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Responsible Institution:China Association for Science and Technology

Sponsored by:Chinese Institute of Electronics
Nanjing University of Aeronautics and Astronautics

ISSN:1004-9037

CN:32-1367/TN

Address:29Yudao Street,Nanjing,China

Telephone:025-84892742

Chief Editor:025-84892742

E-mail:sjcj@nuaa.edu.cn

Post Coder:210016

:China Association for Science and Technology
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  • ISSN 1004-9037
    CN 32-1367/TN
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  • XU Zheng, PAN Zihao, WANG Ning, GUO Daoxing

    2025,40(6):1382-1411, DOI: 10.16337/j.1004-9037.2025.06.002

    Abstract:

    With the increasing of array antennas and the growing complexity of anti-jamming, traditional adaptive beamforming methods often suffer from high computational complexity. Deep learning, with its powerful data-driven capabilities, offers a novel approach to overcoming the performance bottlenecks of traditional adaptive beamforming. This paper provides a systematic review on current studies and development trends of deep learning in array antenna beamforming. First, we revisit the evolution of traditional beamforming algorithms,ranging from the Howells-Applebaum adaptive processor to robust beamforming based on convex optimization. Second, we analyze the innovative applications of deep learning models such as convolutional neural networks (CNNs), recurrent neural networks (RNNs), and long short-term memory (LSTM) networks in beamforming. This review demonstrates that deep learning methods exhibit significant advantages in improving system performance due to their powerful nonlinear modeling capabilities, end-to-end optimization characteristics, and environmental adaptability. Specifically, in mobile communications, deep learning-based beamforming methods substantially enhance the computational efficiency and environmental adaptability of massive multiple input multiple output (MIMO) systems. In radar signal processing, deep learning models effectively improve anti-jamming performance and target detection accuracy. In acoustic signal processing, deep neural networks enable more precise sound source localization and noise suppression. Finally,this paper explores future research directions, including network architecture innovation, real-time processing optimization, robustness enhancement, cross-scenario transfer learning, theoretical foundation deepening, and novel application expansion.

  • ZHAO Qianxi, LIU Jianing, WANG Diwen, TIAN Feng

    2025,40(6):1412-1423, DOI: 10.16337/j.1004-9037.2025.06.003

    Abstract:

    In the terahertz frequency band, utilizing an orthogonal time frequency space (OTFS)-integrated sensing and communication (ISAC) vehicular networking system with superimposed pilots can achieve high-speed data transmission between vehicles alongside high-precision parameter sensing. First, the OTFS signal modulation model, the communication signal model, and the sensing signal model are established and analyzed. At the receiver, a new sensing channel model containing angle information is derived. Subsequently, a coarse angle estimation algorithm based on the uniform grid multiple signal classification (MUSIC) method and a fine angle estimation algorithm based on the golden section MUSIC method are proposed. Finally, a maximum likelihood estimation criterion-based algorithm is proposed for estimating the integer and fractional parts of the channel delay and Doppler shift parameters. Simulation results demonstrate that the proposed algorithms can achieve accurate estimation of sensing parameters such as angle, distance, and velocity.

  • REN Xiaoning, DUAN Hongguang, HUANG Fengxiang, DONG Shikang

    2025,40(6):1424-1433, DOI: 10.16337/j.1004-9037.2025.06.004

    Abstract:

    In non-terrestrial network (NTN) scenarios, to overcome the effect of large Doppler frequency offset on the communication, a channel estimation method based on global information super resolution denoising neural network (GSRDnNet) is proposed. This method considers the channel estimation matrix at the pilot obtained by the least square(LS) estimation algorithm as a low-resolution small-size image and takes it as the input to the neural network. The input data is then processed by the GSRDnNet network to obtain a more accurate high-resolution image with a complete channel response matrix for the time-frequency resource block. Four NTN-tapped delay line (TDL) A,B,C and D channel models are used for simulation verification. Simulation results indicate that GSRDnNet improves mean squared error (MSE) performance by 3.37—8.83 dB compared to the traditional LS algorithm. Compared with the practical channel estimation(PCE) algorithm, the MSE is improved by 2.11—6.06 dB, and compared with the SRCNN+DnCNN method, which requires pre-interpolation processing, the MSE is improved by 1.37—4.40 dB. And compared with super resolution convolutional neural network (SRCNN)+denoising convolutional neural network (DnCNN) ,the input of GSRDnNet network model only considers the channel estimation matrix at the pilot, so it not only has higher estimation accuracy, but also reduces the computational complexity by about 84%.

  • LI Zhili, FU Youhua, SONG YUNCHAO

    2025,40(6):1434-1444, DOI: 10.16337/j.1004-9037.2025.06.005

    Abstract:

    This paper studies the precoding optimization problem for extremely large-scale multiple-input-multiple-output(XL-MIMO) downlink systems under a near-field channel model based on spectral efficiency fairness. The near-field channel model considers the coexistence of line-of-sight (LOS) and non LOS (NLOS) non-stationary mixed channels within the cell, where LOS channels are modeled using spherical wave models, while NLOS channels are modeled using Rayleigh models. The geometric mean of spectral efficiency is used as the optimization target to ensure fairness among users and optimize the overall spectral efficiency of the system. To handle the complex optimization objective function, a first-order Taylor expansion approximation is applied to create a simplified objective function. Subsequently, Lagrangian dual transformation and quadratic transformations are used to transform the original optimization problem into an equivalent one that is easier to solve. Finally, to reduce computational complexity, the projection fast iterative shrinkage threshold algorithm (PFISTA), which combines fast iterative shrinkage thresholding algorithms with projected gradient descent, is employed to solve the equivalent optimization problem. Simulation results show that using the geometric mean as the objective function can reduce differences in spectral efficiencies among users, leading to a balanced improvement in user spectral efficiencies. Moreover, PFISTA achieves comparable performance to existing methods while maintaining lower computational complexity.

  • SHEN Zixuan, XIE Lei, GUO Ming

    2025,40(6):1445-1463, DOI: 10.16337/j.1004-9037.2025.06.006

    Abstract:

    To cope with the challenges of low communication rate and susceptibility to interception of sensing-centric waveforms in integrated sensing and communication (ISAC) systems, this paper designs a multi-channel frequency-hopping transmission architecture based on quadrature phase shift keying (QPSK) and linear frequency modulation (LFM) signals. This architecture transmits multiple LFM signals simultaneously within overlapping spectral bands to enhance the symbol rate. Encrypted communication is achieved by the frequency-hopping characteristics of the LFM subcarriers. Furthermore, the time-division multiplexing (TDM) mechanism of dynamic preambles and data improves the accuracy of path indexing and parameter estimation for multi-path LFM signals. Simulation results show that under identical symbol rate constraints, the proposed multi-channel parallel architecture achieves superior bit error rate (BER) performance compared to traditional single-channel schemes. Specifically, the BER of the four-channel architecture at 0 dB SNR is reduced by one order of magnitude relative to the single-channel scheme. Additionally, the dynamic preamble scheme meets the requirements for path index identification across various scenarios. At 0 dB SNR, the normalized mean square error(NMSE) remains below 10-2. Furthermore, both the proposed symbol demodulation algorithms achieve a BER below 10-2 at 0 dB SNR in their respective scenarios. Moreover, the frequency hopping mechanism significantly enhances the system’s anti-intercept capability. Even with 50% parameter leakage, the probability of accurate recovery (PAR) of signal parameters by the third parties remains suppressed below 7%, validating the robustness and application value of the solution.

  • WANG Jiaqi, WANG Wei

    2025,40(6):1464-1476, DOI: 10.16337/j.1004-9037.2025.06.007

    Abstract:

    Due to the varying distances of different unmanned aerial vehicles (UAVs), the overlapping signals often exhibit different signal-to-noise ratios, and the presence of various interference signals in low-altitude environments further increases the difficulty of identification. To address these problems, this paper proposes a joint detection-separation-identification scheme for overlapping signals from multiple UAVs. The scheme effectively improves the detection and identification performance of overlapping signals with different SNRs through three steps: signal detection, signal separation, and signal identification. First, the YOLO detector is employed to locate potential UAV signals on the time-frequency spectrogram. Then, a data augmentation method based on random deviation is proposed to mitigate the bias in the signal separation process. Subsequently, the bandwidth and duration features of the signals are extracted using a YOLO-based classifier to achieve classification of distinct UAV signals. Finally, to further improve the recognition accuracy of signals from identical UAV models, an enhanced ResNet model integrated with attention mechanisms and an optimized Bagging ensemble learning method are proposed. Experimental results based on publicly available datasets demonstrate that the proposed scheme outperforms existing methods in scenarios where interference signals and UAVs of the same model coexist.

  • LI Guoxin, GAN Qi, CHEN Jin, JIAO Yutao, WANG Haichao, HE Xing

    2025,40(6):1477-1489, DOI: 10.16337/j.1004-9037.2025.06.008

    Abstract:

    To solve the fast pairing and power allocation problem of non-orthogonal multiple access (NOMA) under imperfect serial interference cancellation (SIC) conditions, a deep reinforcement learning-based user pairing and power optimization scheme is proposed. First, this paper considers the scenario of imperfect SIC for multiuser NOMA, and constructs an optimization problem to maximize the system reachable communication rate with user pairing and user transmit power allocation factor as optimization variables. The condition of user pairing using NOMA under the imperfect SIC condition is analyzed, and the user power allocation for the maximum reachable rate under this condition is introduced. Second, the user pairing problem is treated as a combinatorial optimization problem, and a novel user pairing scheme is designed based on the real-time requirement using an improved pointer network. Simulation results show that this scheme can effectively improve the reachable rate of the NOMA system to 99.8% of that of the optimal exhaustive search algorithm. It achieves real-time performance and adapts to the dynamic change of the number of users.

  • ZHU Yinxia, ZHANG Jian, ZHANG Bangning, GUO Daoxing, CHENG Jian

    2025,40(6):1490-1504, DOI: 10.16337/j.1004-9037.2025.06.009

    Abstract:

    With the development of 5G and 6G communications, satellite communication, which can provide the global seamless coverage capability, has become an indispensable and important role. In view of the open channel characteristics of satellite communication, the security of satellite communication has been paid more and more attention, especially the demand for satellite secure communication in national defense and military communication. Based on the architecture of the multi-satellite and terminal multi-antenna (MS-TMA) satellite communication system, from the physical layer of security and covert communication theory, a satellite covert communication strategy is put forward. Theoretical analysis and simulations validate the secure communication performance of the approach. The findings offer significant insights for advancing satellite secure communication theory and the practical applications.

  • TIAN Minghao, LI Pan, YAN Su, WU Xueliang, XU Luping, YAN Bo

    2025,40(6):1505-1517, DOI: 10.16337/j.1004-9037.2025.06.010

    Abstract:

    Currently, LiDAR- and vision-based 3D reconstruction technologies are widely used in terrain scene measurement. Although various 3D imaging methods based on LiDAR and cameras have been developed, each has limitations. RGB-D cameras can capture both color and depth information but often have lower accuracy than LiDAR, while 3D LiDAR provides high-precision spatial data but lacks color information and is typically expensive. This paper proposes a 3D-RGB imaging method based on data-level fusion of a 2D LiDAR and multi-view cameras, integrating 3D-RGB point cloud data from a 2D LiDAR and four cameras from different viewpoints. The method achieves accurate and dense 3D-RGB imaging through 3D-RGB point enhancement, feature plane detection and extraction, and global consistency alignment. First, fusing RGB and point cloud data enhances data quality, while feature plane detection optimizes geometric structure representation. Then, a global consistency alignment strategy reduces accumulated errors and improves overall imaging accuracy. Experimental results show that compared with multi-line LiDAR solutions, the proposed method offers advantages in imaging density and accuracy, with an overall error of less than 0.15 m, demonstrating its potential for 3D reconstruction and environmental surveying in complex environments.

  • ZHAO Shanshan, SHEN Qi, MIAO Jianing

    2025,40(6):1518-1526, DOI: 10.16337/j.1004-9037.2025.06.011

    Abstract:

    Existing multi-station fusion technologies focus on utilizing intuitive features such as echo amplitude correlation and spatial location. However, the comprehensiveness of manual feature extraction is insufficient, which can easily lead to signal resource waste, incomplete feature extraction, and limited generalization of discrimination algorithms. To address this issue, this paper innovatively proposes a jamming identification strategy that integrates multi-radar cooperative detection with convolutional neural network. This approach leverages convolutional neural networks to deeply explore unknown information in echo data, extracting differences between real and false targets in multidimensional deep features, surpassing single spatial correlation differences, and achieving deception jamming identification. Finally, simulation experiments validate the feasibility of the proposed method in resisting deception jamming and analyze the effects of target size, multi-station radar deployment and phase errors on the proposed algorithm.

  • YANG Xiao, YAO Aiqin, SHI Xiling

    2025,40(6):1527-1537, DOI: 10.16337/j.1004-9037.2025.06.012

    Abstract:

    To address the low recognition rates due to the insufficient utilization of original signal timing information in automatic modulation recognition (AMR), this paper proposes a signal pattern recognition algorithm based on frequency domain denoising and temporal convolutional networks (TCN). Experiments are conducted using the standard dataset RML2016.10a, and a frequency domain denoising module (FDDM) is introduced to effectively suppress environmental noise. The I/Q components of the signal are converted into A/P components, followed by vector normalization to enhance stability. Finally, the preprocessed signals are fed into the TCN network for classification recognition. Results indicate that this algorithm achieves an average recognition rate higher than those of models such as gated recurrent unit (GRU), convolutional long short-term memory (LSTM), memory-cost-efficient convolutional neural network (MCNet), cost-efficient hybrid deep learning network (CGDNet), and denoising auto-encoder (DAE) when processing complex modulation schemes like 16 QAM and 64 QAM. Additionally, validation using actual I/Q data collected through the universal software radio peripheral (USRP) demonstrates that the algorithm exhibits good robustness and application potential under additive white Gaussian noise (AWGN) channels.

  • YUAN Chengsheng, ZHANG Xueyuan, ZHOU Zhili, LI Xinting, FU Zhangjie

    2025,40(6):1538-1555, DOI: 10.16337/j.1004-9037.2025.06.013

    Abstract:

    To solve the problem of low accuracy and weak generalization of forged speech detection, a new algorithm based on time-frequency feature fusion is proposed. Firstly, in order to excavate the uneven energy distribution of speech fragments or the abnormal fundamental frequency fluctuation, and extract the subtle difference of semantic coherence, a multi-branch feature fusion network is proposed to excavate the difference traces of true and false speech from the pitch, pitch intensity and energy distribution respectively, so as to better represent the frequency change, amplitude change and peak difference of true and false speeches, and improve the accuracy of forged speech detection. Secondly, the classical coordinate attention mechanism fails to effectively mine the fine-grained differences in the time-frequency domain of speech. Therefore, a time-frequency coordinate attention mechanism is proposed to jointly encode the energy distribution and fundamental frequency fluctuation anomalies from the time domain and the frequency domain respectively, so as to better characterize the common high frequency energy anomalies in the spectral graph and improve the generalization of the model. Finally, an adaptive joint loss optimization function is designed to balance the importance of different branch networks to further improve the model’s ability to learn high frequency energy anomalies and semantic incoherence in forged speech. Performance is evaluated on the logical access (LA) dataset of ASVspoof 2019, and experimental results show that compared with the current methods, the proposed method achieves good performance in both EER(Equal error rate) and mint-DCF(Minimum normalized tandem detection cost function) indicators, which decrease by 0.34% and 0.014, respectively. In addition, when dealing with unknown attack A17, which is extremely difficult to detect, it also show good generalization, where EER and mint-DCF decrease by 3.952 2% and 0.136 4, respectively. When dealing with unknown types of spoofing attacks, it also shows better generalization.

  • ZHAO Ming, CHEN Rui

    2025,40(6):1556-1567, DOI: 10.16337/j.1004-9037.2025.06.014

    Abstract:

    Most existing deep learning-based sound event detection methods adopt the conventional 2D convolution. However, its inherent translation invariance property is incompatible with audio signals, rendering the model incompetent in detecting complex sound events. To address the issue, a hybrid convolutional neural network based on feature fusion is proposed. Specifically, by calculating the distribution of the audio spectrogram and adaptively generating convolutional kernels, the proposed model dynamically extracts local features that maintain physical consistency with the audio signal. Meanwhile, the self-attention mechanism is employed in parallel to capture long-distance feature dependencies of the spectrogram. To eliminate the semantic gap between local and global features, a feature fusion module is designed to effectively integrate these two distinct feature representations. Furthermore, to further enhance the detection performance of the proposed model, an improved bidirectional gated recurrent unit based on a multi-scale attention mechanism is proposed to fully refine the fused feature information, which emphasizes event-related frames and suppresses background frames. Experiment results on the DCASE2020 dataset indicate that the proposed model has achieved an F1-score of 52.57%, which outperforms other existing methods.

  • WANG Haoran, ZHANG Chun, ZHANG Guohui, WANG Wenzhuo, WANG Nana

    2025,40(6):1568-1580, DOI: 10.16337/j.1004-9037.2025.06.015

    Abstract:

    In island wetlands, the acoustic environment is complex, with various noise sources such as wind, rain, and ocean waves. To effectively address these interferences in bird song processing and improve the accuracy of species identification, a noise reduction method based on adaptive Kalman filtering with linear predictive coding (A-KF-LPC) is proposed to tackle the issue of noise interference in real-time bird song monitoring under complex acoustic conditions in island wetlands. The A-KF-LPC filter enhances stability by weighted filtering bird song signals, while also suppressing noise and providing precise estimations of uncertain small segments within the acoustic signals, progressively approximating the real scenario. Simulations verify the performance of the A-KF-LPC filter, demonstrating its effectiveness in noise reduction. Experimental results show that under different signal to noise ratios (SNRs), the A-KF-LPC filtering method is more effective in denoising bird songs compared to traditional Kalman filtering and least mean squares (LMS) adaptive filtering methods. Even under conditions where the signal is fully masked by -10 dB noise, the method can still filter out part of the noise. The A-KF-LPC method proposed in this study holds significant application value in the field of acoustic signal processing, offering an efficient and feasible solution for research on bird species in wetland ecosystems, with potential for broader applications.

  • LI Shaoshan, CHENG Jianhong, ZHANG Zhao, JIN Long

    2025,40(6):1581-1595, DOI: 10.16337/j.1004-9037.2025.06.016

    Abstract:

    Addressing the issues of complex surface defect types and minute defect sizes in electronic components that are difficult to detect, this paper proposes an electronic component surface defect detection model based on an improved reverse distillation network. Firstly, the model adopts an unsupervised learning approach, effectively mitigating the dependency on a large amount of annotated data. Secondly, by employing a reverse distillation network architecture, it alters the traditional mode of one-way guidance from the teacher model to the student model in distillation networks, thereby enhancing the model’s adaptability in surface defect detection tasks. Furthermore, a receptive field attention convolutional module is introduced into the student decoder of the reverse distillation network to bolster the model’s ability to detect minute defects. Finally, the cosine similarity is utilized as the loss function to train the student network and the bottleneck module. Experimental results using a self-constructed dataset for electronic component surface defect detection demonstrate significant improvements in detection accuracy, achieving 86.7% and 89.1% in the area under the receiver operating characteristic curve(AUROC) at the image and pixel levels, respectively, and 69.4% in area under the pre-region-overlap curve (AUPRO) at the pixel level.

  • YANG Zhenzhen, WU Xinyi

    2025,40(6):1596-1607, DOI: 10.16337/j.1004-9037.2025.06.017

    Abstract:

    Cross-modality person re-identification, as a research hotspot in the field of computer vision, aims to solve the challenge of matching pedestrians across varying imaging conditions. Existing methods focus on extracting modality-shared features, but fail to fully mine the detailed features that are crucial for discriminative person identities. To address this issue, a hybrid convolutional enhancement and content-aware attention (HCECA) for cross-modality person re-identification is proposed, which aims to extract pedestrian features with more detailed information. First, a hybrid convolutional enhancement (HCE) module is embedded in the backbone network to capture richer cross-modality feature representation, enhancing the distinctiveness and robustness of the features. Second, a content-aware attention (CA) module is employed to mine rich detailed information, thereby improving the discriminability of pedestrian features. Finally, experiments are performed on the SYSU-MM01 and RegDB datasets. The proposed HCECA attains the Rank-1 accuracy of 72.21% and the mean average preeison(mAP) of 69.89% in the all-search mode on the SYSU-MM01 dataset, while achieving the Rank-1 accuracy of 92.23% and the mAP of 85.08% in the visible-infrared mode on the RegDB dataset. Both results outperform better than those of current cross-modality person re-identification methods.

  • GU Junhao, ZHANG Sunjie, QIN Chendong

    2025,40(6):1608-1624, DOI: 10.16337/j.1004-9037.2025.06.018

    Abstract:

    Although Transformers have made significant progress in 3D point cloud processing, efficiently and accurately learning valuable low-frequency and high-frequency information remains a challenge. Moreover, most existing methods focus primarily on local spatial information, neglecting global spatial information, which leads to information loss. This paper proposes a novel point cloud learning network, referred to as the multi-scale dual-branch dual-attention network. First, in the feature extraction process of the point cloud, compared to methods that search for neighboring points at a fixed scale, the multi-scale K-nearest neighbor(KNN) approach not only preserves local structural details but also more effectively captures global geometric information. Second, this paper introduces a dual-branch dual-attention architecture to extract different spatial features, proposing a dual-attention mechanism combining local window attention and global channel content attention to extract low-frequency and high-frequency information from the network, respectively. Additionally, on this basis, this paper introduces the group-rational Kolmogorov-Arnold (GR-KAN) layer into the classification head, replacing the traditionally used multilayer perceptron (MLP) layer, which allows for more flexible handling of nonlinear features and makes the network more sensitive to complex datasets. Finally, extensive experiments demonstrate that the proposed model achieves an accuracy of 93.8% on the ModelNet40 dataset and 86.5% on the ScanObjectNN dataset, showcasing its superior performance and broad application prospects in 3D point cloud processing.

  • XU Xinzhi, HE Hong

    2025,40(6):1625-1636, DOI: 10.16337/j.1004-9037.2025.06.019

    Abstract:

    Aiming at the problems of high computational complexity and large number of parameters in the current pose estimation model, this paper proposes a lightweight pose estimation algorithm. Firstly, the partial channel encoding (PCE) module is introduced in the feature extraction process, and the local and global features of the image are extracted respectively by combining the advantages of convolutional neural network and visual encoder. Then, the weighted feature fusion is introduced in the process of multi-scale feature fusion to enhance the multi-scale feature fusion ability of the model and avoid the problem of reduced accuracy caused by model lightweight. Then, in the process of regression prediction, the detection head of the human detection and classification parts is shared to improve the recognition efficiency of the model in the pose estimation task. Experimental results show that compared with the basic model, the proposed model reduces the number of parameters by 27% and the amount of computation by 18%, and increases the accuracy by 0.2%. It not only ensures the accuracy of recognition, but also realizes the lightweight of the detection algorithm, providing an effective means to achieve real-time accurate pose estimation.

  • ZHANG Jing, YANG Yuhao, CAO Feng, ZHANG Chao, LI Deyu

    2025,40(6):1637-1649, DOI: 10.16337/j.1004-9037.2025.06.020

    Abstract:

    High-resolution remote sensing image scene classification aims to accurately perceive complex surface scenes, which is significant for the understanding and information extraction of high-resolution remote sensing images. A new scene classification method based on feature aggregated convolution neural network (FACNN) and capsule network(CapsNet), named FACNNCN, is proposed in this paper. For the proposed method, the distinguish ability and robustness of convolutional features for scene classification are enhanced by adding aggregated features. Meanwhile, the spatial relationship between geographic entity and scene is represented based on CapsNet. Therefore, the proposed method can overcome some drawbacks usually found in existing high-resolution remote sensing image scene classification methods based on CNNs. For example, the extracted representative features of scene images are insufficient and the spatial features of geographical objects are lack of consideration. The method proposed in this paper is tested on two public high-resolution remote sensing image scene classification datasets (UC Merced Land-Use and NWPU-RESISC45). Experimental results show that the classification accuracy of FACNNCN is better than those of comparison methods.

  • ZHENG Jiyuan, ZHANG Shaobo, WANG Xin, WANG Xiaobo

    2025,40(6):1650-1660, DOI: 10.16337/j.1004-9037.2025.06.021

    Abstract:

    The vehicle routing problem is a classic combinatorial optimization problem that has been proven to be NP-hard. It is widely applied to the fields of transportation logistics and intelligent manufacturing. However, such problems usually assume the homogeneity of vehicles, making it difficult to characterize the differences in vehicles transportation capabilities for different types of commodities in practical scenarios. To address it, a new heterogeneous vehicle routing problem (HVRP) is proposed. By introducing commodity type attributes and vehicle transportation capability constraints, an integer programming model describing the vehicle-order matching relationship is constructed, with the objective of minimizing the total transportation distance. The service relationship between vehicles and customers is formally described by modeling the transport capability of different vehicle types for various product categories. To achieve efficient optimization of the HVRP, a variable step multi-neighborhood search (VSMNS) algorithm is proposed, along with a solution representation method that combines path encoding with linked-list structures. Finally, comparative experiments are conducted among VSMNS with genetic algorithms, hybrid genetic algorithms and artificial bee colony algorithms on 15 test cases. Experimental results show that not only the VSMNS achieves excellent performance in solution quality, but also its performance advantages become more significant as the problem scale increases. Ablation experiments further verify the contribution of each component in the algorithm, demonstrating the effectiveness and superiority of the designed local operators.

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