the required separation time between aircraft. This information is essential for solving the model and validating the algorithm.

Experiment process

In the dataset description, each aircraft in the n-th dataset is subject to n + 6 constraints. The first six constraints are fixed and primarily define the landing time windows for the aircraft and the corresponding penalty costs, ensuring that aircraft land within reasonable time frames, with penalties applied for violations. The subsequent n constraints involve the landing time intervals between n aircraft, ensuring safe separation, preventing conflicts, and maintaining the safety and feasibility of the scheduling plan.

QEWOA utilizes an innovative population initialization strategy that significantly enhances solution diversity. Coupled with the jump mechanism, this strategy enables whale individuals to quickly adjust their positions and search directions based on real-time environmental information. This adjustment not only improves search efficiency but also enhances solution quality. Furthermore, QEWOA features robust adaptive adjustment capabilities, enabling automatic parameter optimization in response to dynamic problem changes. This adaptability ensures flexibility and robustness in addressing complex real-world challenges in the aircraft landing scheduling process.

In the ALP problem, the primary objectives are to minimize total delay time, reduce fuel consumption, and enhance runway utilization at airports. The constraints include minimum separation times between aircraft, restrictions on takeoff and landing intervals, and penalties for deviations from these arrangements. These objectives and constraints collectively determine the quality of the scheduling plan, necessitating the effective balancing of multi-objective and multi-constraint conditions by the algorithm. By incorporating these objectives and complex constraints into its fitness function, QEWOA uses its robust global and local search capabilities to identify optimal solutions that balance multiple objectives. Specifically, QEWOA effectively manages and optimizes complex constraints, thereby ensuring both the optimality and feasibility of the scheduling plan. These constraints are detailed in Table 1, presented below.

Table 1 Definition of constraints.

$$\:{P}_{total}=\sum\:_{i\in\:I}{(Pb}_{i}+{PA}_{i})$$

(29)

Other constraints are defined as follows⁴⁰ :

The landing time ($\:{x}_{i}$) of each aircraft must ensure continuity with the preceding or succeeding aircraft:

$$\:{x}_{i+1}-{x}_{i}={S}_{i,\:i+1},\:\:\:\:\:\forall\:i\in\:I$$

(30)

The scheduling of all aircraft must be accomplished within the overall operational time frame:

$$\:\begin{array}{c}\text{max}{x}_{i}\\\:i\in\:I\end{array}-\begin{array}{c}\text{min}{x}_{i}\\\:i\in\:I\end{array}\le\:{T}_{max}$$

(31)

No two aircraft are permitted to land on the same runway simultaneously:

$$\:\left|{x}_{i}-{x}_{j}\right|\ge\:{S}_{ij},\:\:\:\:\:\forall\:i,\:j\in\:I,\:\:i\ne\:j,\:\:k\in\:R$$

(32)

And:

$$\:{E}_{i}\le\:{x}_{i}\le\:{L}_{i}\:\:\:\:\:i\in\:I$$

(33)

$$\:0\le\:{TE}_{i}\le\:{T}_{i}-{E}_{i}\:\:\:\:\:i\in\:I$$

(34)

$$\:0\le\:{TL}_{i}\le\:{L}_{i}-{L}_{i}\:\:\:\:\:i\in\:I$$

(35)

$$\:{TE}_{i}\ge\:{T}_{i}-{x}_{i}\:\:\:\:\:i\in\:I$$

(36)

$$\:{TL}_{i}\ge\:{x}_{i}-{T}_{i}\:\:\:\:\:i\in\:I$$

(37)

$$\:{x}_{i}={T}_{i}-{TE}_{i}-{TL}_{i}\:\:\:\:\:i\in\:I$$

(38)

Parameter settings

Among the three key parameters in QEWOA, $\:\eta\:$ controls the step size of the algorithm within the search space. The range of values for $\:\eta\:$ is set between [0.1, 0.5], balancing global exploration and convergence speed. A smaller $\:\eta\:$ value facilitates a fine-grained search, mitigating the risk of missing local optima, while a larger η value accelerates global exploration, making it effective for quickly identifying potential solution spaces in the early stages. The selection of $\:\gamma\:$ within the range [0.5, 1] is guided by the dynamic balance principle of local search. Introducing moderate perturbations enhances the algorithm’s diversity, preventing convergence to local optima, while avoiding excessive perturbations that could cause instability. The value of k, ranging from^5,10, is determined through a trade-off analysis of convergence speed and search accuracy. Lower k values increase the algorithm’s sensitivity to details within the solution space, whereas higher k values expedite iterations and improve efficiency. The parameter ranges described above were determined through a combination of heuristic optimization theory and several preliminary experiments, ensuring that QEWOA achieves an optimal balance between exploration and exploitation, thereby enhancing the overall performance and robustness of the algorithm.

A systematic sensitivity analysis of these three parameters is provided below. In the experiments, a unified computational formula was employed to quantify the impact of each parameter on the algorithm’s output. The specific formula is “Sensitivity value = Output variation rate / Parameter variation rate,” where the output variation rate is calculated from changes in the objective function values under different parameter settings, and the parameter variation rate corresponds to the relative change in the parameter for each experiment. This method allows for the direct quantification of each parameter’s effect on QEWOA’s performance and evaluates its stability under different experimental settings. The sensitivity line chart is presented in Fig. 5.

Fig. 5

Sensitivity analysis line chart.

When $\:\eta\:$ is within the range of [0.1, 0.5], $\:\gamma\:$ is within the range of [0.5, 1], and k is within the range of^5,10, the algorithm demonstrates low sensitivity and minimal fluctuations, with the overall sensitivity value changing relatively smoothly. This indicates that QEWOA maintains stable performance in the face of parameter variations, demonstrating strong robustness.

Experimental results

The experimental results for the small-scale dataset are presented in Fig. 6, which illustrates the number of iterations and the convergence behavior of various algorithms under different aircraft quantities. The figure consists of nine subplots, each illustrating the convergence trends of a simple heuristic method⁴¹, the ADHS algorithm⁴², the ISA algorithm⁴³, a standard selection method⁴⁴, and the proposed algorithm throughout the execution process. The subplots are arranged in a left-to-right, top-to-bottom order.

Fig. 6

Comparison of algorithm convergence on small-scale datasets.

The figure above illustrates that, after using quantum random numbers for population initialization, QEWOA outperforms other algorithms in the early convergence stage across most test cases, highlighting its strengths in global search and local optimization. Additionally, the convergence curve of QEWOA demonstrates relatively high stability, reflecting both the consistency of the algorithm during the optimization process and its stability and robustness when solving the ALP. This performance further supports the contribution of quantum random numbers and the enhanced search mechanism to the algorithm’s performance, allowing it to maintain a relatively high solution efficiency and quality under complex constraints.

For large-scale datasets, QEWOA also demonstrates relatively strong performance, as illustrated in Fig. 7 below. The algorithm exhibits improved convergence in the initial stage compared to other comparison algorithms. Additionally, the jumping mechanism enhances the diversity and flexibility of the search, enabling the algorithm to explore the solution space effectively while avoiding local optima.

Fig. 7

Comparison of algorithm convergence on large-scale datasets.

During the initialization phase, several strategies contributed to the enhanced early iteration performance of QEWOA compared to competing algorithms, as shown in the figure. These include quantum random number-based initialization, which increases population diversity; the uniform distribution of initial solutions, which broadens the coverage of the search space; and the preservation of potential optimal solutions from the outset, which facilitates early convergence. These initialization strategies fundamentally enhance QEWOA’s global search capability and account for its notable performance during the early stages of iteration.

The simple heuristic method proposed by Amir Salehipour⁴¹ is used as the benchmark, and the penalty value calculated by this method is denoted as $\:\lambda\:$. Based on this, the relative penalty $\:{\Delta\:}$ for other algorithms can be determined using the following formula, facilitating a comparative analysis of algorithm performance in addressing the problem:

$$\:{\Delta\:}\left(\%\right)=\frac{{\lambda\:}^{{\prime\:}}-\lambda\:}{\lambda\:}\times\:100\%$$

(39)

If the calculation result is negative, this indicates that the algorithm achieves a lower penalty value than the benchmark, highlighting its superior optimization performance. The corresponding results are presented in Table 2.

Table 2 Penalty value ratios of different algorithms on datasets of varying scales.

The experimental results indicate that initializing the population with quantum random numbers offers certain benefits when handling large-scale datasets. This initialization method enhances the diversity and coverage of the population, enabling the algorithm to explore the solution space more efficiently, thus reducing the overall penalty value. Moreover, the method achieves a balanced approach between global search and local optimization, offering a viable solution for optimization problems with complex constraints, thus further demonstrating its practicality and reliability in large-scale optimization problems.

To further examine the variation between each iteration and the preceding one, this study calculates the average variance of the results across iterations to evaluate the stability of the convergence process. As shown in Fig. 8, the variance values for the majority of datasets exhibit a relatively smooth trend throughout the iterations. This behavior further supports the contribution of quantum random numbers in the population initialization process. The introduction of quantum random numbers not only enhances the initial diversity of the population but also promotes the algorithm’s rapid convergence to lower penalty values in the early stages, thus providing a solid foundation for the subsequent optimization process. This convergence behavior reflects the algorithm’s effectiveness and stability in addressing complex optimization problems.

Fig. 8

Comparison of the average variance of convergence between adjacent iterations.

To comprehensively assess the overall efficiency of the five proposed algorithms, this study employs the Friedman test for statistical analysis. The Friedman test is a non-parametric statistical method used to compare differences in the medians of multiple related samples and is widely applied for evaluating algorithm performance. For each aircraft landing instance, all algorithms were executed, and the total penalty values for each algorithm were recorded. These data were organized into a matrix, with rows representing the test instances and columns representing the different algorithms. The Friedman test was subsequently applied to evaluate the performance differences among the algorithms across various instances.

Specifically, assuming a significance level of α = 0.05, the Friedman test statistic $\:{\chi\:}_{F}^{2}$ was calculated. When the p-value is less than the significance level, it suggests a significant performance difference between the algorithms. The experimental results demonstrate that the p-value obtained from the Friedman test is close to zero, well below 0.05. Therefore, the null hypothesis can be rejected, and it can be concluded that a significant performance difference exists between the five algorithms. To further identify the best-performing algorithm and the specific performance gaps, a post-hoc analysis was conducted. The results reveal that QEWOA performs optimally in most instances, demonstrating its substantial advantage in solving the ALP. The detailed statistical results and algorithm rankings are presented in Table 3.

Table 3 Algorithm ranking.

This analysis not only validates the effectiveness of QEWOA but also highlights the performance differences among various algorithms in solving the ALP problem, providing a scientific basis for selecting the most suitable algorithm. Through the use of Friedman tests and post-hoc analysis, the results clearly demonstrate the relative advantages of different algorithms in improving timeliness, enhancing runway utilization, reducing controller workload under LAITS, and ensuring fairness among airlines. These findings provide valuable insights for future research in scheduling optimization.

Overall, the experimental results suggest that QEWOA demonstrates significant potential in solving the ALP and offers an effective optimization method for landing scheduling in the LAIT field. However, further research and exploration are required to enhance the algorithm’s efficiency, robustness, and adaptability in addressing more complex real-world scenarios.

Conclusion and outlook

This paper presents an improved version of WOA that integrates quantum random numbers and the quantum tunneling effect, leading to the development of QEWOA, which is further enhanced by the collaborative mechanism of ABC. Initializing the population with quantum random numbers enhances population diversity, overcoming the limitations of traditional random number initialization and providing broader coverage of the solution space for global search. The introduction of the quantum tunneling effect simulates the behavior of quantum particles traversing energy barriers, facilitating escape from local optima when individuals become trapped, thereby improving the global search performance of the algorithm. Furthermore, the incorporation of the collaborative mechanism of employed bees, onlooker bees, and scout bees from ABC further enhances the algorithm’s search ability within the local solution space. Experimental results demonstrate that QEWOA, when applied to optimize the landing problem of low-altitude manned aircraft—a practical problem with complex constraints—effectively avoids being trapped in local optima and quickly identifies the global optimum, showcasing its strong practical application value compared to traditional WOA and ABC.

However, despite the promising results obtained by this algorithm in ALP, areas for improvement remain. First, the selection of parameters for the quantum tunneling mechanism significantly influences the algorithm’s performance. The adjustment of tunneling probability and step size control factors is crucial for further improving the algorithm’s performance. Second, although the algorithm integrates quantum mechanics and the ABC mechanism, enhancing its stability and computational efficiency remains an area worth exploring. Future work could focus on integrating quantum computing with other intelligent optimization algorithms, further expanding the algorithm’s application scope and verifying its performance in real-world engineering problems.

Finally, although quantum computing offers new perspectives for optimization algorithms, the generation of quantum random numbers remains constrained by hardware limitations. Implementing quantum computing more efficiently in practical applications remains an ongoing challenge. In the future, combining quantum computing with traditional computational resources in a hybrid computing mode may further enhance optimization efficiency. Moreover, with the continued development of quantum computing technology, the scalability and application prospects of quantum optimization algorithms are expanding. Future research may explore the integration of quantum computing with additional intelligent algorithms, thereby broadening the application fields of optimization algorithms.

In summary, QEWOA, as a novel intelligent optimization algorithm, holds significant development potential and offers new approaches for solving complex optimization problems. In the future, with the continued advancement of quantum computing technology and the integration of more advanced optimization strategies, QEWOA is expected to be applied across a broader range of fields, particularly in aerospace, traffic scheduling, and intelligent manufacturing, offering efficient solutions.