A multi-objective unbalanced transportation optimization method based on hybrid evolutionary algorithm

Abstract

This invention discloses a multi-objective imbalanced transportation optimization method based on a hybrid evolutionary algorithm, belonging to the fields of computational intelligence, logistics engineering, and supply chain management optimization technology. Addressing the transportation optimization problem under conditions of unequal total supply and total demand, this invention constructs a mathematical model with dual objectives: minimizing total transportation cost and minimizing bottleneck supply-demand deviation. A hybrid Pareto evolutionary algorithm, HPEA-SH, combining Gaussian probability learning and the transportation simplex heuristic, is designed. This algorithm ensures solution feasibility through a scenario-dependent normalization repair mechanism, utilizes the statistical characteristics of the first non-dominated frontier elite individuals to construct an adaptive Gaussian probability model for global search, and embeds three representative solutions into the transportation simplex method for local refinement. This invention achieves an effective trade-off between transportation cost and system stability, obtaining a non-dominated solution set with good distribution and convergence, providing multiple optimization decision-making schemes for logistics scheduling and supply chain management.

Description

Technical Field

[0001] This invention relates to the fields of computational intelligence, logistics engineering, and supply chain management optimization, specifically to a multi-objective unbalanced transportation optimization method based on a hybrid evolutionary algorithm. Background Technology

[0002] With the continuous development of manufacturing, commerce and trade, and modern logistics, the role of transportation systems in resource allocation, supply guarantee, and cost control is becoming increasingly prominent. Especially in the process of supply chain collaboration and logistics network operation, the rational formulation of transportation plans directly affects the operational efficiency, service level, and cost-effectiveness of enterprises. Optimizing transportation plans can improve the efficiency of resource allocation in the logistics process, reduce transportation costs, enhance adaptability to demand fluctuations and resource constraints, and help achieve rational scheduling of materials and stable operation of the supply chain. Therefore, transportation optimization has always been an important research topic in the fields of logistics engineering and operations research.

[0003] In the actual operation of logistics and manufacturing enterprises, cost control is always a core issue in transportation management. Enterprises typically aim to reduce transportation costs as much as possible while meeting delivery requirements, thereby improving overall profitability and resource utilization efficiency. However, with the increasing complexity of logistics systems, enterprises are paying more attention to the stability, reliability, and supply-demand balance of the transportation network, in addition to explicit transportation costs. Large backlogs at supply points or severe shortages at demand points not only reduce logistics efficiency but may also trigger additional scheduling costs, delays, and even supply chain disruptions. Therefore, in the process of optimizing transportation plans, in addition to considering total transportation costs, it is also necessary to take into account the robustness of the transportation system and the level of supply-demand balance.

[0004] In real-world transportation scenarios, minimizing transportation costs and minimizing supply-demand deviation are often two conflicting optimization objectives. This is because, in pursuit of lower transportation costs, decision-makers tend to prioritize a small number of low-cost, high-efficiency transportation routes, leading to over-concentration of resources and exposing some nodes to significant supply-demand deviation risks. Conversely, to reduce node deviation and improve system robustness, a more balanced transportation allocation scheme is often required, which may incur additional transportation costs. Therefore, for practical transportation planning, achieving a reasonable trade-off between transportation costs and system stability has significant theoretical and engineering application value.

[0005] Existing imbalanced transportation optimization problems typically involve optimizing transportation allocation schemes under the condition that total supply and total demand are unequal. These problems not only include supply constraints, demand constraints, and non-negativity constraints, but also involve the balance relationships between multiple objectives, resulting in high solution complexity. Generally, there is no single solution that simultaneously optimizes all objectives; instead, a set of non-dominated solutions (Pareto solutions) is needed, allowing decision-makers to choose based on different preferences. For these problems, multi-objective evolutionary algorithms are considered a suitable solution method due to their ability to perform parallel search and maintain population diversity. In particular, typical multi-objective evolutionary algorithms such as NSGA-II, relying on non-dominated sorting and crowding distance mechanisms, have been widely used in multi-objective optimization.

[0006] However, traditional multi-objective evolutionary algorithms still have shortcomings in handling imbalanced transport problems. On the one hand, imbalanced transport problems have strong structural constraints; directly using conventional encoding and random search strategies can easily generate a large number of infeasible solutions, thus reducing search efficiency. On the other hand, although traditional evolutionary algorithms have good global search capabilities, they often suffer from insufficient local mining capabilities and limited convergence accuracy in the later stages of optimization, making it difficult to fully utilize the precise mathematical structure inherent in the transport problem itself. Furthermore, relying solely on conventional crossover and mutation operations for searching makes it difficult to effectively utilize the distribution information implicit in historical high-quality individuals, limiting the ability to guide potentially high-quality regions.

[0007] Therefore, considering the supply-demand imbalance constraints, multi-objective trade-offs, and local structural optimization requirements of multi-objective unbalanced transportation problems, there is an urgent need to design a hybrid optimization method that can simultaneously consider global search capabilities and local fine-grained optimization capabilities. In particular, how to combine probabilistic learning mechanisms with the concept of exact mathematical programming in transportation problems, while maintaining the diversity of solution sets and improving the convergence and feasibility of solutions, has become a key technical issue for improving the intelligence level and optimization quality of transportation scheduling. Summary of the Invention

[0008] To overcome the above-mentioned technical problems, this invention provides a multi-objective imbalanced transport optimization method based on a hybrid evolutionary algorithm.

[0009] To achieve the above technical solution, the present invention includes: S1. Construct a mathematical model for the multi-objective unbalanced transport problem; The multi-objective unbalanced transportation problem has two optimization objectives: minimizing the total transportation cost and minimizing the bottleneck supply-demand deviation. S2. Design a hybrid Pareto evolutionary algorithm based on simplex heuristics to solve the multi-objective unbalanced transport problem; The solution methods include: The imbalance between total supply and total demand is identified using a ratio matrix, and a normalization operation is performed; the decoded transportation plan is then repaired. Gaussian learning is used to generate offspring, and representative solutions are selected by combining the generated offspring with the current parent population through non-dominated sorting and crowding distance; the representative solutions are then locally refined using transport simplex local search. The updated individuals accepted after the current parent population, evolutionary offspring, Gaussian learning offspring, and transport simplex local search are selected by non-dominated sorting and crowding distance.

[0010] Furthermore, the mathematical model expression for the multi-objective unbalanced transport problem is as follows: In the formula, The objective function is to minimize the total transportation cost; For supply point aggregation, Indicates the index of the supply point; For the set of demand points, Index for demand points; Indicates from the supply point To the point of demand The unit transportation cost; For supply points To the point of demand Actual transport volume; Indicates supply point Maximum supply capacity; Indicate demand points The maximum demand.

[0011] Furthermore, the mathematical model for the aforementioned multi-objective unbalanced transport problem is also subject to constraints, including: Case S1: The quantity received at each demand point does not exceed its maximum demand. , ; Case S2: The total shipments from each supply point shall not exceed its maximum supply capacity. , ; All shipment volumes must be non-negative. .

[0012] Furthermore, step S2 includes: S2.1 Set the parameters required for HPEA-SH, including population size. Maximum number of iterations Learning rate Gaussian sampling scaling factor and the maximum number of iterations for transport simplex local refinement for each representative solution. ; S2.2. The initial population is randomly generated using a real-number encoding method based on a proportional matrix. S2.3, Normalization Repair and Decoding: A scenario-dependent normalization method maps candidate solutions to feasible transportation schemes; S2.4 Perform fast non-dominated sorting and crowding distance calculation on the current population to obtain the sorted population; S2.5 Extract elite individuals from the current first non-dominated frontier, construct a diagonal Gaussian probability model, and sample to generate new candidate solutions; S2.6 Perform crossover and mutation operations on the individuals encoded by the proportional matrix to generate new candidate individuals; S2.7 Merge the current parent population, Gaussian-learned offspring, and evolved offspring, and filter them using non-dominated sorting and crowding distance; S2.8 Select three representative solutions: the individual with the lowest total transportation cost, the individual with the lowest bottleneck supply-demand deviation, and the individual in the middle position selected from the first non-dominated frontier individuals other than the first and second representative solutions, sorted by congestion distance. S2.9. Perform heuristic local search refinement using the transport simplex method on the three representative solutions; S2.10. Select the current parent population, evolutionary offspring, Gaussian learning offspring, and updated individuals accepted after the transport simplex local search by non-dominated sorting and crowding distance. Select individuals with smaller non-dominated levels from the current population based on non-dominated level and crowding, and select individuals with larger crowding distances under the same non-dominated level. S2.11, Repeat S2.5~S2.10 until the maximum number of iterations is reached. If the preset termination condition is met, the algorithm terminates and outputs the final high-quality non-dominated solution set. All first non-dominated frontier individuals are extracted from the final population. Each individual includes: total transportation cost, bottleneck supply and demand deviation, and actual transportation decision variables.

[0013] Furthermore, the method of S2.2 specifically involves: employing a proportional matrix-based approach. To represent a transportation allocation scheme, where each element Indicates supply point Assigned to demand points The material proportion factor, and Randomly generate a proportional matrix with the same size as the population. The elements of each matrix are in The population is generated uniformly and randomly within the population to obtain the initial population.

[0014] Furthermore, the method described in S2.3 specifically includes: S2.3.1 Calculate the total supply in the system Total demand Determine whether the current case belongs to Case S1 or Case S2; S2.3.2 If it is Case S1, then perform column normalization on the proportion matrix so that the sum of the elements in each column is 1, that is: ,in, The cardinality of the set of supply points; If we are in Case S2, then we perform row normalization on the proportion matrix so that the sum of the elements in each row is 1, that is: ,in, The cardinality of the set of demand points; S2.3.3, Based on the normalized scaling factor Compared with known demand or supply capacity The product of the actual transportation decision variables is calculated. The calculation formula is as follows: In Case S1: ; In Case S2: ; S2.3.4 Feasibility Repair: Truncate negative values ​​to zero and iteratively correct the total values ​​of each row and column to ensure that the repaired transportation plan meets the non-negative constraints and the supply and demand constraints under the corresponding scenario.

[0015] Furthermore, the method of S2.5 specifically includes: S2.5.1 Initialize the parameters of the Gaussian probability model, including: initial mean vector, initial standard deviation vector and initial scaling factor; S2.5.2 Parameter update, including: The expression for updating the mean vector is: In the formula, This represents the updated mean vector; This represents the total number of individuals in the Gaussian probability model. Indicates its index; The proportional matrix expansion form corresponding to elite individuals; The expression for updating the standard deviation vector is: In the formula, To prevent the standard deviation from degenerating into a positive number of zero; Dynamic scaling factor adjustment: Adjusts the sampling range and scaling factor according to the current search state. The decay rate dynamically decreases with each iteration to facilitate the transition from global exploration to local development. The calculation formula is as follows: In the formula, Indicates the current iteration algebra number; This is the Gaussian sampling scaling factor; S2.5.3, Diagonal Gaussian distribution sampling; Through the updated mean vector Standard deviation vector and scaling factor Construct a diagonal Gaussian distribution model, the expression of which is: Randomly draw sample points from a diagonal Gaussian distribution to generate new candidate solutions for the scaling matrix. .

[0016] S2.5.3, Each sample generated Perform the normalization repair and decoding operations of S2.3 to ensure that the candidate individuals generated by Gaussian learning satisfy the constraints of the unbalanced transport problem.

[0017] Furthermore, the method described in S2.6 specifically includes: S2.6.1 Selection: Select individuals with lower non-dominance levels from the current population based on non-dominance level and crowding degree; for individuals with the same non-dominance level, select individuals with larger crowding distance. S2.6.2 Crossover operation: Perform row swapping, column swapping, or weighted averaging on the proportion matrix of parent individuals; S2.6.3, Mutation Operation: Scale Factor Random perturbations are applied to enhance the algorithm's global search capability and reduce the risk of wind direction getting trapped in local optima; S2.6.4 Normalization Repair and Decoding: Perform the normalization repair and decoding operation of S2.3 on the generated offspring individuals.

[0018] Furthermore, the method of S2.9 specifically includes: S2.9.1 For each representative solution, treat it as a basic feasible solution to the transportation problem, and calculate the dual variable using the potential method: the potential at the supply point. and demand point potential Then, the test numbers for non-basic variables are calculated. ; If it exists Then, the improved path is identified and the basic feasible solution is updated according to the closed-loop rule; Repeat the update process until the maximum number of iterations for local refinement is reached or no further improvement can be made; S2.9.2, Controlled compromise acceptance mechanism; The locally refined solution needs to be accepted: whether it satisfies Pareto dominance or whether the refined solution meets the preset target improvement conditions. With the target degradation threshold condition If the threshold condition is met, the refined solution is accepted, and the accepted solution is re-encoded back into the proportional matrix form so that it can be put back into the evolutionary population.

[0019] The beneficial effects of this invention are: This invention proposes a multi-objective unbalanced transportation model considering the bottleneck supply-demand deviation, providing a new approach to balancing economic costs and system stability in complex logistics scheduling. It proposes a solution based on the hybrid evolutionary algorithm HPEA-SH, which improves the search efficiency and solution quality of traditional algorithms under complex constraints by combining the local refinement capability of the transportation simplex method and the global guidance capability of adaptive Gaussian probability learning. By modeling the statistical distribution of the first non-dominated frontier elite individuals, the algorithm can be guided to search for potentially high-quality regions, thereby obtaining a set of relatively uniform and representative non-dominated solutions within an acceptable computational time, providing decision support for enterprise transportation planning and supply chain scheduling management. Attached Figure Description

[0020] Figure 1 This is an overall flowchart of the present invention; Figure 2 This is a schematic diagram of the encoding scheme of the present invention; Figure 3 This is a schematic diagram of the improved local search path for closed loops according to the present invention. Detailed Implementation

[0021] The present invention will be further described in detail below with reference to specific embodiments.

[0022] The multi-objective unbalanced transportation problem is essentially a multi-objective extension of the transportation problem in scenarios where total supply and total demand are unequal. This problem originates from real-world logistics scheduling and supply chain management scenarios: when a set of goods needs to be transported from multiple supply points to multiple demand points, and the total supply and total demand are not equal, an unbalanced transportation optimization problem arises. Due to differences in geographical location, transportation modes, and transportation routes, the unit transportation cost between different supply points and different demand points varies. Reducing total transportation costs is a crucial objective for minimizing logistics expenses. Simultaneously, in actual logistics operations, decision-makers also need to consider the stability of supply and demand balance. This is because under conditions of supply fluctuations or sudden demand changes, if some nodes bear extreme imbalances, it will lead to a fragile logistics chain and increase the risk of system failure. Therefore, comprehensively considering transportation costs and supply-demand deviations, and formulating reasonable transportation plans, helps to achieve efficient operation of the logistics system and improve its resilience.

[0023] See Figures 1 to 3 A multi-objective imbalanced transport optimization method based on a hybrid evolutionary algorithm includes the following steps: S1. Construct a mathematical model for the multi-objective unbalanced transport problem (MOUTP); This model comprehensively considers the practical constraint of the imbalance between the total supply and demand points, and takes minimizing the total transportation cost and minimizing the bottleneck supply-demand deviation (SDD) as dual optimization objectives. It introduces a bottleneck deviation function to characterize the imbalance state of extreme nodes, thereby improving the stability of the transportation network under complex supply and demand environments. The expression includes: In the formula, The objective function is to minimize the total transportation cost; For supply point aggregation, Indicates the index of the supply point; For the set of demand points, Index for demand points; Indicates from the supply point To the point of demand The unit transportation cost; For supply points To the point of demand The actual transport volume is a decision variable, and ; Indicates supply point Maximum supply capacity; Indicate demand points The maximum demand; Setting constraints includes: Case S1: The amount received at each demand point does not exceed its maximum demand, when the total supply exceeds the total demand (i.e., Then we have: , ; Case S2: The total shipments from each supply point do not exceed its maximum supply capacity, when the total supply is not greater than the total demand (i.e., ...). Then we have: , ; And it is required that all transport volumes are non-negative, that is: .

[0024] S2. Design a simplex-inspired hybrid Pareto evolutionary algorithm (HPEA-SH) to solve the unbalanced transport problem; The HPEA-SH algorithm employs a hybrid architecture consisting of a global search driven by Gaussian learning and a local search based on the transport simplex method (TSM). The algorithm mainly consists of two parts: (1) Scenario-dependent normalization and repair mechanism for unbalanced transportation constraints: The transportation allocation relationship is represented by a proportional matrix encoding scheme. By identifying the imbalance between total supply and total demand, row normalization or column normalization is performed adaptively. When the total supply is greater than the total demand, column normalization is performed on the proportional matrix, and the transportation plan is calculated based on the demand. When the total supply is not greater than the total demand, row normalization is performed on the proportional matrix, and the transportation plan is calculated based on the supply. On this basis, the decoded transportation plan is repaired to obtain a feasible transportation scheduling plan that satisfies the supply and demand constraints, thereby improving the structural feasibility of candidate solutions and reducing the invalid search space.

[0025] (2) Hybrid evolutionary strategy integrating distributed learning and local refinement of transportation simplex: Gaussian learning operator is used to extract the statistical distribution characteristics of the elite solutions of the current first non-dominated frontier, an adaptive diagonal Gaussian probability model is established, and new candidate solutions are generated based on the probability model to guide the algorithm to perform a global search for potential high-quality regions; at the same time, three representative solutions are selected from the current first non-dominated frontier, including the solution with the minimum total transportation cost, the solution with the minimum bottleneck supply-demand deviation, and an intermediate compromise solution. The transportation simplex heuristic operator is embedded in the three representative solutions, and the individuals are locally refined by calculating the test number of non-basic variables and performing closed-loop update of basic feasible solutions; combined with the controlled compromise acceptance mechanism, when the locally refined solution meets the preset target improvement and target deterioration threshold conditions, the solution is accepted and re-encoded back into the evolutionary population; after iterative search, the algorithm maintains the diversity of the solution set through non-dominated sorting and congestion distance, and finally retains the high-quality non-dominated solution set, providing decision-makers with multiple trade-offs between cost and supply-demand balance.

[0026] Includes the following steps: S2.1, HPEA-SH key parameter settings; Set key parameters in HPEA-SH, including population size. Maximum number of iterations Learning rate Gaussian sampling scaling factor and the maximum number of iterations for transport simplex local refinement for each representative solution. .

[0027] S2.2 Population initialization; An initial population is randomly generated using a real-number encoding method based on a proportional matrix, wherein the encoding method includes: First, based on the characteristics of the unbalanced transport problem, a proportional matrix-based approach is adopted. To represent a transportation allocation scheme, where each element Indicates supply point Assigned to demand points The material proportion factor, and This encoding method, by establishing proportional relationships rather than directly transporting quantities, can maintain the structural information of the solution during the evolution process and facilitate subsequent normalization and decoding operations that depend on the execution scenario. Secondly, random generation A proportional matrix The elements of each matrix are in The population is generated uniformly and randomly within the population to obtain the initial population.

[0028] S2.3, Normalization Repair and Decoding; Due to the proportional matrix generated during the evolution process The physical constraints of the transportation problem may not be fully satisfied. Therefore, this invention proposes a scenario-dependent normalization method to map candidate solutions to feasible transportation schemes, including the following steps: S2.3.1 Scene Recognition; Total supply in the calculation system Total demand Determine whether the current case belongs to Case S1 or Case S2; S2.3.2, Perform normalization; If we are in Case S1, then we perform column normalization on the proportion matrix so that the sum of the elements in each column is 1, that is: ,in, The cardinality of the set of supply points; If we are in Case S2, then we perform row normalization on the proportion matrix so that the sum of the elements in each row is 1, that is: ,in, The cardinality of the set of demand points; S2.3.3, mapped to a transportation plan; Based on the normalized scaling factor Compared with known demand or supply capacity The product of the actual transportation decision variables is calculated. The calculation formula is as follows: In Case S1: ; In Case S2: ; S2.3.4, Feasibility Repair; The algorithm repairs negative transport volumes, numerical perturbations, or slight imbalances that appear after decoding: negative values ​​are truncated to zero, and the total values ​​of each row and column are iteratively corrected to ensure that the repaired transport schemes satisfy the non-negativity constraints and the supply and demand constraints of the corresponding scenario. This restricts candidate solutions to the feasible region and improves the evolution efficiency of the algorithm. Finally, a population of feasible transport schemes is obtained, where each individual contains two objective values. and .

[0029] S2.4, Non-dominated sorting and crowding distance calculation; The current population is sorted using fast non-dominated sorting and crowding distance calculation. Individual quality is then evaluated using non-dominated rank and crowding distance, specifically including: Population stratification based on Pareto dominance; Pareto dominance definition (minimization problem): If... and Then it is called Dominate ,in, To solve In the Function values ​​on each target To solve In the Function values ​​on each target; The formula for calculating congestion distance is as follows: In the formula, Represents an individual The degree of congestion; and They represent the target Above and Individual The objective values ​​of two adjacent solutions; and Representing the target The maximum and minimum values ​​in the current population.

[0030] S2.5, Gaussian learning for offspring generation; Extract elite individuals from the current first non-dominated frontier, statistically analyze their distribution characteristics, construct a diagonal Gaussian probability model, and sample to generate new candidate solutions, including the following steps: S2.5.1 Initialization of Gaussian probability model parameters; The initial mean vector is set as the empirical distribution center of each decision dimension in the current population; the initial standard deviation vector is set as the empirical dispersion of each decision dimension in the current population; the initial scaling factor is used to control the sampling range; where the mean vector and the standard deviation vector both correspond to the decision variable space after the proportional matrix encoding is expanded. S2.5.2, Update Gaussian probability model parameters; Extract all individuals from the first-level non-dominated solutions (Rank-1) of the current iteration population to form an elite individual set. Let there be a total of Individuals; elite individuals collectively reflect the distribution characteristics of the current Pareto frontier in the decision space. By modeling the statistics of the elite individual set in each decision dimension, a diagonal Gaussian probability model for subsequent sampling is obtained to guide the algorithm to search for potential high-quality regions. The proportional matrix corresponding to each elite individual is expanded as follows: Then the expression for updating the mean vector is: In the formula, This represents the updated mean vector; The expression for updating the standard deviation vector is: In the formula, To prevent the standard deviation from degenerating into a positive number of zero; Dynamic scaling factor adjustment: Adjusts the sampling range and scaling factor according to the current search state. The decay rate dynamically decreases with each iteration to facilitate the transition from global exploration to local development. The calculation formula is as follows: In the formula, Indicates the current iteration algebra number; The scaling factor is Gaussian sampling, which is set to 0.05 in this invention; S2.5.3, Diagonal Gaussian distribution sampling; Through the updated mean vector Standard deviation vector and scaling factor The sampling logic for constructing a diagonal Gaussian distribution model and generating Gaussian learning offspring from the Gaussian probability model is as follows: in, This represents the mean vector obtained from statistics of elite individuals. This represents the sampling scaling factor that changes dynamically with the iteration process. It is the identity matrix; In this invention, the expression is: Randomly draw sample points from a diagonal Gaussian distribution to generate new candidate solutions for the scaling matrix. ; S2.5.4, Feasibility post-processing; Each sample generated Perform normalization repair and decoding operations to ensure that the candidate individuals generated by Gaussian learning satisfy the constraints of the unbalanced transport problem.

[0031] S2.6, Generation of offspring through evolution; To introduce diversity into the search space while preserving the superior structural information of the parent generation, the algorithm performs crossover and mutation operations on the individuals encoded by the proportion matrix to generate new candidate individuals. The crossover operator is used to integrate the distribution proportion characteristics of the parent individuals, and the mutation operator is used to perturb the proportion factor, thereby enhancing the algorithm's global search capability and reducing the risk of getting trapped in local optima. Specifically, the algorithm includes the following steps: S2.6.1, Selection; Based on the non-dominance level and crowding degree, individuals with a lower non-dominance level are selected from the current population, while individuals with a larger crowding distance are selected within the same non-dominance level. S2.6.2, Cross Operation; The crossover operator is applied to the proportion matrix of the parent individuals. The distribution proportion characteristics of the parent individuals can be combined and achieved through row swapping, column swapping, or weighted averaging. S2.6.3, Mutation Operation; Scale factor Random perturbations are applied to enhance the algorithm's global search capability and reduce the risk of wind direction getting trapped in local optima; S2.6.4, Normalization Repair and Decoding; Perform normalization repair and decoding operations on the generated offspring individuals.

[0032] S2.7, Merging and Non-Dominant Screening; The parent population, Gaussian-learned offspring, and evolutionary offspring are merged and then filtered using non-dominated sorting and crowding distance.

[0033] S2.8 Select three representative solutions; Based on S2.7, three representative solutions are selected from the current first non-dominated front as the objects of the local search of the transport simplex. The first representative solution: total transportation cost The smallest individual; The second representative solution: Bottleneck supply-demand deviation The smallest individual; The third representative solution is an intermediate compromise solution, selected from the first non-dominated frontier individuals other than the first and second representative solutions, sorted by crowding distance (this invention selects the middle position; if two individuals appear in the middle position, their average value is taken).

[0034] S2.9, Local search of the transport simplex; Performing a transport simplex method (TSM) heuristic local search refinement on the three representative solutions involves the following steps: S2.9.1 For each representative solution, treat it as a basic feasible solution to the transportation problem, and calculate the dual variables using the potential method. (Supply point potential) and (Demand point potential), and then calculate the test numbers for non-basic variables: ; If it exists If the improvement path is identified and the basic feasible solution is updated according to the closed-loop rule, the above process is repeated until the maximum number of iterations of local refinement is reached or no further improvement can be made, thereby performing local refinement on the representative solution. S2.9.2, Controlled compromise acceptance mechanism; The locally refined solution needs to be accepted: whether it satisfies Pareto dominance or whether the refined solution meets the preset target improvement conditions. With the target degradation threshold condition If the threshold condition is met, the refined solution is accepted, and the accepted solution is re-encoded back into the proportional matrix form so that it can be put back into the evolutionary population.

[0035] S2.10, Final Merging and Next Generation Screening; The current parent population, evolutionary offspring, Gaussian-learned offspring, and updated individuals accepted after the transport simplex local search are filtered by non-dominated sorting and crowding distance. Individuals with lower non-dominated levels are selected from the current population based on non-dominated level and crowding, while individuals with larger crowding distances are selected within the same non-dominated level.

[0036] S2.11, Iterative judgment; Repeat steps S2.5 through S2.10 until the maximum number of iterations is reached. The algorithm terminates when a preset termination condition is met (such as convergence of the solution set or no further improvement in the objective value), outputting the final high-quality non-dominated solution set. All first-non-dominated frontier individuals are extracted from the final population, each individual including: total transportation cost. Bottleneck supply-demand deviation and actual transportation decision variables .

[0037] This invention establishes a multi-objective unbalanced transportation problem model with the optimization objectives of minimizing total transportation cost and minimizing bottleneck supply-demand deviation (SDD). Furthermore, a hybrid Pareto evolutionary algorithm (HPEA-SH) combining transportation simplex heuristic and Gaussian probability learning is designed to solve this problem, thereby achieving an effective trade-off between transportation cost and system stability, obtaining a non-dominated solution set with good distribution and convergence, and providing decision support for logistics scheduling managers.

[0038] To verify the effectiveness of the proposed algorithm HPEA-SH, this invention was compared with three mainstream multi-objective optimization algorithms: MOEA / D, MOGLS, and NSGA-II. The comparison experiments were conducted under the same computational environment and termination conditions, with all algorithms run independently 21 times on each test case. A total of 100 test cases were used, covering two imbalanced scenarios, Scenario S1 and Scenario S2, with problem sizes ranging from... arrive .

[0039] To evaluate the quality of the approximate Pareto fronts obtained by each algorithm, this embodiment uses three evaluation metrics: convergence metric GD, inverse generation distance metric IGD, and dominance metric. Among them, the smaller the values ​​of GD and IGD, the closer the solution set obtained by the algorithm is to the reference front. A larger value indicates a higher proportion of solutions in the obtained solution set that are not dominated by the reference front, i.e., better dominance performance. The comparison results of each algorithm are shown in Tables 1 to 6, where bold text indicates the better results on the corresponding examples.

[0040] Table 1: Comparison Results of GD Indicators under Scenario S1 Table 2 Comparison of IGD indices under Scenario S1 Table 3 Comparison results of Ω index under Scenario S1 As shown in Tables 1 to 3, under the test conditions of Scenario S1, the proposed HPEA-SH achieved good results on medium-to-large-scale examples. Regarding the GD metric, HPEA-SH achieved superior Mean values ​​in 7 out of 10 scale groups, with its advantage gradually becoming apparent from the 6×13 scale. Regarding the IGD metric, although HPEA-SH was not optimal on some small-scale examples, its Mean value consistently lagged behind MOEA / D and NSGA-II from the 10×20 scale onwards. Regarding the Ω metric, HPEA-SH's Mean value was generally higher than the comparative algorithms on medium-to-large-scale examples. Furthermore, the total NB values ​​for GD, IGD, and Ω metrics for HPEA-SH under Scenario S1 were 77, 58, and 74, respectively. On the largest scale example of 30×100, the Mean GD, Mean IGD, and Mean Ω of HPEA-SH were 0.168095, 0.188880, and 0.225475, respectively. The above results show that, in the Scenario S1 scenario, the method proposed in this invention can achieve better solution set quality.

[0041] Table 4 Comparison of GD Indicators under Scenario S2 Table 5 Comparison of IGD indices under Scenario S2 Table 6 Comparison of Ω index under Scenario S2 As shown in Tables 4 to 6, HPEA-SH further improved its performance under Scenario S2 testing conditions. In the GD metric, HPEA-SH achieved superior Mean values ​​in 8 out of 10 scale groups; in the IGD metric, it achieved superior Mean values ​​in 6 scale groups; and in the Ω metric, it achieved superior Mean values ​​in 8 scale groups. The corresponding total NB values ​​were 74 for GD, 57 for IGD, and 71 for Ω, which are generally higher than the other three comparison algorithms. In the largest example of 30×100, HPEA-SH's Mean GD, Mean IGD, and Mean Ω were 0.049664, 0.081067, and 0.326455, respectively. Furthermore, under Scenario S2, HPEA-SH's p-values ​​for the Mean column in the GD, IGD, and Ω metrics were all less than 0.05 compared to the three comparison algorithms. The above results demonstrate that the proposed method achieves good solution performance in highly unbalanced and large-scale testing scenarios.

[0042] In summary, the HPEA-SH proposed in this invention can improve algorithm convergence while ensuring solution set diversity, and exhibits superior comprehensive solution capabilities in medium-to-large-scale and highly imbalanced scenarios. By integrating scenario-dependent normalization repair, elite statistics-driven Gaussian probability learning, and a local refinement mechanism based on the transportation simplex method, this invention can obtain a set of relatively uniform and representative non-dominated solutions within an acceptable computational time, providing decision support for logistics transportation planning and supply chain scheduling management.

[0043] The specific embodiments of the present invention have been described in detail above with reference to the accompanying drawings and examples. However, the present invention is not limited to the above embodiments. Within the scope of knowledge possessed by those skilled in the art, various changes can be made without departing from the spirit of the present invention.

Claims

1. A multi-objective imbalanced transport optimization method based on a hybrid evolutionary algorithm, characterized in that, Includes the following steps: S1. Construct a mathematical model for the multi-objective unbalanced transport problem; The multi-objective unbalanced transportation problem has two optimization objectives: minimizing the total transportation cost and minimizing the bottleneck supply-demand deviation. S2. Design a hybrid Pareto evolutionary algorithm based on simplex heuristics to solve the multi-objective unbalanced transport problem; The solution methods include: The imbalance between total supply and total demand is identified using a ratio matrix, and a normalization operation is performed; the decoded transportation plan is then repaired. Gaussian learning is used to generate offspring, and representative solutions are selected by combining the generated offspring with the current parent population through non-dominated sorting and crowding distance; the representative solutions are then locally refined using transport simplex local search. The updated individuals accepted after the current parent population, evolutionary offspring, Gaussian learning offspring, and transport simplex local search are selected by non-dominated sorting and crowding distance.

2. The multi-objective imbalanced transport optimization method based on a hybrid evolutionary algorithm according to claim 1, characterized in that: The mathematical model expression for the multi-objective unbalanced transport problem is as follows: In the formula, The objective function is to minimize the total transportation cost; For supply point aggregation, Indicates the index of the supply point; For the set of demand points, Index for demand points; Indicates from the supply point To the point of demand The unit transportation cost; For supply points To the point of demand Actual transport volume; Indicates supply point Maximum supply capacity; Indicate demand points The maximum demand.

3. The multi-objective imbalanced transport optimization method based on a hybrid evolutionary algorithm according to claim 2, characterized in that: The mathematical model for the aforementioned multi-objective unbalanced transport problem is further constrained by the following conditions: Case S1: The quantity received at each demand point does not exceed its maximum demand. ， ； Case S2: The total shipments from each supply point shall not exceed its maximum supply capacity. ， ； All shipment volumes must be non-negative. .

4. The multi-objective imbalanced transport optimization method based on a hybrid evolutionary algorithm according to claim 3, characterized in that: The steps in S2 include: S2.1 Set the parameters required for HPEA-SH, including population size. Maximum number of iterations Learning rate Gaussian sampling scaling factor and the maximum number of iterations for transport simplex local refinement for each representative solution. ; S2.

2. The initial population is randomly generated using a real-number encoding method based on a proportional matrix. S2.3, Normalization Repair and Decoding: A scenario-dependent normalization method maps candidate solutions to feasible transportation schemes; S2.4 Perform fast non-dominated sorting and crowding distance calculation on the current population to obtain the sorted population; S2.5 Extract elite individuals from the current first non-dominated frontier, construct a diagonal Gaussian probability model, and sample to generate new candidate solutions; S2.6 Perform crossover and mutation operations on the individuals encoded by the proportional matrix to generate new candidate individuals; S2.7 Merge the current parent population, Gaussian-learned offspring, and evolved offspring, and filter them using non-dominated sorting and crowding distance; S2.8 Select three representative solutions: the individual with the lowest total transportation cost, the individual with the lowest bottleneck supply-demand deviation, and the individual in the middle position selected from the first non-dominated frontier individuals other than the first and second representative solutions, sorted by congestion distance. S2.

9. Perform heuristic local search refinement using the transport simplex method on the three representative solutions; S2.

10. Select the current parent population, evolutionary offspring, Gaussian learning offspring, and updated individuals accepted after the transport simplex local search by non-dominated sorting and crowding distance. Select individuals with smaller non-dominated levels from the current population based on non-dominated level and crowding, and select individuals with larger crowding distances under the same non-dominated level. S2.11, Repeat S2.5~S2.10 until the maximum number of iterations is reached. If the preset termination condition is met, the algorithm terminates and outputs the final high-quality non-dominated solution set. All first non-dominated frontier individuals are extracted from the final population. Each individual includes: total transportation cost, bottleneck supply and demand deviation, and actual transportation decision variables.

5. The multi-objective imbalanced transport optimization method based on a hybrid evolutionary algorithm according to claim 4, characterized in that: The specific method of S2.2 is as follows: using a proportional matrix-based approach. To represent a transportation allocation scheme, where each element Indicates supply point Assigned to demand points The material proportion factor, and Randomly generate a proportional matrix with the same size as the population. The elements of each matrix are in The population is generated uniformly and randomly within the population to obtain the initial population.

6. The multi-objective imbalanced transport optimization method based on a hybrid evolutionary algorithm according to claim 5, characterized in that: The method described in S2.3 specifically includes: S2.3.1 Calculate the total supply in the system Total demand Determine whether the current case belongs to Case S1 or Case S2; S2.3.2 If it is Case S1, then perform column normalization on the proportion matrix so that the sum of the elements in each column is 1, that is: ,in, The cardinality of the set of supply points; If we are in Case S2, then we perform row normalization on the proportion matrix so that the sum of the elements in each row is 1, that is: ,in, The cardinality of the set of demand points; S2.3.3, Based on the normalized scaling factor Compared with known demand or supply capacity The product of the actual transportation decision variables is calculated. The calculation formula is as follows: In Case S1: ; In Case S2: ; S2.3.4 Feasibility Repair: Truncate negative values ​​to zero and iteratively correct the total values ​​of each row and column to ensure that the repaired transportation plan meets the non-negative constraints and the supply and demand constraints under the corresponding scenario.

7. The multi-objective imbalanced transport optimization method based on a hybrid evolutionary algorithm according to claim 6, characterized in that: The specific methods of S2.5 include: S2.5.1 Initialize the parameters of the Gaussian probability model, including: initial mean vector, initial standard deviation vector and initial scaling factor; S2.5.2 Parameter update, including: The expression for updating the mean vector is: In the formula, This represents the updated mean vector; This represents the total number of individuals in the Gaussian probability model. Indicates its index; The proportional matrix expansion form corresponding to elite individuals; The expression for updating the standard deviation vector is: In the formula, To prevent the standard deviation from degenerating into a positive number of zero; Dynamic scaling factor adjustment: Adjusts the sampling range and scaling factor according to the current search state. The decay rate dynamically decreases with each iteration to facilitate the transition from global exploration to local development. The calculation formula is as follows: In the formula, Indicates the current iteration algebra number; This is the Gaussian sampling scaling factor; S2.5.3, Diagonal Gaussian distribution sampling; Through the updated mean vector Standard deviation vector and scaling factor Construct a diagonal Gaussian distribution model, the expression of which is: Randomly draw sample points from a diagonal Gaussian distribution to generate new candidate solutions for the scaling matrix. ; S2.5.3, Each sample generated Perform the normalization repair and decoding operations of S2.3 to ensure that the candidate individuals generated by Gaussian learning satisfy the constraints of the unbalanced transport problem.

8. The multi-objective imbalanced transport optimization method based on a hybrid evolutionary algorithm according to claim 7, characterized in that: The method described in S2.6 specifically includes: S2.6.1 Selection: Select individuals with lower non-dominance levels from the current population based on non-dominance level and crowding degree; for individuals with the same non-dominance level, select individuals with larger crowding distance. S2.6.2 Crossover operation: Perform row swapping, column swapping, or weighted averaging on the proportion matrix of parent individuals; S2.6.3, Mutation Operation: Scale Factor Random perturbations are applied to enhance the algorithm's global search capability and reduce the risk of wind direction getting trapped in local optima; S2.6.4 Normalization Repair and Decoding: Perform the normalization repair and decoding operation of S2.3 on the generated offspring individuals.

9. A multi-objective imbalanced transport optimization method based on a hybrid evolutionary algorithm according to claim 8, characterized in that: The method described in S2.9 specifically includes: S2.9.1 For each representative solution, treat it as a basic feasible solution to the transportation problem, and calculate the dual variable using the potential method: the potential at the supply point. and demand point potential Then, the test numbers for non-basic variables are calculated. ; If it exists Then, the improved path is identified and the basic feasible solution is updated according to the closed-loop rule; Repeat the update process until the maximum number of iterations for local refinement is reached or no further improvement can be made; S2.9.2, Controlled compromise acceptance mechanism; The locally refined solution needs to be accepted: whether it satisfies Pareto dominance or whether the refined solution meets the preset target improvement conditions. With the target degradation threshold condition If the threshold condition is met, the refined solution is accepted, and the accepted solution is re-encoded back into the proportional matrix form so that it can be put back into the evolutionary population.

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