Decision tree-mathematical programming hierarchical machining scheduling method considering multi-objective tradeoff

Abstract

The embodiment of the invention provides a decision tree-mathematical programming hierarchical machining scheduling method considering multi-objective tradeoff. The method comprises the following steps: establishing an optimization model containing a plurality of objective functions, and constructing a decision tree structure; generating a multi-target tradeoff function based on the decision tree structure, and calculating gradient information of the multi-target tradeoff function; performing partition regulation and control on an optimization space based on the trade-off function gradient so as to identify and avoid a local degradation region; and executing mathematical programming solution in the regulated optimization space to obtain an optimal machining scheduling scheme. Through the scheme of the embodiment of the invention, the problem of how to carry out partition regulation and control on the multi-target optimization space according to the trade-off function gradient so as to solve the problem of local deterioration of the target function can be solved.

Description

Technical Field

[0001] This application relates to the optimization algorithm decision tree, specifically to a decision tree-mathematical programming hierarchical machining scheduling method that considers multiple objective trade-offs. Background Technology

[0002] The hierarchical machine processing scheduling method, which considers multi-objective trade-offs and employs mathematical programming, is an optimization approach that combines decision trees with mathematical programming. It aims to achieve a more scientific and rational scheduling scheme in machine processing by constructing a hierarchical decision structure that comprehensively considers multiple conflicting optimization objectives, such as production efficiency, cost, and resource utilization. This method introduces a trade-off function to dynamically adjust the relative importance of different objectives, thereby finding a better solution in the multi-objective optimization space. However, a key challenge of this method is how to partition and regulate the multi-objective optimization space based on the gradient of the trade-off function to address the problem of solution quality degradation caused by local deterioration of the objective function. Further exploration of effective partitioning strategies and regulation mechanisms is needed to improve the algorithm's stability and global search capability. Summary of the Invention

[0003] In view of this, embodiments of the present disclosure provide a hierarchical machining scheduling method based on decision tree-mathematical programming that considers multiple objective trade-offs, which at least partially solves the problems existing in the prior art.

[0004] A hierarchical machining scheduling method based on decision trees and mathematical programming, considering multi-objective trade-offs, includes: Establish an optimization model containing multiple objective functions and construct a decision tree structure; A multi-objective trade-off function is generated based on the decision tree structure, and its gradient information is calculated; The optimization space is partitioned and controlled based on the gradient of the aforementioned tradeoff function in order to identify and avoid localized degradation regions. Mathematical programming is performed in the optimized space after adjustment to obtain the optimal machining scheduling scheme.

[0005] In one specific implementation, generating the multi-objective trade-off function based on the decision tree structure further includes: Extract the path weight α_i of each node in the decision tree; Calculate the correlation between the path and the objective function θ_i = ∑(w_j g_j), where w_j represents the weight coefficient of the j-th target, and g_j represents the target value under this path; Construct a multi-objective weighted function f_w = ∑(α_i) θ_i); The conditional statement f_w>threshold_w is used to determine whether the path is a good path.

[0006] In one specific implementation, the step of partitioning and controlling the optimization space based on the gradient of the tradeoff function further includes: Obtain the gradient matrix G; Calculate the maximum principal curvature κ = max(σ(G)) at each point, where σ(G) represents the singular value of G; Construct a threshold Δ = γ max(κ), where γ is an adjustment coefficient ranging from 1 to 5; If κ>Δ, then this region is considered to be in a locally deteriorated region and is marked.

[0007] In one specific implementation, the step of partitioning and controlling the optimization space based on the gradient of the tradeoff function further includes: Calculate the gradient change ΔG at the current solution point; Construct the neighborhood distance d_n = ∑|x_i x_j| / N, where x_i and x_j are the coordinates of the nearest solution points, and N is the neighborhood size; The stable gradient trend G_avg = (G + G_prev) / 2 is calculated using the moving average formula. According to the discriminant ΔG>λ d_n determines whether to activate the control strategy, and λ is the sensitivity factor.

[0008] In one specific implementation, the step of partitioning and controlling the optimization space based on the gradient of the tradeoff function further includes: Define the local convergence index ρ = (G_prev G_current) / G_prev; Set the initial convergence threshold ε = 0.5; The convergence metric ε_new = ε is updated using an exponential decay method. e^{β t}, where β is the decay coefficient and t is the number of iterations; If |ρ|>ε_new, then a local adjustment mechanism is triggered to avoid risk points.

[0009] In one specific implementation, the step of partitioning and controlling the optimization space based on the gradient of the tradeoff function further includes: Establish the dynamic gradient interval [G_min, G_max]; A fuzzy variable μ is introduced to represent whether the current region is a transition zone; The boundary determination formula is constructed as G_boundary = (G_max + G_min) / 2, where μ ∈ [0, 1] represents the degree of region transition; If μ ≥ τ, then this region is designated as a high-sensitivity region, and the sampling frequency is increased.

[0010] In one specific implementation, the step of partitioning and controlling the optimization space based on the gradient of the tradeoff function further includes: Local terrain characteristics are described using a quadratic function f(x) = ax^2 + bx + c, where a, b, and c are obtained by fitting gradient information. Construct a minimum value determination formula and a criterion for min(f(x)) ≤ 0; Introduce a penalty factor λ, such that the objective function f_p = f(x) + λ penalty_factor; By using the constraint f_p ≤ threshold, we ensure that we avoid local minima.

[0011] In one specific implementation, the step of partitioning and controlling the optimization space based on the gradient of the tradeoff function further includes: Set the tolerance threshold for gradient change as T_tol = ΔG_avg δ, where ΔG_avg is the historical average change value, and δ is the dynamic tolerance factor; If ΔG > T_tol in N consecutive runs, it is considered an unstable state; Triggering partitioning strategy: Divide regions into dense and sparse categories based on their density p; For dense areas, an adaptive mesh refinement algorithm is enabled to reduce the risk of deviation.

[0012] In one specific implementation, the step of partitioning and controlling the optimization space based on the gradient of the tradeoff function further includes: Calculate the point cloud density ρ_density = 1 / dist_mean in the multi-object space, where dist_mean is the average distance of the nearest neighbor; Establish the condition for determining uneven distribution: dist_mean > σ threshold; Introducing the Laplacian operator ∇²f in non-uniform regions to detect local bending directions; Combine the ∇²f symbol to determine if there is a potential depression; if so, trigger the adjustment step.

[0013] In one specific implementation, the step of partitioning and controlling the optimization space based on the gradient of the tradeoff function further includes: The smooth transition region is constructed using the Sigmoid function: S(t) = 1 / (1 + e^{k(t t_c)}), where k controls the slope and t_c is the critical value; The activation threshold S(t) > S_threshold is set to determine the activation condition; Use the activation point as the partition boundary and record the region name; Determine whether to merge or split based on the shape parameter σ of the active region.

[0014] This disclosure provides a hierarchical machining scheduling method based on decision tree and mathematical programming that considers multi-objective trade-offs. The method includes: establishing an optimization model containing multiple objective functions and constructing a decision tree structure; generating multi-objective trade-off functions based on the decision tree structure and calculating their gradient information; partitioning and regulating the optimization space based on the gradients of the trade-off functions to identify and avoid local degradation regions; and performing mathematical programming in the regulated optimization space to obtain the optimal machining scheduling scheme. This method addresses the problem of how to partition and regulate the multi-objective optimization space based on the gradients of the trade-off functions to solve the problem of local degradation of objective functions. Attached Figure Description

[0015] In the accompanying drawings, unless otherwise specified, the same reference numerals throughout the various drawings denote the same or similar parts or elements. These drawings are not necessarily drawn to scale. It should be understood that these drawings depict only some embodiments disclosed in this application and should not be construed as limiting the scope of this application.

[0016] Figure 1 This is a flowchart of a hierarchical machining scheduling method based on decision trees and mathematical programming that considers multiple objective trade-offs. Figure 2 This is a further flowchart based on the generation of multi-objective tradeoff functions using a decision tree structure; Figure 3 This is a further flowchart based on partitioning and controlling the optimization space according to the gradient of the trade-off function; Figure 4 This is a further flowchart based on partitioning and controlling the optimization space according to the gradient of the trade-off function; Figure 5 This is a further flowchart based on partitioning and controlling the optimization space according to the gradient of the trade-off function; Figure 6 This is a further flowchart based on partitioning and controlling the optimization space according to the gradient of the trade-off function; Figure 7This is a further flowchart based on partitioning and controlling the optimization space according to the gradient of the trade-off function; Figure 8 This is a further flowchart based on partitioning and controlling the optimization space according to the gradient of the trade-off function; Figure 9 This is a further flowchart based on partitioning and controlling the optimization space according to the gradient of the trade-off function; Figure 10 This is a further flowchart based on the partitioning and control of the optimization space using the gradient of the trade-off function. Detailed Implementation

[0017] The embodiments of this disclosure will now be described in detail with reference to the accompanying drawings.

[0018] The following specific examples illustrate the implementation of this disclosure. Those skilled in the art can easily understand other advantages and effects of this disclosure from the content disclosed in this specification. Obviously, the described embodiments are only a part of the embodiments of this disclosure, and not all of them. This disclosure can also be implemented or applied through other different specific embodiments, and the details in this specification can also be modified or changed based on different viewpoints and applications without departing from the spirit of this disclosure. It should be noted that, in the absence of conflict, the following embodiments and features in the embodiments can be combined with each other. Based on the embodiments in this disclosure, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this disclosure.

[0019] It should be noted that various aspects of embodiments within the scope of the appended claims are described below. It will be apparent that the aspects described herein can be embodied in a wide variety of forms, and any particular structure and / or function described herein is merely illustrative. Based on this disclosure, those skilled in the art will understand that one aspect described herein can be implemented independently of any other aspect, and two or more of these aspects can be combined in various ways. For example, any number of aspects set forth herein can be used to implement the device and / or practice the method. Additionally, this device and / or method can be implemented using structures and / or functionalities other than one or more of the aspects set forth herein.

[0020] It should also be noted that the illustrations provided in the following embodiments are only schematic representations of the basic concept of this disclosure. The drawings only show the components related to this disclosure and are not drawn according to the number, shape and size of the components in actual implementation. In actual implementation, the form, quantity and proportion of each component can be arbitrarily changed, and the layout of the components may also be more complex.

[0021] Furthermore, specific details are provided in the following description to facilitate a thorough understanding of the examples. However, those skilled in the art will understand that the described aspects can be practiced without these specific details.

[0022] Next, referring to the accompanying drawings, the specific implementation steps and operation process of the decision tree-mathematical programming hierarchical machining scheduling method considering multi-objective trade-offs of the present invention are described, including establishing an optimization model containing multiple objective functions and constructing a decision tree structure. In this stage, it is first necessary to clarify the objective functions involved in machining scheduling, such as minimizing production cycle time, reducing energy consumption, and optimizing equipment utilization. Then, based on these objectives, a multi-level mathematical programming model is constructed, and a decision tree structure is built to organize and manage the trade-off relationships between these objectives, guiding the weight adjustment of different objectives during the solution process. Specifically, in a machining environment, such as when a workshop needs to arrange the processing sequence of multiple workpieces and needs to consider machine tool availability and workpiece delivery time requirements, these factors can be incorporated into the decision tree and hierarchically divided according to priority or degree of mutual influence, laying the foundation for subsequent trade-off analysis. This stage helps to comprehensively consider various constraints, providing data support and theoretical basis for subsequent steps.

[0023] Next, a multi-objective tradeoff function is generated based on the decision tree structure, and its gradient information is calculated. This step aims to define a tradeoff function that reflects the interrelationships between objective functions by modeling each level in the decision tree, and to calculate the gradient distribution of this tradeoff function in the optimization space using numerical methods (such as finite difference methods or analytical methods). For example, assuming a node in the decision tree involves a tradeoff between machine load and workpiece delay rate, a convex combination form of the tradeoff function can be constructed by considering the mathematical relationship between these two variables, and its derivative with respect to each objective function can be obtained. Such gradient information not only helps identify which objective functions may face a downward trend during the optimization process, but also provides a quantitative basis for subsequent spatial partitioning. This operation ensures that the algorithm has a more accurate directional guidance capability when exploring the optimization space.

[0024] Subsequently, the optimization space is partitioned and controlled based on the gradient of the tradeoff function to identify and avoid locally degraded regions. This is to effectively address the local optimum problem caused by conflicts between multiple objective functions. Through gradient information analysis, the entire optimization space can be divided into several sub-regions, with each region divided based on the change in response to the gradient of the objective function. In specific implementation, when the gradient value exceeds a set threshold, it means that the region may have a tendency to degrade locally. At this time, the search for the region can be strengthened or the strategy adjusted to avoid falling into an invalid region. For example, if a certain interval shows a trend of increased energy consumption but no decrease in completion time during the optimization process, it is judged to be a potential suboptimal region based on the gradient of the tradeoff function, and the adjacent regions are refined to find improvement solutions. This mechanism significantly enhances the global exploration capability of the algorithm, while improving scheduling efficiency and quality.

[0025] Finally, mathematical programming is performed in the adjusted optimization space to obtain the optimal machining scheduling scheme. After the above processing, the scheduling problem is transformed into an efficient search problem within the modified optimization region. Combining traditional mathematical programming techniques (such as mixed-integer linear programming, dynamic programming, or heuristic search) for computation yields the optimal solution that satisfies all objective function requirements. For example, in a manufacturing plant, applying this method can accurately predict the job scheduling between different machines, thereby maximizing resource utilization while shortening the overall production cycle and reducing energy waste. This integrated method effectively solves the trade-off problem in multi-objective optimization and improves the overall scheduling effect.

[0026] The technical solution of this invention can balance performance indicators and computational feasibility in processing scheduling scenarios with multiple complex constraints. It achieves efficient optimization through hierarchical decision-making and gradient guidance, significantly improving the adaptability and stability of the scheduling scheme. It has important practical application value and research prospects.

[0027] Next, the generation of a multi-objective tradeoff function based on a decision tree structure according to the present invention will be further described. The first step is to extract the path weight α_i of each node in the decision tree. This step is used to quantify the influence of each path in the decision tree. The value range of α_i is generally [0,1], and its optimal value is as close to 1 as possible, indicating that the path has a higher priority globally. Path weights help in subsequent analysis of the contribution of each path to the objective tradeoff.

[0028] Calculate the correlation between the path and the objective function θ_i = ∑(w_j The second step is g_j), where w_j represents the weight coefficient of the j-th objective, and g_j represents the objective value under this path. This formula means that it comprehensively measures the performance of each path across multiple objectives through a weighted summation. w_j is typically in the range [0,1], with a sum equal to 1, ensuring reasonable weight allocation; the value of g_j is defined according to the specific objective, such as processing time, energy consumption, etc. The purpose of this formula is to reflect the differences in importance between different objectives.

[0029] Construct a multi-objective weighted function f_w = ∑(α_i) Step θ_i) is the third step, used to summarize the overall evaluation of each path. This function integrates path weights and correlations to form a unified evaluation index. The value of f_w reflects the performance of the overall scheduling scheme under multi-objective conditions; a higher value indicates a better path.

[0030] The final step is to determine whether a path is a good path based on the conditional statement f_w > threshold_w, where threshold_w is a preset threshold. If the condition is met, the path is considered to meet the optimization criteria and can be used for subsequent scheduling. This condition helps to select paths with good overall performance and improves the reliability of the final scheduling results.

[0031] For example, in machining scheduling, assuming there's a need to balance production time and energy consumption, a decision tree generates multiple process paths, each corresponding to different processing time and energy consumption values. The goal correlation degree of each path is calculated based on weight coefficients, and a weighted average is obtained by combining the path weights to arrive at a comprehensive score. If a path's score exceeds a set threshold, it can be selected as the preferred path for execution, achieving multi-objective trade-off optimization.

[0032] This technical solution effectively improves the adaptability and accuracy of the scheduling method in complex multi-objective scenarios by introducing path weights and combining them with multi-objective weighting functions. It also enhances the algorithm's ability to respond to real-world constraints, which is conducive to improving overall manufacturing efficiency and resource utilization.

[0033] Next, the invention further describes the partitioning and control of the optimization space based on the gradient of the tradeoff function. Obtaining the gradient matrix G involves first calculating the gradient of the tradeoff function at each point in the optimization space to reflect the direction and intensity of the function's change at that point. The elements of G represent the derivative changes in each dimension, typically forming an n-dimensional matrix where n represents the number of variables. This step extracts information about the function's sensitive regions for subsequent analysis of its curvature properties.

[0034] Calculate the maximum principal curvature κ = max(σ(G)) at each point, where σ(G) represents the singular value of G. κ reflects the degree of local curvature of the function at a point; a higher maximum principal curvature value indicates that the region where that point is located is more likely to have nonlinear or extreme variation characteristics. By calculating the maximum principal curvature, the stability differences between different regions can be preliminarily determined, serving as a basic indicator for the next step of partitioning.

[0035] Construct a threshold Δ = γ The formula max(κ) represents the threshold value, where γ is an adjustment coefficient ranging from 1 to 0.5. A larger γ value indicates a region that tolerates higher levels of variation, while a smaller value more strictly distinguishes between different levels of variation. The purpose of this formula is to set a dynamic threshold based on the overall optimal trend of change, preventing overfitting or oversimplification. The optimal value may depend on the trade-offs in the practical application, with common values ​​ranging from approximately 0.7 to 0.8.

[0036] If κ > Δ, then this region is considered to be in a locally degraded region and is marked. This condition is used to determine whether intervention is needed in the current optimization path to prevent entering inefficient or infeasible sub-regions. In hierarchical machining scheduling using decision trees, this avoids selecting overly complex paths and ensures stable operation of the production process.

[0037] For example, in a machining scheduling scenario, if the arrangement of a certain workpiece increases both the overall processing time and energy consumption, the gradient of the corresponding trade-off function may exhibit high curvature. After detecting this situation, the system marks the affected area and prioritizes adjusting such paths subsequently, improving overall scheduling efficiency and energy efficiency. This method achieves better multi-objective coordination and resource allocation by precisely dividing spatial segments with varying degrees of change, thereby improving system adaptability and robustness.

[0038] Next, the invention further describes the partitioning and control of the optimization space based on the gradient of the tradeoff function. First, the gradient change ΔG at the current solution point is calculated. ΔG represents the gradient difference between the current solution and the previous solution, and its value range is generally [−1, 1]. A large absolute value indicates a significant gradient change. This calculation is used to identify the stability near the solution point. For example, in machining scheduling, if there are obvious contradictions in the optimization objectives of the machining sequence, ΔG can reflect the balance relationship between the objectives.

[0039] Next, we construct the neighborhood distance d_n = ∑|x_i − x_j| / N, where x_i and x_j are the coordinates of neighboring solutions, and N is the number of solutions in the neighborhood. This parameter measures the distribution density of solutions within a local region and the average distance between neighboring solutions. Generally, the larger N is, the more stable the value of d_n, indicating a more balanced region. For example, in scheduling problems, if all neighboring solutions of a certain solution are close in distance, it indicates that the region may be close to a local optimum.

[0040] Subsequently, the stable gradient trend G_avg = (G + G_prev) / 2 is calculated using the moving average formula, where G is the current gradient and G_prev is the gradient of the previous period, with a value in the range [-1, 1]. G_avg can smooth short-term fluctuations and enhance the continuity of the trend. This method can reduce noise interference and provide a more reliable basis for control. For example, when certain process paths change frequently during scheduling, G_avg helps to maintain a stable state.

[0041] According to the discriminant ΔG>λ `d_n` determines whether to activate the control strategy, where `λ` is a sensitivity factor, typically set within the range of [0.5, 1.5], to adjust the system's sensitivity to changes. This judgment evaluates whether the solution gradient change exceeds the expected fluctuation range. For example, if the solution points in a certain region change drastically due to uneven resource allocation, the judgment will trigger the control mechanism to adjust subsequent operations.

[0042] Through the above steps, this technical solution can effectively identify the regions in the optimization space that need to be adjusted, improve the scheduling performance under multi-objective trade-offs, avoid getting trapped in local optima, and enhance the overall robustness of the algorithm.

[0043] Next, the invention further describes the partitioning and control of the optimization space based on the gradient of the tradeoff function. First, a local convergence index ρ = (G_prev G_current) / G_prev is defined. This formula measures the relative proportion of the change in the target space between the current iteration and the previous iteration, where G_prev represents the value of the tradeoff function in the previous iteration, and G_current represents the value of the tradeoff function in the current iteration. The value of ρ ranges from 0 to infinity. If ρ is close to 1, it indicates a faster convergence speed; if ρ is small, it indicates smaller changes and a potential tendency towards stability. This index is set to assess the trend of change during the optimization process, providing a quantitative basis for subsequent judgments.

[0044] Secondly, an initial convergence threshold ε = 0.5 is set. This parameter serves as the initial standard for determining whether to trigger the adjustment mechanism. A value of 0.5 aims to balance convergence and stability, preventing premature triggering of adjustments and unnecessary consumption of computational resources. By setting a reasonable starting point, the relationship between the algorithm's initial exploration and later optimization can be balanced.

[0045] Next, the convergence metric ε_new = ε is updated using an exponential decay method. e^{β ε_new is defined as follows: ε_new ≈ t. Here, β is a coefficient controlling the decay rate, typically a negative value less than 0 to ensure that ε_new continuously decreases as the iteration number t increases. t represents the number of iterations, reflecting the stages the optimization process has passed through. This update mechanism allows the threshold to gradually tighten as iterations progress, improving the sensitivity of risk point identification while avoiding situations where a fixed threshold would lead to unsuitability.

[0046] Subsequently, if |ρ| > ε_new, a local adjustment mechanism is triggered to avoid risk points. When the current change ratio exceeds the current convergence threshold, it indicates that the system may have entered an unstable region or that there is a significant conflict between multiple objectives. This mechanism can effectively adjust the search strategy, keeping the algorithm away from positions that may get stuck in suboptimal solutions or cannot be further optimized, thereby improving the overall search efficiency and accuracy.

[0047] For example, in one embodiment, considering a multi-objective trade-off decision tree-mathematical programming hierarchical machining scheduling problem, if in a certain iteration it is found that the trade-off between total delay and energy consumption after task arrangement changes significantly (i.e., |ρ| exceeds the current threshold), an automatic local adjustment strategy is adopted, such as reallocating the priority of certain tasks or modifying equipment selection, thereby avoiding excessive delay or energy waste and achieving better overall performance. This technical solution can effectively enhance the robustness and optimization capability of the algorithm in dynamic environments.

[0048] Next, the invention further describes the partitioning and control of the optimization space based on the gradient of the tradeoff function. First, a dynamic gradient interval [G_min, G_max] is established. This interval represents the range of variation of the gradient of the tradeoff function in the optimization space. G_min represents the minimum possible gradient value, typically between 0 and 1, while G_max represents the maximum possible gradient value, also possibly between 0 and 1. Its optimal value is determined by the problem complexity and actual computational resources. Setting this interval helps identify regions with different sensitivities in the optimization space, thereby improving algorithm efficiency. For example, in machining scheduling, this interval can be dynamically adjusted based on changes in historical task completion time and cost.

[0049] A fuzzy variable μ is introduced to represent whether the current region is a transition zone. The value of μ ranges from [0, 1]. When μ is close to 0, it indicates that the region is in a stable region, and when it is close to 1, it indicates that the region is in a transition zone with significant changes. This allows the model to capture the characteristics of different regions more precisely, avoiding errors or redundant calculations caused by single processing. In one embodiment, μ is calculated based on a combination of task priority and resource matching degree.

[0050] A boundary determination formula, G_boundary = (G_max + G_min) / 2, is constructed to determine the boundary between high-sensitivity and low-sensitivity regions. This formula makes partitioning more flexible and adaptable to different optimization needs. If μ ≥ τ, this region is classified as a high-sensitivity region, and the sampling frequency is increased. τ is a preset threshold, typically 0.5, used to distinguish between high-sensitivity and non-high-sensitivity regions. For example, when scheduling multiple tasks, frequent sampling of high-sensitivity regions can significantly improve decision-making accuracy.

[0051] This technical solution achieves intelligent partitioning of the optimization space through gradient analysis, improves the adaptability and accuracy of the scheduling process, reduces unnecessary computational burden, and effectively addresses multi-objective trade-offs.

[0052] Next, the invention further describes the partitioning and control of the optimization space based on the gradient of the trade-off function. First, a quadratic function is used to describe the local terrain characteristics: f(x) = ax^2 + bx + c, where a, b, and c are obtained by fitting gradient information. a reflects the curvature of the curve, b represents the current gradient direction, and c is the initial value of the function at that point. By fitting the gradient data of key points in the optimization space, a quadratic function is constructed, enabling the model to more accurately reflect the changing trends of the local terrain. For example, in the machining scheduling problem, when dealing with multiple objective functions for different workpieces, the gradient data of a specific process point can be used to construct this quadratic function, thus providing a foundation for subsequent operations.

[0053] Next, we construct a criterion for determining the minimum value: min(f(x)) ≤ 0. By calculating the minimum value of this quadratic function within a local range, we determine whether a potential local minimum exists in that region. If this value is less than or equal to zero, it may indicate a local optimum trap. For example, in multi-objective scheduling optimization, if the minimum value of the combination of total processing time and energy consumption for a certain path satisfies the condition, we need to be wary of the possibility of a suboptimal solution in this region to prevent the algorithm from prematurely converging to a local optimum.

[0054] Subsequently, a penalty factor λ is introduced, making the objective function f_p = f(x) + λ `penalty_factor`. `λ` is a parameter used to adjust the severity of the penalty, typically taking a small value greater than zero, such as 0.1 or 0.5. `penalty_factor` is designed based on the risk of local extrema, used to enhance the attraction of areas far from unstable regions. This adjustment makes the objective function more robust throughout the search process.

[0055] Finally, the constraint f_p ≤ threshold ensures that local minima are avoided. The threshold is typically set as a safety boundary value to limit the penalized objective function from exceeding this upper limit, thus guiding the search towards a better global region. For example, in machining scheduling, if the objective function of a certain scheme exceeds the threshold after adjustment, it indicates that the scheme may be trapped in local optimization and should be excluded from further optimization. This setting helps improve the quality and stability of the scheduling results, thereby achieving better overall performance.

[0056] Next, the invention further describes the partitioning and control of the optimization space based on the gradient of the tradeoff function. First, a tolerance threshold for gradient change, T_tol = ΔG_avg, is set. δ, where ΔG_avg represents the historical average change value, which is derived from the gradient difference calculation results of the trade-off function in the past few runs. It is generally recommended to take the data from the most recent 10 to 50 runs to balance stability and response speed; δ is the dynamic tolerance factor, which is usually set between 0.5 and 1.5 to adjust the overall sensitivity. The optimal value can be selected through experimental verification; the setting of T_tol is intended to provide a dynamically adjustable reference line for the optimization space to avoid frequent triggering of partitioning strategies or misjudging unstable states.

[0057] Subsequently, if ΔG > T_tol in N consecutive runs, it is considered an unstable state. The value of N can be set to 3 to 10 depending on the system complexity to reduce the misjudgment rate caused by instantaneous fluctuations; ΔG is the difference between the gradient of the current tradeoff function and the previous time step, used to reflect the stability of the optimization process.

[0058] Next, a partitioning strategy is triggered, dividing the optimization space into dense and sparse categories based on the region density p. The calculation method of p is based on the ratio of the number of solutions within a region to the size of the space. Dense regions represent areas with high solution concentration and require more detailed processing, while sparse regions can have their constraints appropriately relaxed.

[0059] Finally, for dense regions, an adaptive mesh refinement algorithm is employed to reduce the risk of bias. This algorithm improves search accuracy by locally refining the mesh, thereby reducing the potential for local convergence problems in multi-objective optimization.

[0060] For example, in machining scheduling tasks, when the arrangement of certain processes involves multiple objective trade-offs leading to significant optimization fluctuations, the system identifies these fluctuations and determines them as unstable states. It then divides the current solution space into dense and sparse regions and performs refinement operations on the dense regions, thereby ensuring a more balanced, stable, and efficient scheduling scheme. This technical solution improves the robustness of the decision tree-mathematical programming joint method and achieves optimization quality assurance under dynamic trade-offs.

[0061] Next, the invention further describes the partitioning and control of the optimization space based on the gradient of the tradeoff function. First, the point cloud density ρ_density = 1 / dist_mean in the multi-objective space is calculated, where dist_mean is the average distance to the nearest neighbor. dist_mean represents the average distance from each point in the multi-objective space to its nearest neighbor. The smaller this value, the denser the point cloud; the larger ρ_density, the more concentrated the distribution of regions in the optimization space. The range of ρ_density depends on the point cloud distribution, and its optimal value should be within a moderate range to ensure that it is neither too concentrated nor too sparse, thereby improving the stability of the algorithm.

[0062] Next, we establish the condition for determining uneven distribution: dist_mean > σ. The threshold, where σ represents the standard deviation of the distribution, and threshold is a sensitivity adjustment factor, typically set between 1.5 and 2.5. This formula can be used to determine whether the point cloud is unevenly distributed, thereby identifying areas that may require adjustment.

[0063] In non-uniform regions, the Laplacian operator ∇²f is introduced to detect local curvature directions. ∇²f represents the curvature change of the function f at a certain point, and its positive or negative value can reflect the convexity / concavity characteristics. The sign of ∇²f is used to determine whether there are potential depressions. If so, an adjustment step is triggered to ensure that local optima or unreasonable regions are avoided during multi-objective spatial optimization.

[0064] For example, in a hierarchical machining scheduling method based on decision tree-mathematical programming that considers multiple objective trade-offs, if there is a significant conflict between workpiece scheduling time and resource utilization in a certain region, the above method can be used to identify and adjust that region, thereby improving the overall optimization effect of the scheduling scheme. This adjustment method can more precisely balance multiple objective variables in a multidimensional optimization space, reduce unnecessary resource waste, and improve scheduling efficiency and feasibility.

[0065] Next, the invention further describes the partitioning and control of the optimization space based on the gradient of the trade-off function. First, a smooth transition region is constructed according to a preset multi-objective trade-off function. The space is then segmented using the Sigmoid function S(t) = 1 / (1 + e^{k(t_c)}), where k is a slope parameter that adjusts the steepness of the transition, typically ranging from 1 to 50, with the optimal value adjusted based on the actual scheduling complexity. t_c is the decision boundary point, usually corresponding to the time threshold of a key processing indicator, such as the median of the process completion time. This function makes the partitioning of the optimization region more continuous and natural, avoiding the instability caused by traditional discrete partitioning.

[0066] Next, an activation threshold S(t) > S_threshold is set, where S_threshold is typically a value between 0.5 and 0.8 to distinguish different regions. This setting ensures that only regions with sufficiently obvious gradient changes are activated, thereby reducing unnecessary subcategories.

[0067] Then, the activation points are used as partition boundaries, and the names of each region are recorded, so that different sub-regions can be modeled and optimized independently, which facilitates the formulation of subsequent hierarchical scheduling strategies.

[0068] Finally, based on the shape parameters σ of the activated regions, such as variance or density indices, it is determined whether regions need to be merged or split, thereby improving the adaptability of the scheduling method to dynamic environments. For example, in the scheduling tasks of a machining workshop, the delivery time and resource requirements of different work orders exhibit a non-linear relationship. This technology can automatically identify several regions with similar performance, reducing model complexity and improving decision-making efficiency. This process effectively balances multiple conflicting objectives, improving the quality and executability of the overall scheduling scheme.

[0069] The hierarchical machining scheduling method based on decision tree and mathematical programming, considering multi-objective trade-offs, of this invention includes the following steps: First, a mathematical programming model is constructed, encompassing multiple optimization objectives, typically including production efficiency, equipment utilization, process stability, and cost control. Then, a framework based on a decision tree structure is established to systematically express the relationships between different optimization objectives and possible priority combinations. Within this framework, a multi-objective trade-off function is generated to quantify the interactions and potential conflicts between different optimization objectives, and its gradient information is calculated to reflect the changing trends and sensitivity of the objective space. Next, the entire optimization space is partitioned and controlled based on the gradient information; that is, the optimization problem is divided into several local regions according to the gradient characteristics of the trade-off function, and the optimization paths and feasible solution sets in each region are dynamically adjusted. During this process, regions where local degradation may occur (such as local minima, inefficient regions, etc.) are identified and avoided, ensuring that the optimization direction always moves towards the overall optimum or Pareto front. Finally, in the effectively controlled optimization space, a mathematical programming algorithm is used to solve for the final machining scheduling scheme, thereby improving the robustness and adaptability of the scheduling strategy while satisfying multiple optimization objectives.

[0070] To address the problem of local degradation of the objective function by partitioning and regulating the multi-objective optimization space based on the gradient of the tradeoff function, this invention employs a gradient-driven space partitioning mechanism. By calculating the partial derivatives of the tradeoff function with respect to different variables, the optimization potential and risk level in each direction can be determined. Based on this information, the optimization space is divided into different sub-regions according to their physical distribution or logical correlation, and targeted regulation strategies are applied to each sub-region. For example, if gradient directions converge in a certain region, leading to local optima, a penalty term can be introduced or the objective weights adjusted to guide the search out of that region; conversely, if gradient fluctuations are significant in another region, convergence can be ensured by refining the boundaries or strengthening local optimization techniques. This dynamic partitioning and regulation mechanism enables the entire optimization process to adaptively adjust strategies, reducing the risk of local degradation while improving the algorithm's global optimization capability.

[0071] In the above embodiments, implementation can be achieved, in whole or in part, through software, hardware, firmware, or any combination thereof. When implemented in software, it can be implemented, in whole or in part, as a computer program product. A computer program product includes one or more computer instructions. When the computer program instructions are loaded and executed on a computer, all or part of the flow or function according to this application is generated. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device. The computer instructions can be stored in a computer-readable storage medium or transferred from one computer-readable storage medium to another.

[0072] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of this application. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of those different embodiments or examples.

[0073] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one of that feature. In the description of this application, "a plurality of" means two or more, unless otherwise explicitly specified.

[0074] Any process or method description in the flowchart or otherwise herein can be understood as representing a module, segment, or portion of code comprising one or more executable instructions for implementing a particular logical function or process. Furthermore, the scope of the preferred embodiments of this application includes additional implementations in which functions may be performed not in the order shown or discussed, including substantially simultaneously or in reverse order depending on the functionality involved.

[0075] The logic and / or steps represented in the flowchart or otherwise described herein, for example, can be considered as a sequenced list of executable instructions for implementing logical functions, and can be embodied in any computer-readable medium for use by, or in conjunction with, an instruction execution system, apparatus or device (such as a computer-based system, a processor-included system or other system that can fetch and execute instructions from, an instruction execution system, apparatus or device).

[0076] It should be understood that various parts of this application can be implemented using hardware, software, firmware, or a combination thereof. In the above embodiments, multiple steps or methods can be implemented using software or firmware stored in memory and executed by a suitable instruction execution system. All or part of the steps of the methods in the above embodiments can be implemented by a program instructing related hardware, the program being stored in a computer-readable storage medium, which, when executed, includes one or a combination of the steps of the method embodiments.

[0077] Furthermore, the functional units in the various embodiments of this application can be integrated into a processing module, or each unit can exist physically separately, or two or more units can be integrated into a module. The integrated module can be implemented in hardware or as a software functional module. If the integrated module is implemented as a software functional module and sold or used as an independent product, it can also be stored in a computer-readable storage medium. This storage medium can be a read-only memory, a disk, or an optical disk, etc.

[0078] The above are merely specific embodiments of this application, but the scope of protection of this application is not limited thereto. Any person skilled in the art can easily conceive of various variations or substitutions within the technical scope disclosed in this application, and these should all be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.

Claims

1. A hierarchical machining scheduling method based on decision tree and mathematical programming considering multi-objective trade-offs, characterized in that... include: Establish an optimization model containing multiple objective functions and construct a decision tree structure; A multi-objective trade-off function is generated based on the decision tree structure, and its gradient information is calculated; The optimization space is partitioned and controlled based on the gradient of the aforementioned tradeoff function in order to identify and avoid localized degradation regions. Mathematical programming is performed in the optimized space after adjustment to obtain the optimal machining scheduling scheme.

2. The hierarchical machining scheduling method based on decision tree mathematical programming considering multi-objective trade-offs as described in claim 1, characterized in that, The generation of the multi-objective tradeoff function based on the decision tree structure further includes: Extract the path weight α_i of each node in the decision tree; Calculate the correlation between the path and the objective function θ_i = ∑(w_j g_j), where w_j represents the weight coefficient of the j-th target, and g_j represents the target value under this path; Construct a multi-objective weighted function f_w = ∑(α_i) θ_i); The path is judged as a good path based on the condition f_w > threshold_w.

3. The hierarchical machining scheduling method based on decision tree mathematical programming considering multi-objective trade-offs as described in claim 2, characterized in that, The step of partitioning and regulating the optimization space based on the gradient of the tradeoff function further includes: Obtain the gradient matrix G; Calculate the maximum principal curvature κ = max(σ(G)) at each point, where σ(G) represents the singular value of G; Construct a threshold Δ = γ max(κ), where γ is an adjustment coefficient ranging from 1 to 0.5; If κ > Δ, then this region is considered to be in a locally deteriorated region and is marked.

4. The hierarchical machining scheduling method based on decision tree mathematical programming considering multi-objective trade-offs according to claim 3, characterized in that, The step of partitioning and regulating the optimization space based on the gradient of the tradeoff function further includes: Calculate the gradient change ΔG at the current solution point; Construct the neighborhood distance d_n = ∑|x_i x_j| / N, where x_i and x_j are the coordinates of the nearest solution points, and N is the neighborhood size; The stable gradient trend G_avg = (G + G_prev) / 2 is calculated using the moving average formula. According to the discriminant ΔG > λ d_n determines whether to activate the control strategy, and λ is the sensitivity factor.

5. The hierarchical machining scheduling method based on decision tree mathematical programming considering multi-objective trade-offs according to claim 4, characterized in that, The step of partitioning and regulating the optimization space based on the gradient of the tradeoff function further includes: Define the local convergence index ρ = (G_prev G_current) / G_prev; Set the initial convergence threshold ε = 0.5; The convergence metric ε_new = ε is updated using an exponential decay method. e^{β t}, where β is the decay coefficient and t is the number of iterations; If |ρ| > ε_new, then a local adjustment mechanism is triggered to avoid risk points.

6. The hierarchical machining scheduling method based on decision tree mathematical programming considering multi-objective trade-offs according to claim 5, characterized in that, The step of partitioning and regulating the optimization space based on the gradient of the tradeoff function further includes: Establish the dynamic gradient interval [G_min, G_max]; A fuzzy variable μ is introduced to represent whether the current region is a transition zone; The boundary determination formula is constructed as G_boundary = (G_max + G_min) / 2, where μ ∈ [0, 1] represents the degree of region transition; If μ ≥ τ, then this region is designated as a high-sensitivity region, and the sampling frequency is increased.

7. The hierarchical machining scheduling method based on decision tree mathematical programming considering multi-objective trade-offs as described in claim 6, characterized in that, The step of partitioning and regulating the optimization space based on the gradient of the tradeoff function further includes: Local terrain characteristics are described using a quadratic function f(x) = ax^2 + bx + c, where a, b, and c are obtained by fitting gradient information. Construct a minimum value determination formula and a criterion for min(f(x)) ≤ 0; Introduce a penalty factor λ, such that the objective function f_p = f(x) + λ penalty_factor; By using the constraint f_p ≤ threshold, we ensure that we avoid local minima.

8. The hierarchical machining scheduling method based on decision tree mathematical programming considering multi-objective trade-offs according to claim 7, characterized in that, The step of partitioning and regulating the optimization space based on the gradient of the tradeoff function further includes: Set the tolerance threshold for gradient change as T_tol = ΔG_avg δ, where ΔG_avg is the historical average change value, and δ is the dynamic tolerance factor; If ΔG > T_tol in N consecutive runs, it is considered an unstable state; Triggering partitioning strategy: Divide regions into dense and sparse categories based on their density p; For dense areas, an adaptive mesh refinement algorithm is enabled to reduce the risk of deviation.

9. The hierarchical machining scheduling method based on decision tree mathematical programming considering multi-objective trade-offs as described in claim 8, characterized in that, The step of partitioning and regulating the optimization space based on the gradient of the tradeoff function further includes: Calculate the point cloud density ρ_density = 1 / dist_mean in the multi-object space, where dist_mean is the average distance of the nearest neighbor; Establish the condition for determining uneven distribution: dist_mean > σ threshold; Introducing the Laplacian operator ∇²f in non-uniform regions to detect local bending directions; Combine the ∇²f symbol to determine if there is a potential depression; if so, trigger the adjustment step.

10. The hierarchical machining scheduling method based on decision tree mathematical programming considering multi-objective trade-offs according to claim 9, characterized in that, The step of partitioning and regulating the optimization space based on the gradient of the tradeoff function further includes: The smooth transition region is constructed using the Sigmoid function: S(t) = 1 / (1 + e^{k(t t_c)}), where k controls the slope and t_c is the critical value; The activation threshold S(t) > S_threshold is set to determine the activation condition; Use the activation point as the partition boundary and record the region name; Determine whether to merge or split based on the shape parameter σ of the active region.

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