A logistics distribution multi-objective combination optimization method based on graph filtering and cross attention mechanism

Abstract

The application discloses a logistics distribution multi-objective combination optimization method based on graph filtering and a cross attention mechanism, and comprises the following steps: a neural model GF-MOCA for multi-objective combination optimization is built; instance data of a logistics distribution multi-objective combination optimization problem is acquired, including characteristic vectors of n distribution nodes; after decomposition processing, a plurality of scalarized sub-problems are output; the characteristic vectors of the distribution nodes and weight vectors in the instance data are input into the neural model GF-MOCA for training; the neural model GF-MOCA iteratively predicts the distribution node selection probability distribution at each time in a self-recurrence mode, and samples or selects nodes according to the distribution node selection probability distribution, gradually constructs the visiting sequence of the distribution nodes, and finally forms a solution sequence representing a complete distribution scheme. The application can quickly generate a high-quality Pareto distribution scheme set covering different target preferences by using a single model, and meets the needs of real-time decision-making of logistics distribution.

Description

Technical Field

[0001] This invention belongs to the interdisciplinary field of artificial intelligence and operations research, and relates to a multi-objective combined optimization method for logistics distribution based on graph filtering and cross-attention mechanism. Background Technology

[0002] In actual logistics operations, route planning often requires balancing multiple conflicting optimization objectives, such as total transportation cost versus delivery time, and vehicle mileage versus customer satisfaction. Shortening delivery time usually means increasing vehicle investment or detour costs, while reducing transportation costs may extend the waiting time for some customers. Mathematically, such problems can be reduced to Multi-Objective Combinatorial Optimization Problems (MOCOPs), which involve optimizing multiple conflicting objectives in a discrete decision space. The goal is to find a set of Pareto optimal solutions representing the trade-offs between different objectives, allowing scheduling decision-makers to choose based on actual business preferences. Since MOCOPs are NP-hard problems (i.e., nondeterministic polynomial-time hard problems), the computational complexity of exact algorithms increases exponentially with the size of the delivery nodes. Given the order scale of dozens or even hundreds of delivery points in real-world logistics scenarios, exact solutions are difficult to complete within an acceptable timeframe. Therefore, heuristic algorithms and learning-based approximation methods have become the mainstream solution approaches.

[0003] In recent years, neural combinatorial optimization methods based on deep reinforcement learning (DRL) have shown great potential in solving multi-objective optimization problems (MOCOPs). Mainstream methods typically employ a decomposition-based strategy, breaking down a MOCOP into multiple scalarized subproblems defined by different weight vectors, and then using a neural network to learn a solution strategy. Specifically, each subproblem is represented by a weight vector λ, which defines a linear weighted sum of multiple objective functions, thus transforming the multi-objective problem into a single-objective optimization problem.

[0004] Traditional multi-objective combinatorial optimization methods mainly include metaheuristic algorithms, such as the Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA / D) and the Non-dominated Sorting Genetic Algorithm II (NSGA-II) with an elitist retention strategy. These methods approximate the Pareto front through iterative search and population evolution, and theoretically can obtain good approximate solutions. However, their computational cost is usually very high, making it difficult to meet the needs of real-time decision-making. This is also an important background for the rise of neural combinatorial optimization methods.

[0005] The existing technologies that are closest to this invention mainly include the following two: (1) CNH (Conditional Neural Hypernetwork) method: This method dynamically adjusts model parameters through a conditional hypernetwork, using weight vectors as conditional inputs to generate model parameters, thus enabling a single model to handle multiple sub-problems. Although this method can achieve dynamic adjustment of model parameters, its fusion of weights and node information is a simple conditional generation method, which fails to fully capture the complex dependency relationship between weight vectors and node features. In actual logistics and distribution scenarios, when the scheduling decision-maker gives a certain target preference (such as cost priority or time priority), the model has difficulty in accurately establishing the relationship between this preference and the spatial distribution and demand distribution of each delivery node, resulting in a deviation between the generated delivery plan and the decision-maker's true preference. Under some preference settings, the quality of the plan decreases significantly, making it difficult to provide a set of candidate delivery plans that cover various preferences and have balanced quality.

[0006] (2) WE-CA (Weight Embedding with Cross-Attention) method: This method encodes weight vectors into the problem representation and uses a conditional attention mechanism to fuse weight information with node information. Although this method improves the solution quality to some extent, the direct fusion strategy it uses (such as splicing or addition) often destroys the integrity of the original node features, leading to feature interference, which affects the convergence and solution performance of the model. In addition, the standard multi-head attention mechanism used by this method can only capture first-order structural information and lacks effective ability to capture higher-order structural dependencies and long-distance dependencies between nodes. Specifically in delivery route planning: On the one hand, the key features such as the location and demand of delivery nodes are distorted after being interfered with by weight information, and the model is prone to generating low-quality delivery routes that detour and intersect; on the other hand, the model has difficulty perceiving the cluster structure (such as multiple delivery points in the same park or the same block) and long-distance associations across areas that are common in the delivery network, and cannot make area-level global route planning, resulting in high total mileage and total cost of the delivery plan.

[0007] In summary, existing methods for solving neural MOCOP have several common shortcomings: (1) Insufficient fusion of weight vectors and node features. Existing neural MOCO methods typically employ simple additive aggregation or conditional attention mechanisms to fuse weight vectors and node information. These simple fusion strategies have the following drawbacks: on the one hand, directly concatenating or additively fusing weight vectors and node vectors will destroy the integrity of the original node features, leading to feature interference; on the other hand, this direct fusion method is difficult to effectively capture the complex interdependencies between weights and nodes, resulting in poor joint representation learning quality under different weight preferences, ultimately leading to poor solution performance. Reflected in actual delivery operations, this means that the model cannot consistently generate high-quality delivery solutions under different target preferences: under some preference settings, delivery routes are redundant and the solution costs are significantly higher, the overall quality of the Pareto solution set is poor, and scheduling decision-makers find it difficult to select delivery routes that truly meet business needs.

[0008] (2) Standard attention mechanisms lack the ability to capture high-order structural information. Existing methods generally use the standard multi-head attention mechanism as the core computational unit of the encoder. However, the standard attention mechanism can only capture the first-order interaction between nodes, that is, it can only represent the direct relationship between pairs of nodes. In actual logistics and distribution networks, delivery nodes often exhibit obvious clustered distribution (such as multiple delivery points in the same business district, community or park). High-quality delivery routes usually need to perceive such multi-hop neighbor relationships and global topological features, and carry out overall planning in the form of "first focus on serving one area, and then go to the next area". The standard attention mechanism has limited ability to perceive such high-order structural dependencies and long-distance dependencies. The receptive field of the model is limited to local geometric neighbors, which makes the generated delivery routes prone to back-and-forth between areas and redundancy in overall mileage.

[0009] (3) Increased model complexity leads to oversmoothing. When the number of encoder layers increases or complex fusion modules (such as cross attention) are introduced, node features tend to be consistent after multiple layers of propagation, which is the so-called "oversmoothing" phenomenon. For logistics and distribution problems, oversmoothing means that the model cannot effectively distinguish delivery nodes with different locations and demand, loses the ability to distinguish key topological information, reduces the pertinence of route decisions, and significantly reduces the quality of delivery solutions.

[0010] (4) Existing methods suffer from insufficient model training and inference efficiency. Existing neural MOCO methods typically require training a separate model for each subproblem defined for each weight vector, resulting in training costs and storage overhead increasing linearly with the number of subproblems. In logistics and distribution applications, this means that for every additional combination of target preferences added by the decision-maker, an additional model needs to be trained and stored, leading to high system deployment and maintenance costs and making it difficult to implement in actual scheduling systems. Furthermore, while traditional heuristic algorithms (such as MOEA / D, NSGA-II, etc.) can theoretically achieve a good Pareto front approximation, their iterative search process typically takes hours, which cannot match the real-time decision-making requirements of logistics and distribution tasks, such as same-day order placement, same-day route scheduling, or even minute-level dynamic adjustments. Summary of the Invention

[0011] To address the problems of insufficient fusion of weight preference information and delivery node features, inadequate capture of high-order structural information, and difficulty in meeting real-time scheduling requirements when solving multi-objective optimization problems in logistics and distribution using existing neural multi-objective combinatorial optimization methods, this invention aims to provide a multi-objective combinatorial optimization method for logistics and distribution based on graph filtering and cross-attention mechanisms. This method enables the rapid generation of a high-quality Pareto delivery solution set covering different objective preferences using a single model, thereby meeting the needs of real-time decision-making in logistics and distribution.

[0012] To achieve the above objectives, the present invention provides the following technical solution: A multi-objective combined optimization method for logistics distribution based on graph filtering and cross-attention mechanism includes the following steps: Step 1: Construct the neural model GF-MOCA for multi-objective combinatorial optimization; The neural model GF-MOCA includes a graph-filtered multi-head attention module (GFMA) and a cross-attention-based weight embedding module, both embedded in a unified encoder-decoder architecture. The encoder consists of stacked L identical coding layers, each containing a cross-attention-based weight embedding sublayer, a GFMA sublayer, and a feedforward network sublayer. The decoder also employs the GFMA mechanism, progressively generating the selection probability of each candidate node based on the encoder's output vector sequence and the historical states of currently selected nodes. The graph-filtered multi-head attention module (GFMA) includes an unit term w0I, a first-order term w1Ā, and higher-order approximation terms. The three branches, weighted and combined using learnable parameters, are configured to: take the initial attention weights as input and generate an enhanced attention matrix through a learnable graph filtering transformation; the weight embedding module based on cross-attention is configured to embed the weight vector... As a special node embedding, the weight embedding is dynamically updated by interacting with graph node features through a cross-attention mechanism at each layer of the encoder. Step 2: Obtain instance data of the multi-objective combined optimization problem of logistics distribution. The instance data comes from the actual logistics distribution task to be optimized, including feature vectors of n distribution nodes. After decomposition, multiple scalar sub-problems are output. The decomposition process is as follows: a set of weight vectors is predefined, each weight vector representing a preference trade-off of the logistics scheduling decision-maker for each optimization objective. The multi-objective problem is transformed into multiple single-objective scalar sub-problems represented by the weight vectors using a weighted sum decomposition method. The feature vectors include at least the location coordinates of each distribution node in the metric space corresponding to each of the multiple optimization objectives, the distribution center node information, and the cargo demand of each distribution node. Step 3: Train the neural model GF-MOCA: Combine the feature vectors and weight vectors of each delivery node in the instance data described in Step 2. The input neural model GF-MOCA iteratively predicts the node selection probability distribution at each time step using an autoregressive approach, and samples or selects nodes accordingly, progressively constructing the access order of delivery nodes, ultimately forming a solution sequence representing the complete delivery plan. The solution process for each sub-problem is modeled as a Markov decision process; at each time step, the neural model GF-MOCA determines the next access node based on the current environmental state and outputs the probability distribution on the set of available delivery nodes. The strategy is parameterized by the neural network. Generate the complete solution π = ( using an autoregressive approach) , ..., The probability of generating a solution is: In the formula π represents a problem instance, which is the instance data composed of all the feature vectors of the delivery nodes obtained in step 2; π represents the complete solution sequence generated by the model. This represents the delivery node selected at time step t; Let represent the sequence of nodes selected in the first t-1 time steps; T is the length of the solution sequence; θ is the learnable parameter of the neural network. (·) represents the probability output by the strategy determined by the parameter θ; Step 4: Apply the trained neural model GF-MOCA to actual logistics and distribution tasks: Obtain actual logistics and distribution instance data, including the feature information of distribution centers and each distribution node under each optimization objective metric space, and set a set of weight vectors according to the logistics scheduling decision-maker's preference for each optimization objective; input the instance data and each weight vector into the trained neural model GF-MOCA, and the model outputs a complete distribution node access sequence, i.e., a distribution plan, for each weight vector in an autoregressive manner; all distribution plans together constitute the Pareto optimal solution set of this logistics and distribution instance, from which the logistics scheduling decision-maker can select the final delivery plan to be executed according to actual business needs.

[0013] Furthermore, step 2 includes the following process: Step 2.1: Obtain instance data of a multi-objective combinatorial optimization problem, including feature vectors of n nodes; Step 2.2: Predefine a set of weight vectors {λ1, λ2, ..., λ} n}, where each weight vector λ∈ Satisfies: All components of the weight vector are non-negative, i.e., λ m ≥0, and the sum of all components is 1, i.e., Σλ m =1, M is the target quantity. This represents the m-th component, i.e., the weight assigned to the m-th optimization objective. Each weight vector corresponds to a scalar subproblem. Step 2.3: Use a weighted sum decomposition method to transform the multi-objective problem into multiple single-objective sub-problems; for problems consisting of weight vectors... The subproblem to be represented has a scalarized objective function as follows: in Let π be the m-th objective function, and let π represent a feasible solution to the problem, i.e., a complete sequence of delivery node visits.

[0014] Furthermore, the GF-MOCA neural model training process in step 3 includes: Step 3.1: Input the query vector q, key vector k, and value vector v into the graph filtering-based multi-head attention module GFMA. First, calculate the original attention score and obtain the initial attention weight matrix through Softmax. Then, use the weight matrix output by Softmax as a signal, and after graph filtering transformation, output the optimized attention matrix and the weighted summation result. Step 3.2: Transfer the weight vector Considered as a special node embedding, it is related to the original features of the node. The weight embedding module based on cross-attention is input together; through linear projection, cross-attention mechanism and feedforward network, weight and node information are dynamically fused in each layer of the encoder to synchronously generate updated weight embedding and node embedding. Step 3.3: The updated weight embedding is concatenated with the original node embedding of the same layer to obtain the enhanced embedding set. The enhanced embedding set is first processed by the graph filtering multi-head attention mechanism, residual connection and instance normalization of the GFMA sub-layer to obtain the intermediate node embedding, and then sent to the feedforward network sub-layer. Combined with residual connection and instance normalization, the final node representation of the current layer is output. After L-layer iterative update, the encoder outputs the final weight embedding and node embedding. Step 3.4: The node embedding output by the encoder is used as input and sent to the decoder. The decoder calculates the probability distribution of the currently selectable delivery nodes step by step in an autoregressive manner, and determines the next delivery node to be visited in sequence according to the probability distribution, until all delivery nodes have been visited, thus constructing a complete delivery node visit sequence, which yields the complete delivery route scheme corresponding to the current subproblem.

[0015] Furthermore, step 3.1 includes the following process: Step 3.1.1: Given a query q, a key k, and a value v, calculate the initial attention weight matrix Ā using dot product and softmax operations: Where score is the initial attention score matrix; q and k are the query vector and key vector, respectively. is the dimension of the key vector; mask is the mask matrix, used to set the score of the delivery node that has been visited or does not meet the constraints to negative infinity, so as to ensure the feasibility of the generated delivery plan; This is the initial attention weight matrix after Softmax normalization; Step 3.1.2, will The input signal is fed into the graph filter, which is defined as a matrix polynomial containing two low-order terms and one high-order term: Where I is the identity matrix and K is the order (hyperparameter) of the graph filter. , , These are learnable parameters; Step 3.1.3: For higher-order terms, use a first-order Taylor expansion at a=1 and combine it with the finite difference method to approximate the derivative, resulting in a simplified formula: Step 3.1.4: Use the attention matrix after graph filtering transformation to perform a weighted summation on the value vector v, producing the output of each attention head: .

[0016] Furthermore, step 3.2 includes the following process: Step 3.2.1: Project the weight vector and the original features of each delivery node linearly into the latent space using the trainable parameter matrix to obtain the initial weight embedding and the initial node embedding. in , For a learnable projection matrix, , For the corresponding bias vector, For the initial embedding of the weight vector λ, For the initial embedding of the i-th delivery node, Let be the original features of the i-th node, and n be the number of delivery nodes; Step 3.2.2: In each layer l of the encoder, the embedding matrix of the previous layer... Layer normalization is performed to obtain Generate query Q, key K, and value V through independent linear transformations: In this context, query Q is only embedded by the normalized weights. This generation allows weight information to selectively focus on node features. , , This is the linear transformation matrix that can be learned by this layer; Step 3.2.3: Fuse the output of the cross-attention with the original weight embedding through a residual connection to obtain the intermediate vector. : Step 3.2.4, Intermediate Vector The process is performed through a feedforward sublayer, and the final output of the current layer is obtained through residual connections and layer normalization. .

[0017] Furthermore, step 3.3 includes the following process: (a) GFMA sublayer: The embedding set is processed using the graph filtering-based multi-head attention mechanism described in step 3.1, and the intermediate node embeddings are obtained through residual connections and instance normalization: Where IN(·) represents the instance normalization operation; This represents the enhanced embedding set obtained by concatenating the updated weighted embeddings with the original node embeddings at the same level. Let i be the i-th embedding vector in this set. (·) represents multi-head attention operation based on graph filtering; (b) Feedforward sublayer: The intermediate embeddings are processed through the feedforward network and then through residual connections and instance normalization to obtain the final node representation of the current layer: Where FF(·) represents the operation of the feedforward network sublayer. This represents the final representation of the i-th node output from the l-th layer. Through L-layer iterative updates, the encoder ultimately outputs the weight embedding. and node embedding ..., .

[0018] Furthermore, the decoder process in each decoding step t described in step 3.4 includes: (1) Embedded from context Generate "glimpse" vectors using GFMA layers. ; where context embedding The average value of the final embedding of each delivery node output by the encoder is concatenated with the state embedding of the currently constructed partial solution; (2) Calculate the compatibility score: = -∞, if node i has been visited in The compatibility score for delivery node i; (·) is the hyperbolic tangent function; The projection matrix is ​​learnable; For embedding dimension or scaling factor; The embedding of node i in the final output of the encoder; Then, the Softmax function is applied to the compatibility scores of all nodes to obtain the probability distribution for selecting the next node: Where C is a constant hyperparameter, which is set to 10 by default and is used to control the entropy of the probability distribution; (3) Based on the above probability distribution, the decoder selects the next node and repeats this process until a complete solution is constructed.

[0019] Furthermore, the training of the neural model GF-MOCA employs the REINFORCE algorithm combined with the POMO framework: weight vectors are sampled from a Dirichlet distribution in each training batch; multiple solution trajectories are constructed in parallel from different starting delivery nodes; multi-objective rewards are weighted and scalarized based on the current weight vector, and the mean of the rewards of all parallel solution trajectories on the same instance is used as the baseline to estimate the advantage; the model parameters are updated using the Adam optimizer; the training objective is to maximize the expected reward J(θ) = E[R(π)], where the reward R(π) is defined as the negative scalarized objective function value; the REINFORCE algorithm is used and a baseline is introduced to estimate the gradient. Where B is the batch size. This is the baseline value used to reduce gradient variance.

[0020] Compared with the prior art, the present invention has the following beneficial effects: The cross-attention-based weight embedding module of this invention treats the weight vector as an independent special node embedding, processing it and the delivery node features as two separate sequences: in each layer of the encoder, the query vector is generated solely by the weight embedding, while the delivery node features participate in the calculation as keys and values. Since the weight information is no longer directly concatenated or added to the node features, the integrity of the original features of the delivery node, such as its location and demand, is preserved, avoiding feature interference problems caused by fusion operations in existing methods. Simultaneously, the weight embedding continuously and selectively aggregates the node information most relevant to the current preference during layer-by-layer cross-attention calculations, enabling the model to fully capture the complex interdependencies between weight preferences and delivery nodes. Therefore, the model can learn high-quality joint representations under different target preferences, thus stably generating delivery plans that align with the decision-maker's intentions under various preference settings.

[0021] The graph-based multi-head attention module GFMA of this invention treats the attention matrix as a graph structure in graph signal processing, and performs filtering transformation on it through three learnable branches: unit term, first-order term, and higher-order approximation term. The unit term preserves the node's own information, the first-order term describes the direct relationships between delivery nodes, and the higher-order approximation term describes the multi-hop indirect relationships between nodes through a higher power approximation of the attention matrix. The three branches are adaptively weighted and combined through learnable parameters. Therefore, the model's receptive field is no longer limited to the first-order direct relationships between nodes, but can capture higher-order structural dependencies and long-range interactions in the delivery network (such as cluster relationships between delivery points within a region and relationships between different regions). Furthermore, the preservation of node's own information by the unit term effectively alleviates the oversmoothing phenomenon in deep networks. The above mechanism ensures the preservation and enhancement of key graph structure information during encoding and decoding, enabling the model to make region-level, global route decisions and improve the solution quality of delivery schemes.

[0022] To address the calculation of higher-order approximations, this invention employs a first-order Taylor expansion and finite difference method to approximate higher powers of the attention matrix. This simplifies the higher-order exponentiation operations, which originally required multiple matrix multiplications, into a linear combination that only requires the attention matrix and its square, significantly reducing the overall computational complexity. This allows the graph filtering mechanism to be embedded into existing attention architectures without significantly increasing computational overhead, ensuring the model's scalability and inference speed on large-scale delivery instances. Because the graph filtering mechanism captures the structured relationships between delivery nodes, rather than memories bound to a specific problem size, the model trained by this invention not only performs excellently on problems of the same size as during training but also generalizes to multi-objective combinatorial optimization problems of different sizes and types, adapting to the varied order sizes and problem types in actual logistics operations.

[0023] This invention embeds weight vectors as input conditions into the encoder, enabling a single neural network to handle all sub-problems defined by different weight vectors without requiring separate model training for each preference. This eliminates the problem of training costs and storage overhead increasing linearly with the number of sub-problems. Furthermore, since the weight vectors participate in cross-attention calculations as continuous vectors, the model can generalize to weight vectors not seen during training. Decision-makers can adjust target preferences at any time after deployment according to business needs without retraining the model.

[0024] After the model of this invention is trained, a complete Pareto delivery solution set can be generated by performing only one forward inference for a new logistics delivery instance. The inference time is in the second range, which has an order-of-magnitude speed advantage compared with traditional heuristic algorithms (such as MOEA / D, NSGA-II, etc.) that require several hours of iterative search. It can meet the decision-making needs of real-time routing and dynamic adjustment in logistics delivery scenarios. Attached Figure Description

[0025] Figure 1 This is a diagram of the overall architecture of GF-MOCA of the present invention; Figure 2 The diagram shows the structure of a graph-based multi-head attention model (GFMA). Figure 3 This is a performance comparison chart of the present invention and the comparative method under different problem scales; Figure 4 This is a comparison diagram of the Pareto front obtained by the present invention and the comparative method on the KroAB benchmark example; Figure 5 Comparison of ablation experiments of various GF-MOCA variants on Bi-TSP200 samples; Figure 6 This is a sensitivity analysis plot for the filter order K. Detailed Implementation

[0026] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0027] The multi-objective combined optimization method for logistics distribution based on graph filtering and cross-attention mechanism described in this embodiment includes the following steps: Step 1: Construct a neural model GF-MOCA for multi-objective combinatorial optimization. like Figure 1As shown, the neural model GF-MOCA includes a graph-filtered multi-head attention module (GFMA) and a cross-attention-based weight embedding module, both embedded within a unified encoder-decoder architecture. In the encoder, the cross-attention-based weight embedding module first interacts with the input node features and the preference weight vectors in multi-objective optimization, aggregating node information into the weight space. Subsequently, this fused representation is fed into the graph-filtered multi-head attention module (GFMA), which re-propagates the contextual information, incorporating weight semantics, back to each node vector and dynamically assigns attention weights to different nodes. After this processing, the encoder finally outputs a context-aware vector sequence containing both node structure information and target preference information. The encoder consists of stacked L identical coding layers. Each coding layer sequentially includes a cross-attention-based weight embedding sublayer, a GFMA sublayer, and a feedforward network sublayer, each supplemented with residual connections and normalization. The feedforward network sublayer is composed of multilayer perceptrons used to perform nonlinear transformations on the attention output.

[0028] In the decoder, the GFMA mechanism is also used. Based on the vector sequence output by the encoder and the historical state of the currently selected nodes, the selection probability of each candidate node is gradually generated, thereby completing the autoregressive construction of the multi-objective combinatorial optimization solution.

[0029] like Figure 2 As shown, the multi-head attention module GFMA based on graph filtering includes an unit term w0I, a first-order term w1Ā, and higher-order approximation terms. The three branches are weighted and combined using learnable parameters. The internal structure of the graph-filter-based multi-head attention module (GFMA) includes three steps: standard attention calculation, graph filtering transformation, and weighted summation output. The core idea of ​​this module is to use the initial attention weights calculated via the Softmax function as input, rather than directly using them, to generate a more expressive attention matrix through a learnable graph filtering transformation process.

[0030] The core idea of ​​the weight embedding module based on cross-attention is: to embed the weight vector... Treated as a special node embedding, the weight embedding is dynamically updated at each layer of the encoder using a cross-attention mechanism, so that the weight information can fully interact with all node information.

[0031] Step 2: Obtain instance data for the multi-objective combinatorial optimization problem. The instance data comes from the actual logistics and distribution task to be optimized, including feature vectors of n distribution nodes. After decomposition, multiple scalar sub-problems are output. The decomposition process is as follows: a set of weight vectors is predefined, each weight vector representing a preference trade-off between the logistics scheduling decision-maker and each optimization objective. A weighted sum decomposition method is used to transform the multi-objective problem into multiple single-objective scalar sub-problems represented by the weight vectors. The feature vector includes at least the location coordinates of each delivery node in the metric space corresponding to each of the multiple optimization objectives, the information of the delivery center node, and the cargo demand of each delivery node.

[0032] Step 2.1: Obtain instance data for multi-objective combined optimization problems in logistics and distribution, including feature vectors of n delivery nodes. For example, in the multi-objective traveling salesman problem in logistics and distribution scenarios, the n nodes correspond to n delivery points to be served, and the feature vector of each node contains its coordinate information in multiple objective metric spaces (such as two sets of two-dimensional coordinates corresponding to transportation distance cost and transportation time cost respectively); for multi-objective vehicle routing problems with capacity constraints, the instance data also includes distribution center (warehouse) nodes and the cargo demand of each delivery node. Step 2.2: Predefine a set of weight vectors {λ1, λ2, ..., λ} n}, where each weight vector λ∈ Satisfies: All components of the weight vector are non-negative, i.e., λ m ≥0, and the sum of all components is 1, i.e., Σλ m =1, M is the target quantity. This represents the m-th component, i.e., the weight assigned to the m-th optimization objective. Each weight vector corresponds to a scalar subproblem. Step 2.3: Using a weighted sum decomposition method, the multi-objective problem is transformed into multiple single-objective sub-problems. For problems consisting of weight vectors... The subproblem to be represented has a scalarized objective function as follows: in Let π be the m-th objective function, and let π represent a feasible solution to the problem, i.e., a complete sequence of delivery node visits.

[0033] Step 3: Combine the feature vectors and weight vectors of each delivery node obtained in Step 2. The input neural model GF-MOCA iteratively predicts the node selection probability distribution at each time step using an autoregressive approach, and samples or selects nodes accordingly to gradually construct the access order of delivery nodes, ultimately forming a solution sequence representing the complete delivery solution. The solution process for each sub-problem is modeled as a Markov Decision Process (MDP). At each time step, the neural model GF-MOCA determines the next access node based on the current environmental state (including the delivery node feature vectors described in step 2.1 and the constructed partial solutions) and outputs a probability distribution over the set of available delivery nodes. The strategy is parameterized by the neural network. Generate the complete solution π = ( using an autoregressive approach) , ..., The probability of generating a solution is: In the formula π represents a problem instance, which is the instance data consisting of all the feature vectors of the delivery nodes obtained in step 2.1; π represents the complete solution sequence generated by the model. This represents the delivery node selected at time step t; Let represent the sequence of nodes selected in the first t-1 time steps; T is the length of the solution sequence; θ is the learnable parameter of the neural network. (·) represents the probability output by the strategy determined by the parameter θ.

[0034] The training of the neural model GF-MOCA employs the REINFORCE algorithm combined with the POMO framework: weight vectors are sampled from the Dirichlet distribution in each training batch; multiple solution trajectories are constructed in parallel from different starting delivery nodes; multi-objective rewards are weighted and scalarized according to the current weight vector, and the mean of the rewards of all parallel solution trajectories on the same instance is used as the baseline to estimate the advantage; the Adam optimizer is used to update the model parameters.

[0035] The training objective is to maximize the expected reward J(θ) = E[R(π)], where the reward R(π) is defined as the negative scalarized objective function value. The REINFORCE algorithm is used, and a baseline is introduced to estimate the gradient: Where B is the batch size. This is the baseline value used to reduce gradient variance.

[0036] The specific training process of the neural model GF-MOCA includes: Step 3.1: Input the query vector q, key vector k, and value vector v into the graph-filter-based multi-head attention module GFMA. First, calculate the original attention score and obtain the initial attention weight matrix through Softmax. Then, use the weight matrix output by Softmax as the signal, perform graph filtering transformation, and output the optimized attention matrix and the weighted summation result. Step 3.1.1: Calculate the initial attention weight matrix. Given a query q, a key k, and a value v, calculate the initial attention weight matrix K using a dot product and softmax operation: Where score is the initial attention score matrix; q and k are the query vector and key vector, respectively. is the dimension of the key vector; mask is the mask matrix, used to set the score of the delivery node that has been visited or does not meet the constraints to negative infinity, so as to ensure the feasibility of the generated delivery plan; This is the initial attention weight matrix after Softmax normalization; Step 3.12: Apply graph filtering transformation. Input Ā as the input signal into the graph filter. The graph filter is defined as a matrix polynomial containing two low-order terms and one high-order term: Where I is the identity matrix and K is the order (hyperparameter) of the graph filter. , , These are learnable parameters that can adaptively adjust the filtering characteristics based on downstream tasks.

[0037] Step 3.13, Approximate calculation of higher-order terms. Since direct calculation... The cost is high. A simplified formula is obtained by using a first-order Taylor expansion at a=1 combined with the finite difference method to approximate the derivative: This approximation only requires calculating Ā and Ā², avoiding higher-order matrix exponentiation operations and significantly reducing computational complexity.

[0038] Step 3.1.4, Weighted Summation Output. The attention matrix after graph filtering transformation is used to perform a weighted summation on the value vector v, producing the output of each attention head: Step 3.2: Transfer the weight vector Considered as a special node embedding, it is related to the original features of the node. The input is a weight embedding module based on cross-attention; through linear projection, cross-attention mechanism and feedforward network, weight and node information are dynamically fused in each layer of the encoder, and updated weight embedding and node embedding are generated synchronously. Step 3.2.1: Initialize the embedding. Using a trainable parameter matrix, linearly project the weight vector and the original features of each delivery node into the latent space to obtain the initial weight embedding and the initial node embedding: in , For a learnable projection matrix, , For the corresponding bias vector, For the initial embedding of the weight vector λ, For the initial embedding of the i-th delivery node, Let be the original features of the i-th node, and n be the number of delivery nodes.

[0039] Step 3.2.2, Cross-attention calculation. In each layer l of the encoder, the embedding matrix of the previous layer is calculated... Layer normalization is performed to obtain Then, query Q, key K, and value V are generated through independent linear transformations: In this context, query Q is only embedded by the normalized weights. This generation allows weight information to selectively focus on node features. , , This is the linear transformation matrix that can be learned by this layer.

[0040] Step 3.2.3: Attention Output and Residual Connection. The output of the cross-attention and the original weight embedding are fused through a residual connection to obtain an intermediate vector. : Step 3.2.4: Feedforward network processing. Intermediate vector. The processing is performed through a feedforward network sublayer (containing a multilayer perceptron MLP), and the final output of the current layer is obtained through residual connections and layer normalization: Step 3.3: The updated weight embeddings are concatenated with the original node embeddings of the same layer to obtain an enhanced embedding set. This set is first processed by the graph filtering multi-head attention mechanism, residual connections, and instance normalization of the GFMA sublayer to obtain intermediate node embeddings, which are then fed into the feedforward network sublayer. Combined with residual connections and instance normalization, the final node representation of the current layer is output. After L layers of iterative updates, the encoder outputs the final weight embeddings and node embeddings. (a) GFMA sublayer: The embedding set is processed using the graph filtering-based multi-head attention mechanism described in step 3.1, and the intermediate node embeddings are obtained through residual connections and instance normalization: Where IN(·) represents the instance normalization operation; This represents the enhanced embedding set obtained by concatenating the updated weighted embeddings with the original node embeddings at the same level. Let i be the i-th embedding vector in this set. (·) represents multi-head attention operation based on graph filtering; (b) Feedforward sublayer: The intermediate embeddings are processed through the feedforward network and then through residual connections and instance normalization to obtain the final node representation of the current layer: Where FF(·) represents the operation of the feedforward network sublayer. This is the final representation of the i-th node output from the l-th layer.

[0041] Through L-layer iterative updates, the encoder ultimately outputs the weight embedding. and node embedding ..., .

[0042] Step 3.4: The node embedding output by the encoder is used as input and fed into the decoder. The decoder calculates the probability distribution of the currently available delivery nodes step by step in an autoregressive manner, and determines the next delivery node to be visited according to the probability distribution, until all delivery nodes have been visited, thus constructing a complete delivery node visit sequence, which yields the complete delivery route solution corresponding to the current subproblem. In each decoding step t: (1) Embedded from context Generate "glimpse" vectors using GFMA layers. The context embedding hc is composed of the global graph embedding output by the encoder (i.e., the mean of the final embeddings of each delivery node) and the state embeddings of the currently constructed partial solutions (such as the embeddings of the first visited node and the current node). (2) Calculate the compatibility score: = -∞ (if node i has been visited) in The compatibility score for delivery node i; (·) is the hyperbolic tangent function; The projection matrix is ​​learnable; For the embedding dimension (scaling factor); The embedding of node i in the final output of the encoder; This is the "glimpse" vector.

[0043] Then, the Softmax function is applied to the compatibility scores of all nodes to obtain the probability distribution for selecting the next node: Where C is a constant hyperparameter (default value is 10), used to control the entropy of the probability distribution.

[0044] (3) Based on the above probability distribution, the decoder selects the next node and repeats this process until a complete solution is constructed.

[0045] Training strategy Training was performed using the REINFORCE algorithm combined with the POMO (Policy Optimization with Multiple Optima) framework. (1) Weight sampling: In each training batch, the weight vector is sampled from the Dirichlet distribution.

[0046] (2) POMO expansion: Multiple agents start from different starting nodes and generate solutions in parallel.

[0047] (3) Multi-objective reward scalarization: Based on the current weight vector, the multi-objective reward is weighted and or Chebyshev decomposed to obtain the scalarized reward.

[0048] (4) Advantage estimation: Use the POMO-style group baseline, i.e., the mean of all agent rewards as the baseline.

[0049] (5) Data augmentation: 64x augmentation (8 symmetric transformations such as flipping / rotation are applied independently to the coordinate pairs of each target to obtain 8×8=64x augmented samples) to improve the generalization ability of the model.

[0050] (6) Parameter update: The Adam optimizer and multi-step learning rate scheduler are used to update the parameters.

[0051] Step 4: Apply the trained neural model GF-MOCA to actual logistics and distribution tasks: Obtain actual logistics and distribution instance data, including the feature information (such as coordinates, demand, etc.) of the distribution center and each distribution node in each optimization objective metric space, and set a set of weight vectors according to the logistics scheduling decision-maker's preference for each optimization objective; input the instance data and each weight vector into the trained neural model GF-MOCA, and the model outputs a complete distribution node access sequence, i.e., a distribution plan, for each weight vector in an autoregressive manner; all distribution plans together constitute the Pareto optimal solution set of this logistics and distribution instance, and the logistics scheduling decision-maker can select the final execution distribution plan from them according to actual business needs (such as cost priority, time priority, or a compromise between the two).

[0052] Experimental section: (1) Solving the Bi-Objective Traveling Salesman Problem (BI-TSP) All experiments were conducted on a workstation equipped with an NVIDIA RTX 3090 GPU and implemented using the PyTorch framework. For a bi-objective traveling salesman problem with 20 nodes, each node contains 4 coordinate values ​​(corresponding to two-dimensional coordinates in the two object spaces). The model parameters were set as follows: embedding dimension of 128, encoder layers of 6, attention heads of 8, and graph filter order K=4. A weighted summation method was used, sampling weight vectors from the Dirichlet distribution Dir(1,1). Training was performed using the Adam optimizer with a learning rate of 1e-4 for 200 epochs. After training, the model could generate an approximate solution set covering the Pareto front within 7 seconds, achieving a hypervolume metric (HV) of 0.6272, which is superior to the 0.6270 of the WE-CA method and the CNH method.

[0053] (2) Solving the Bi-Objective Capacity Vehicle Routing Problem (BI-CVRP) For a dual-objective capacity vehicle routing problem involving 20 customer nodes and 1 warehouse node, where each node contains coordinates and demand information, the model adds warehouse node embedding and remaining capacity tracking dimensions to the model described in Example 1. The model is trained with 128-dimensional embedding and a 6-layer encoder. On the Bi-CVRP20 benchmark, the proposed method achieves a hypervolume metric of 0.4295, outperforming the WE-CA method's 0.4290 and the CNH method's 0.4287. It also demonstrates superior performance compared to the baseline methods on Bi-CVRP50 and Bi-CVRP100.

[0054] (3) Solving the Triple-Objective Traveling Salesman Problem (TRI-TSP) For a Traveling Salesman Problem (TSP) with three conflicting objectives, each node contains six coordinates (two-dimensional for each of the three objective spaces), and the weight vector is three-dimensional, sampled from the Dir(1,1,1) distribution. On the Tri-TSP20 benchmark, our method achieves a hypervolume index of 0.4712, matching the best traditional method WS-LKH and significantly outperforming neural baseline methods such as DRL-MOA (0.4699) and PMOCO (0.4693). Significant advantages are also demonstrated on Tri-TSP50 and Tri-TSP100.

[0055] (4) Comparison and analysis of experimental results To fully verify the performance of the algorithm, nine representative test scenarios covering different problem types and node sizes were selected for comparative experiments. The hypervolume index and computation time of each algorithm were recorded. The experimental results are shown in Table 1. Note: The larger the HV, the better the solution set quality.

[0056] Table 1. Hypervolume metrics and computation time for different methods in typical benchmark scenarios. As shown in Table 1, in the nine representative scenarios selected, the HV of GF-MOCA is higher than that of CNH and WE-CA; this advantage remains stable as the node size increases from 20 to 100. Furthermore, the computation time shows that GF-MOCA can complete the solution in just a few seconds to tens of seconds, far superior to the hour-long computation time required by MOEA / D and NSGA-II. These results demonstrate that graph filtering and cross-attention mechanisms can improve the convergence and diversity of the solution set, maintain good solution quality under different problem types and node sizes, and possess significant computational efficiency advantages.

[0057] Regarding computational efficiency, Table 1 shows that on the Bi-TSP20 instance, GF-MOCA, MOEA / D, and NSGA-II took 7 seconds, 1.7 hours, and 6.0 hours, respectively; after conversion, our method is approximately 874 times and 3086 times faster, respectively. Other scenarios also show differences in the order of seconds and hours, indicating that our method significantly reduces computational overhead while maintaining a high HV (High Performance Value), making it more suitable for rapid decision-making in logistics and distribution.

[0058] Cross-scale generalization analysis: Models uniformly trained on mixed-scale instances of Bi-TSP from n=20 to 100 were tested on instances of n=20, 50, 100, 150, and 200, as shown. Figure 3 As shown, the radar graph contours of GF-MOCA under all five scale conditions are outside of WE-CA, PMOCO, and CNH; the algorithm still maintains its performance advantage on n=150 and n=200 instances, which are beyond the training scale range. This demonstrates that the weight-node relationships and high-order graph structures learned by the model can be transferred to larger-scale logistics and distribution problems, exhibiting good cross-scale generalization performance.

[0059] Pareto Frontier Analysis: such as Figure 4 As shown, for the KroAB100, KroAB150, and KroAB200 instances, the solution set generated by GF-MOCA provides more comprehensive coverage in the locally magnified region; its HV values ​​are 0.6955, 0.6915, and 0.7306, respectively, all superior to WE-CA's 0.6952, 0.6913, and 0.7304, and CNH's 0.6947, 0.6892, and 0.7250. When the problem size increases, GF-MOCA still maintains the highest HV value, indicating that graph filtering is beneficial for improving solution set coverage, and the algorithm has good scale adaptability.

[0060] Ablation experiment analysis: such as Figure 5As shown, in the Bi-TSP200 instance test, the HV values ​​of OnlyGFMA, OnlyCross, GFMA+CA, WE-CA, and GF-MOCA are 0.7253, 0.7266, 0.7262, 0.7264, and 0.7269, respectively. OnlyCross outperforms WE-CA, indicating that cross-attention can reduce the interference caused by the direct fusion of weight information and node features. The relatively low index of OnlyGFMA indicates that the GFMA module needs to rely on an effective weight-node interaction mechanism. The complete model achieves the best HV result, proving that cross-attention and GFMA have a synergistic optimization effect in preference modeling and high-order graph structure perception.

[0061] Sensitivity analysis of graph filter order: Figure 6 This paper demonstrates the HV variation of the graph filter order (hyperparameter) K in the model of this invention within the range of 3 to 10 for four types of problems. When K=4, the HV of Bi-TSP20, Tri-TSP20, Bi-CVRP20, and Bi-KP20 are 0.6272, 0.4707, 0.4295, and 0.3561, respectively. Further increasing K leads to a general stabilization of the indicators for each problem, with a slight decrease in HV under certain settings. Higher-order graph filters increase computational overhead; therefore, K=4 achieves a better trade-off between solution quality and computational complexity.

Claims

1. A multi-objective combined optimization method for logistics distribution based on graph filtering and cross-attention mechanism, characterized in that, Includes the following steps: Step 1: Construct the neural model GF-MOCA for multi-objective combinatorial optimization; The neural model GF-MOCA includes a graph-filter-based multi-head attention module GFMA and a cross-attention-based weight embedding module, which are embedded together in a unified encoder-decoder architecture. The encoder consists of stacked L layers with the same coding structure. Each coding layer contains a weighted embedding sublayer based on cross-attention, a GFMA sublayer, and a feedforward network sublayer. In the decoder, the GFMA mechanism is also used to generate the selection probability of each candidate node step by step based on the vector sequence output by the encoder and the historical state of the currently selected node. The graph-filter-based multi-head attention module (GFMA) includes an identity term w0I, a first-order term w1Ā, and higher-order approximation terms. The three branches, weighted and combined using learnable parameters, are configured to take the initial attention weights as input and generate an enhanced attention matrix through a learnable graph filtering transformation. The cross-attention-based weight embedding module is configured to embed the weight vector... As a special node embedding, the weight embedding is dynamically updated by interacting with graph node features through a cross-attention mechanism at each layer of the encoder. Step 2: Obtain instance data of the multi-objective combined optimization problem of logistics distribution. The instance data comes from the actual logistics distribution task to be optimized, including the feature vectors of n distribution nodes. After decomposition processing, multiple scalar sub-problems are output. The decomposition processing is as follows: a set of weight vectors are predefined, each weight vector representing the logistics scheduling decision-maker's preference trade-off for each optimization objective. The multi-objective problem is transformed into multiple single-objective scalar sub-problems represented by weight vectors using a weighted sum decomposition method. The feature vector includes at least the location coordinates of each delivery node in the metric space corresponding to each of the multiple optimization objectives, the information of the delivery center node, and the cargo demand of each delivery node; Step 3: Train the neural model GF-MOCA: Combine the feature vectors and weight vectors of each delivery node in the instance data described in Step 2. The input neural model GF-MOCA iteratively predicts the node selection probability distribution at each time step in an autoregressive manner, and samples or selects nodes accordingly to gradually construct the access order of delivery nodes, ultimately forming a solution sequence representing the complete delivery plan. The solution process for each sub-problem is modeled as a Markov decision process. At each time step, the neural model GF-MOCA determines the next access node based on the current environmental state and outputs a probability distribution across the set of available delivery nodes. The strategy is parameterized by the neural network. Generate the complete solution π = ( using an autoregressive approach) , ..., The probability of generating a solution is: In the formula π represents a problem instance, which is the instance data composed of all the feature vectors of the delivery nodes obtained in step 2; π represents the complete solution sequence generated by the model. This represents the delivery node selected at time step t; Let represent the sequence of nodes selected in the first t-1 time steps; T is the length of the solution sequence; θ is the learnable parameter of the neural network. (·) represents the probability output by the strategy determined by the parameter θ; Step 4: Apply the trained neural model GF-MOCA to actual logistics and distribution tasks: Obtain actual logistics and distribution instance data, including the feature information of distribution centers and each distribution node under each optimization objective metric space, and set a set of weight vectors according to the logistics scheduling decision-maker's preference for each optimization objective; Input the instance data and each weight vector into the trained neural model GF-MOCA respectively, and the model outputs a complete distribution node access sequence, i.e. a distribution plan, for each weight vector in an autoregressive manner; The set of Pareto optimal solutions for this logistics delivery instance is composed of all delivery options, from which logistics scheduling decision-makers can select the final delivery solution to be executed based on actual business needs.

2. The multi-objective combined optimization method for logistics distribution based on graph filtering and cross-attention mechanism according to claim 1, characterized in that, Step 2 includes the following process: Step 2.1: Obtain instance data of a multi-objective combinatorial optimization problem, including feature vectors of n nodes; Step 2.2: Predefine a set of weight vectors {λ1, λ2, ..., λ} n }, where each weight vector λ∈ Satisfies: All components of the weight vector are non-negative, i.e., λ m ≥0, and the sum of all components is 1, i.e., Σλ m =1, M is the target quantity. This represents the m-th component, i.e., the weight assigned to the m-th optimization objective. Each weight vector corresponds to a scalar subproblem. Step 2.3: Use a weighted sum decomposition method to transform the multi-objective problem into multiple single-objective sub-problems; for problems consisting of weight vectors... The subproblem to be represented has a scalarized objective function as follows: in Let π be the m-th objective function, and let π represent a feasible solution to the problem, i.e., a complete sequence of delivery node visits.

3. The multi-objective combined optimization method for logistics distribution based on graph filtering and cross-attention mechanism according to claim 2, characterized in that, The process of training the neural model GF-MOCA in step 3 includes: Step 3.1: Input the query vector q, key vector k, and value vector v into the graph filtering-based multi-head attention module GFMA. First, calculate the original attention score and obtain the initial attention weight matrix through Softmax. Then, use the weight matrix output by Softmax as a signal, and after graph filtering transformation, output the optimized attention matrix and the weighted summation result. Step 3.2: Transfer the weight vector Considered as a special node embedding, it is related to the original features of the node. The weight embedding module based on cross-attention is input together; through linear projection, cross-attention mechanism and feedforward network, weight and node information are dynamically fused in each layer of the encoder to synchronously generate updated weight embedding and node embedding. Step 3.3: The updated weight embedding is concatenated with the original node embedding of the same layer to obtain the enhanced embedding set. The enhanced embedding set is first processed by the graph filtering multi-head attention mechanism, residual connection and instance normalization of the GFMA sub-layer to obtain the intermediate node embedding, and then sent to the feedforward network sub-layer. Combined with residual connection and instance normalization, the final node representation of the current layer is output. After L-layer iterative update, the encoder outputs the final weight embedding and node embedding. Step 3.4: The node embedding output by the encoder is used as input and sent to the decoder. The decoder calculates the probability distribution of the currently selectable delivery nodes step by step in an autoregressive manner, and determines the next delivery node to be visited in sequence according to the probability distribution, until all delivery nodes have been visited, thus constructing a complete delivery node visit sequence, which yields the complete delivery route scheme corresponding to the current subproblem.

4. The multi-objective combined optimization method for logistics distribution based on graph filtering and cross-attention mechanism according to claim 3, characterized in that, The process of step 3.1 includes: Step 3.1.1: Given a query q, a key k, and a value v, calculate the initial attention weight matrix Ā using dot product and softmax operations: Where score is the initial attention score matrix; q and k are the query vector and key vector, respectively. is the dimension of the key vector; mask is the mask matrix, used to set the score of the delivery node that has been visited or does not meet the constraints to negative infinity, so as to ensure the feasibility of the generated delivery plan; This is the initial attention weight matrix after Softmax normalization; Step 3.1.2, The input signal is fed into the graph filter, which is defined as a matrix polynomial containing two low-order terms and one high-order term: Where I is the identity matrix and K is the order (hyperparameter) of the graph filter. , , These are learnable parameters; Step 3.1.3: For higher-order terms, use a first-order Taylor expansion at a=1 and combine it with the finite difference method to approximate the derivative, resulting in a simplified formula: Step 3.1.4: Use the attention matrix after graph filtering transformation to perform a weighted summation on the value vector v, producing the output of each attention head: 。 5. The multi-objective combined optimization method for logistics distribution based on graph filtering and cross-attention mechanism according to claim 4, characterized in that, The process in step 3.2 includes: Step 3.2.1: Project the weight vector and the original features of each delivery node linearly into the latent space using the trainable parameter matrix to obtain the initial weight embedding and the initial node embedding. in , The projection matrix is ​​learnable. , For the corresponding bias vector, For the initial embedding of the weight vector λ, For the initial embedding of the i-th delivery node, Let be the original features of the i-th node, and n be the number of delivery nodes; Step 3.2.2: In each layer l of the encoder, the embedding matrix of the previous layer... Layer normalization is performed to obtain Generate query Q, key K, and value V through independent linear transformations: In this context, query Q is only embedded by the normalized weights. This generation allows weight information to selectively focus on node features. , , This is the linear transformation matrix that can be learned by this layer; Step 3.2.3: Fuse the output of the cross-attention with the original weight embedding through a residual connection to obtain the intermediate vector. : Step 3.2.4, Intermediate Vector The process is performed through a feedforward sublayer, and the final output of the current layer is obtained through residual connections and layer normalization. 。 6. The multi-objective combined optimization method for logistics distribution based on graph filtering and cross-attention mechanism according to claim 5, characterized in that, The process in step 3.3 includes: (a) GFMA sublayer: The embedding set is processed using the graph filtering-based multi-head attention mechanism described in step 3.1, and the intermediate node embeddings are obtained through residual connections and instance normalization: Where IN(·) represents the instance normalization operation; This represents the enhanced embedding set obtained by concatenating the updated weighted embeddings with the original node embeddings at the same level. Let i be the i-th embedding vector in this set. (·) represents multi-head attention operation based on graph filtering; (b) Feedforward sublayer: The intermediate embeddings are processed through the feedforward network and then through residual connections and instance normalization to obtain the final node representation of the current layer: Where FF(·) represents the operation of the feedforward network sublayer. This represents the final representation of the i-th node output from the l-th layer. Through L-layer iterative updates, the encoder ultimately outputs the weight embedding. and node embedding ..., .

7. The multi-objective combined optimization method for logistics distribution based on graph filtering and cross-attention mechanism according to claim 6, characterized in that, Step 3.4 describes the decoder's process in each decoding step t, which includes: (1) Embedded from context Generate "glimpse" vectors using GFMA layers. ; where context embedding The average value of the final embedding of each delivery node output by the encoder is concatenated with the state embedding of the currently constructed partial solution; (2) Calculate the compatibility score: = -∞, if node i has been visited in The compatibility score for delivery node i; (·) is the hyperbolic tangent function; The projection matrix is ​​learnable; For embedding dimension or scaling factor; The embedding of node i in the final output of the encoder; Then, the Softmax function is applied to the compatibility scores of all nodes to obtain the probability distribution for selecting the next node: Where C is a constant hyperparameter, which is set to 10 by default and is used to control the entropy of the probability distribution; (3) Based on the above probability distribution, the decoder selects the next node and repeats this process until a complete solution is constructed.

8. The multi-objective combined optimization method for logistics distribution based on graph filtering and cross-attention mechanism according to claim 7, characterized in that, The training of the neural model GF-MOCA adopts the REINFORCE algorithm combined with the POMO framework: in each training batch, weight vectors are sampled from the Dirichlet distribution; multiple solution trajectories are constructed in parallel from different starting delivery nodes; the multi-objective rewards are weighted and scalarized according to the current weight vector, and the mean of the rewards of all parallel solution trajectories on the same instance is used as the baseline to estimate the advantage. Update the model parameters using the Adam optimizer; The training objective is to maximize the expected reward J(θ) = E[R(π)], where the reward R(π) is defined as the negative scalarized objective function value; the REINFORCE algorithm is used and a baseline is introduced to estimate the gradient. Where B is the batch size. This is the baseline value used to reduce gradient variance.

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