Water resource suitable carrying capacity assessment method based on double-layer interval programming

Abstract

The application discloses a water resource suitable bearing capacity evaluation method based on double-layer interval programming, and comprises the following steps: acquiring resource and environment configuration interval data set of a specified region, and constructing a double-layer interval multi-objective programming model; solving the model to extract optimal decision variables and optimal target evaluation value, and acquiring comprehensive sensitivity indexes of each decision variable based on the optimal decision variables; determining self-adaptive tolerance of each decision variable based on the comprehensive sensitivity indexes, and constructing a decision variable membership function; constructing satisfaction constraints by using the decision variable membership function and the optimal target evaluation value, combining the comprehensive sensitivity indexes to execute sequential optimization solving, and obtaining final decision variable interval solution reflecting water resource bearing state of the specified region. The application can effectively reconcile conflicts between hierarchical targets, and improve solving precision and robustness of the evaluation model under multiple uncertain boundary conditions.

Description

Technical Field

[0001] This invention relates to the field of water resources planning and management technology, and in particular to a method for assessing the suitability carrying capacity of water resources based on two-level interval planning. Background Technology

[0002] Accurately assessing the suitable carrying capacity of regional water resources is fundamental to achieving optimal allocation of water resources and coordinated development among multiple sectors. In complex regional water resource systems, not only is overall efficiency optimization at the macro-management level involved, but it is also constrained by production efficiency and environmental capacity at the micro-use level. Furthermore, various input parameters often exhibit significant numerical fluctuations due to statistical errors, observation conditions, and natural variations. Therefore, constructing an optimization model that can realistically reflect the game relationships among multiple levels of stakeholders and effectively handle multidimensional uncertainties is of significant engineering application value for scientifically identifying the carrying capacity boundaries of the system.

[0003] Currently, conventional fuzzy programming or traditional two-level multi-objective evaluation models are commonly used for multi-level water resource optimization problems involving uncertainty. These methods typically construct a multi-dimensional evaluation system, then directly set fixed boundary buffer ranges for constraints and objective functions, and use conventional minimum aggregation operators to transform multi-level objectives into single-level single objectives for overall optimization calculation. However, when facing complex situations with extremely uneven distribution of benefits among hierarchical objectives or significant differences in the activity ranges of various system variables, this approach, relying on static empirical parameters and uniform dimensionality reduction, struggles to reflect the differences in the intensity of different variables' influence on the system output. In handling sharp conflicts, it can easily lead to the excessive sacrifice of key objectives or the inability of the solution process to converge.

[0004] Existing methods for reconciling the inherent conflicts in multi-level complex systems still suffer from significant shortcomings in adapting to the dynamic characteristics of the systems and in robustness in complex game-theoretic environments. How to objectively and precisely allocate the system's equilibrium adjustment space under multiple uncertain boundaries, and ensure the stability and reliability of the multi-level optimization process, is a common technical challenge that urgently needs to be addressed in this field. Therefore, it is necessary to investigate a method that can improve the robustness of evaluation and solution for complex multi-level systems. Summary of the Invention

[0005] Therefore, this application provides a water resources suitability carrying capacity assessment method based on two-layer interval planning, in order to solve the above-mentioned problems of the prior art.

[0006] To achieve the above objectives, the first aspect of this application provides a method for assessing the suitability carrying capacity of water resources based on two-level interval planning, comprising:

[0007] Acquire and construct a two-layer interval multi-objective programming model based on the resource and environmental configuration interval data set of a specified area;

[0008] The two-level interval multi-objective programming model is solved to extract the optimal decision variables and the optimal objective evaluation values, and the comprehensive sensitivity index of each decision variable is obtained based on the optimal decision variables.

[0009] The adaptive tolerance of each decision variable is determined based on the comprehensive sensitivity index, and the membership function of the decision variable is constructed.

[0010] Satisfaction constraints are constructed using the membership functions of decision variables and the optimal objective evaluation value. Sequential optimization is then performed in conjunction with the comprehensive sensitivity index to obtain the final decision variable interval solution reflecting the water resource carrying capacity of the specified area.

[0011] A second aspect of this application provides a water resource suitability carrying capacity assessment device based on two-layer interval planning, comprising:

[0012] The data acquisition module is used to acquire a set of resource and environmental configuration data for a specified area.

[0013] The model building module is used to construct a two-level interval multi-objective programming model based on the resource and environment configuration interval data set;

[0014] An independent solution module is used to solve a two-level interval multi-objective programming model and extract the optimal decision variables and the optimal objective evaluation value.

[0015] The sensitivity assessment module is used to obtain the comprehensive sensitivity index of each decision variable based on the optimal decision variable;

[0016] The tolerance configuration module is used to determine the adaptive tolerance of each decision variable based on the comprehensive sensitivity index and to construct the membership function of the decision variable.

[0017] The sequential optimization module is used to construct satisfaction constraints using the membership function of decision variables and the optimal target evaluation value, and to perform sequential optimization solution in combination with the comprehensive sensitivity index to obtain the final decision variable interval solution reflecting the water resource carrying capacity of the specified area.

[0018] This device is used to perform a water resource suitability carrying capacity assessment method based on two-layer interval planning proposed in this invention.

[0019] A third aspect of this application provides an electronic device, comprising:

[0020] Memory, which stores executable programs;

[0021] A processor is used to run the program, wherein the program executes a water resource suitability carrying capacity assessment method based on two-layer interval planning proposed in this invention.

[0022] The fourth aspect of this application provides a computer-readable storage medium storing computer program instructions, which, when executed by a processor, implement a water resource suitability carrying capacity assessment method based on two-layer interval planning proposed in this invention.

[0023] Based on the above technical solutions, conflicts between hierarchical objectives can be effectively reconciled, and the solution accuracy and robustness of the evaluation model under multiple uncertainty boundary conditions can be improved. Attached Figure Description

[0024] The accompanying drawings, which form part of this application, are used to provide a further understanding of this application. The illustrative embodiments and descriptions of this application are used to explain this application and do not constitute an undue limitation of this application. In the drawings:

[0025] Figure 1 This is a schematic diagram of a water resource suitability carrying capacity assessment method based on two-layer interval planning provided in this application.

[0026] Figure 2 This is a schematic diagram of the process for solving a two-level interval multi-objective programming model and extracting the optimal decision variables and the optimal objective evaluation value, provided in this application.

[0027] Figure 3 This is a schematic diagram of the adaptive tolerance process for determining each decision variable based on a comprehensive sensitivity index, provided in this application.

[0028] Figure 4 This is a schematic diagram of the process for obtaining the upper-layer relaxation amount and the lower-layer relaxation amount provided in this application.

[0029] Figure 5 This is a schematic diagram of the process for constructing the membership function of decision variables provided in this application. Detailed Implementation

[0030] Unless otherwise stated, the specific values ​​of the parameters in this specification can be adaptively adjusted according to the data scale and accuracy requirements in the actual application scenario. Those skilled in the art can determine the appropriate values ​​without creative effort.

[0031] Example 1

[0032] This embodiment provides a method for assessing the suitability carrying capacity of water resources based on two-level interval planning, such as... Figure 1 As shown, the method may include the following steps:

[0033] Step 101: Obtain the set of resource and environment configuration interval data for the specified area.

[0034] Specifically, the resource and environmental allocation interval dataset can refer to a set of basic assessment parameters containing uncertainties within the system. In this embodiment, due to limitations of objective conditions or statistical errors, system parameters are usually difficult to obtain precise point values ​​and are instead presented in interval form. The resource and environmental allocation interval dataset specifically includes the water resource supply, water demand, land use area, socio-economic statistical indicators, and water environmental capacity of each computing unit. This step provides a physical parameter basis containing uncertainty boundaries for subsequent model building.

[0035] Furthermore, in practical engineering applications, the aforementioned set can be obtained by retrieving pre-stored historical hydrological yearbook data and combining it with hydrological elements collected in real time by current sensors. For example, the available surface water volume in a certain area can be represented as an array of intervals with clear lower and upper limits, thereby characterizing the fluctuation range between the dry and wet seasons.

[0036] Step 102: Based on the resource and environment configuration interval data set, construct a two-layer interval multi-objective programming model.

[0037] In this embodiment, the two-level interval multi-objective programming model is a mathematical optimization model with a leader-follower hierarchical game structure, where parameters and decision variables exhibit interval characteristics. Specifically, the model comprises an upper-level interval model and a lower-level interval model. The upper-level interval model is typically oriented towards comprehensive regional benefits, for example, setting the objective function to maximize the regional population carrying capacity, with the balance of total water resource supply and demand and land physical boundaries as core constraints. The lower-level interval model focuses on the synergy between production and the environment, setting the objective function to maximize agricultural food output and minimize non-point source pollutant emissions, and is controlled by specific agricultural water quotas and environmental pollution carrying capacity constraints.

[0038] The two-tier model structure is adopted because traditional single-tier multi-objective optimization cannot reflect the priority differences between macro-level planning managers and micro-level resource users. Through the two-tier architecture, the upper-level managers prioritize the allocation of global resources, while the lower-level users perform local optimizations within their given resource quotas, thus realistically simulating the hierarchical mechanism of resource allocation in reality.

[0039] Step 103: Solve the two-level interval multi-objective programming model, extract the optimal decision variables and the optimal objective evaluation value, and obtain the comprehensive sensitivity index of each decision variable based on the optimal decision variables.

[0040] In this step, the two-layer model needs to be dimensionality-reduced to obtain the baseline state. Specifically, the objective functions and constraints of the upper and lower layers are split into lower-boundary sub-models and upper-boundary sub-models based on the interval boundaries, and optimization is performed independently for each. After the solution is completed, the decision results that enable the objective function to reach its extreme value in each independent state are extracted; these are the optimal decision variables. Simultaneously, the evaluation value achieved by the objective function at this point is recorded; this is the optimal objective evaluation value.

[0041] Furthermore, the comprehensive sensitivity index is used to quantify the overall impact of small fluctuations in each decision variable on the values ​​of the objective functions at both the upper and lower levels. By applying small perturbations near the optimal decision variables in the upper-level interval model and calculating the rate of change of the objective function, it is possible to identify which variables play a decisive role in the system state. Obtaining this index provides objective data support for subsequent differentiated allocation of adjustment space, avoiding the subjective bias caused by manually specifying parameters.

[0042] Step 104: Determine the adaptive tolerance of each decision variable based on the comprehensive sensitivity index, and construct the membership function of the decision variables.

[0043] Furthermore, the adaptive tolerance of each decision variable is determined based on the comprehensive sensitivity index and the optimal decision variable, and the membership function of the decision variable is constructed.

[0044] Specifically, adaptive tolerance refers to the acceptable range by which system state variables deviate from their optimal ideal values ​​during the fuzzy programming solution process. In this embodiment, the traditional approach of using a uniform fixed tolerance for all variables is abandoned, and a mapping mechanism is established between the comprehensive sensitivity index and the tolerance range. The basic principle of the mapping is negative correlation control, that is, for highly sensitive variables with large comprehensive sensitivity index values, an extremely narrow adaptive tolerance is automatically assigned to them to strictly lock their value range and prevent small deviations from causing drastic deterioration of the objective function; for low-sensitivity variables, a wider adaptive tolerance is assigned to release adjustment space for the model to compromise and balance.

[0045] Based on this, the optimal decision variable is used as the geometric peak center, and the adaptive tolerance is used as the geometric half-width to construct the membership function of the decision variable. Specifically, this function is a linear piecewise line function where the output is 1 when the decision variable is exactly at its optimal value, decreases linearly with increasing offset, and drops directly to 0 when the tolerance boundary is exceeded. This function completes the quantitative transformation from physical tolerance to fuzzy logic evaluation value.

[0046] Step 105: Construct satisfaction constraints using the membership function of decision variables and the optimal target evaluation value, and perform sequential optimization by combining the comprehensive sensitivity index to obtain the final decision variable interval solution reflecting the water resource carrying capacity of the specified area.

[0047] In this embodiment, the satisfaction constraint can be used to regulate the bottom line of the model during the compromise process. By normalizing the optimal target evaluation value with the target value of the current solution, parameters reflecting the degree to which the optimal expectation is achieved are obtained. To maintain the physical meaning of the two-layer architecture, sequential optimization is performed. Sequential optimization is an optimization process that proceeds in stages according to hierarchical priority.

[0048] Specifically, priority is given to maximizing the satisfaction of the upper-level objectives. While ensuring that the satisfaction of the upper-level objectives is not compromised excessively, the satisfaction of the lower-level objectives is optimized. Then, the membership degree of the decision variables themselves is weighted and optimized by combining the comprehensive sensitivity index.

[0049] Through the aforementioned multi-stage ordered calculations and hierarchical concessions, the model finds a feasible solution space that satisfies global constraints amidst conflicts among various parties. The final decision variable interval solution output contains the upper and lower limits of the population or industrial scale that the region can support under the predetermined water use structure, and also directly maps the suitable carrying capacity of the region's water resources. Compared to traditional single-point prediction, this interval solution provides a clear risk elasticity buffer.

[0050] According to one aspect of this application, a method for assessing water resource suitability carrying capacity based on two-level interval planning may further include the following steps:

[0051] Obtain the set of resource and environment configuration range data for each computing unit in the specified region;

[0052] A two-level interval multi-objective programming model is constructed based on the resource and environment allocation interval data set. The two-level interval multi-objective programming model includes an upper interval model that reflects the comprehensive benefits of the region and a lower interval model that reflects the production benefits and environmental load.

[0053] The upper-level interval model and the lower-level interval model are solved independently to extract the optimal decision variables and optimal target evaluation values ​​for the corresponding levels.

[0054] Based on the optimal decision variables of the upper-level interval model, perturbation calculation is performed to obtain the comprehensive sensitivity index of each decision variable for the two-level interval multi-objective programming model.

[0055] Based on the comprehensive sensitivity index, the baseline adjustment space determined by the difference between the optimal decision variables between layers is mapped and controlled to determine the adaptive tolerance of each decision variable, and the membership function of the decision variable is constructed based on the adaptive tolerance.

[0056] While maintaining the hierarchical priority sequence, satisfaction constraints are constructed using the membership function of decision variables and the optimal target evaluation value. Based on the comprehensive sensitivity index, the inter-level concession range is adjusted to perform multi-stage sequential optimization to obtain the final decision variable interval solution reflecting the regional water resource carrying capacity.

[0057] Example 2

[0058] Based on the above embodiment 1, the establishment of the two-layer interval multi-objective programming model, the pre-solution of the underlying physical constraints, and the dimensionality reduction decomposition process of the model are further described in detail.

[0059] In one possible implementation, constructing a two-level interval multi-objective programming model based on a set of resource and environmental allocation interval data may include the following steps:

[0060] Step 201: Construct an upper-level interval model using the resource and environment allocation interval data set. The upper-level interval model uses an upper-level objective function to represent the comprehensive benefits of the region and includes corresponding upper-level constraints.

[0061] Specifically, the decision variables for the upper-level interval model are configured as the population size of each computational unit and the water quota for each sector. The regional comprehensive benefit is specifically manifested as the total population carrying capacity of the entire region under predetermined resource conditions. When constructing the objective function, the population size interval variables of each computational unit are summed.

[0062] φ 1_± =∑(x pop,k_± );

[0063] Where, φ 1_± x represents the value of the upper-level objective function, symbolizing the range of the region's total population carrying capacity. pop,k_± Let be the population size interval variable of the k-th computational unit, and ∑() be the summation function operation across computational units.

[0064] Furthermore, upper-level constraints can refer to physical limitations imposed from a macro-planning perspective. These specifically include constraints on water resource supply and demand balance and land carrying capacity limits. The system determines that the total water consumption quota of each water-using sector within each calculation unit does not exceed the system's total available water supply range, and stores all parameters in mathematical inequality form.

[0065] Step 202: Construct a lower-level interval model using the resource and environment allocation interval data set. The lower-level interval model represents production efficiency and environmental load with the lower-level objective function and includes the corresponding lower-level constraints.

[0066] In this embodiment, the lower-level interval model focuses on the execution efficiency of micro-level activities within the region. Decision variables are configured as agricultural planting area allocation and irrigation or industrial water consumption for each industry. The lower-level model has two objective functions with different directions. The first objective is to maximize production efficiency.

[0067] φ 2_± =∑(y k_± *A agr,k_± );

[0068] Where, φ 2_± Let y be the value of the first objective function in the lower layer, representing the range of total grain output. k_± Let A be the interval parameter for the unit area yield of the k-th calculation unit. agr,k_± This represents the range of agricultural land area for the corresponding calculation unit.

[0069] The secondary objective at the lower level is to minimize the environmental load, and its expression is:

[0070] φ env_± =∑(E agr,k_± +E ind,k_± );

[0071] Where, φ env_± E represents the value of the second objective function at the lower level, characterizing the range of total regional pollution emissions. agr,k_± E represents the range of agricultural non-point source pollution emissions for the k-th computational unit. ind,k_± This represents the range of industrial pollution emissions for the k-th calculation unit.

[0072] Correspondingly, lower-level constraints may include ensuring that agricultural and industrial water consumption does not exceed the sectoral water quotas assigned by the upper-level interval model, and that the total amount of all pollution emissions does not exceed the maximum pollution carrying capacity of the water environment. The upper and lower-level models form a mathematically coupled hierarchical relationship through shared resource quota variables.

[0073] Optionally, before solving the two-level interval multi-objective programming model, the following may also be included:

[0074] The upper-level and lower-level constraints are transformed into deterministic boundaries to construct a joint constraint space.

[0075] Before performing sensitivity analysis and subsequent calculations, it is necessary to extract the activity range of the decision variables under extreme physical conditions. All constraint inequalities with interval coefficients are subjected to boundary extremum processing. For constraint inequalities with less than or equal to signs, the lower bound of the absolute value of the interval of the coefficient variables on the left is truncated, and the upper bound of the interval of the resource constraint parameters on the right is truncated, forming the most relaxed deterministic physical boundary of the constraints. After this transformation, the original fuzzy interval intersection is mapped to a deterministic convex polyhedron in the real number field, i.e., the joint constraint space.

[0076] Furthermore, under the joint constraint space, extreme value optimization is performed independently for each decision variable in advance to determine the maximum and minimum values ​​of the corresponding decision variable under all constraints.

[0077] For a total of n decision variables, construct and execute 2n independent linear programming tasks. In any task, maximize or minimize a single decision variable as the objective function, with constraints defined by a joint constraint space. After execution, extract the values ​​at which the objective calculation reaches the upper and lower boundaries, respectively, as the maximum and minimum values.

[0078] Based on this, the feasible region width of each decision variable is obtained by taking the maximum and minimum values. The pre-calculated feasible region width is used to provide a global physical scale reference for each decision variable.

[0079] R j =x jub -x jlb ;

[0080] Among them, R j x is the width of the feasible region of the j-th decision variable in the pre-calculation. jub For the maximum value of the corresponding decision variable, x jlb This represents the minimum value of the corresponding decision variable.

[0081] Obtaining the feasible region width is to extract a unified physical scale for variables of different dimensions and orders of magnitude in the system. This scale will serve as a global reference baseline for resisting numerical perturbation failures and tolerance shrinkage degradation in subsequent operations.

[0082] One possible implementation involves solving a two-level interval multi-objective programming model to extract the optimal decision variables and optimal objective evaluation values, such as... Figure 2 As shown, it includes:

[0083] Step 1: For the upper-level interval model and the lower-level interval model, introduce regularization factors to eliminate the difference in dimensions and construct dimensionless upper-level evaluation functions and lower-level evaluation functions.

[0084] In this step, the ideal point normalization algorithm is used to unify objectives containing different physical dimensions. For upper-level objective functions that exhibit maximization characteristics, they are transformed into minimization evaluation functions.

[0085] f 1_± =(φ 1_ideal_± -φ 1_± ) / (φ 1_ideal_± -φ 1_worst_± );

[0086] Among them, f 1_± For the upper-level evaluation function, φ 1_ideal_± φ represents the range of optimal ideal values ​​achieved when the objective at this layer is independently optimized. 1_worst_± This represents the worst-case range when the objective function degenerates to the worst-case condition. When the original objective function reaches its optimum, the output value of this evaluation function is 0.

[0087] For a lower-level interval model containing two sub-objectives, after mapping in the same way, the inverse proportion of the number of objectives is used as an equal weighting factor to perform an algebraic sum operation on the two objectives, resulting in a lower-level evaluation function with a unified scale.

[0088] Step 2: Decompose the two-level interval multi-objective programming model into an upper-level interval programming model and a lower-level interval programming model according to the hierarchy.

[0089] Specifically, the two-level interval multi-objective programming model is decoupled hierarchically into a structurally independent upper-level interval programming model and a lower-level interval programming model.

[0090] Since the two-layer coupling presents non-convex and nonlinear computational obstacles, the nested constraint calls between layers are blocked in the computation graph structure, decoupling the original model into two structurally independent single-layer interval programming models, so as to provide local gradient evaluation reference points in the current step.

[0091] Step 3: Based on the upper-level evaluation function and the lower-level evaluation function, the upper-level interval programming model and the lower-level interval programming model are respectively decomposed into the lower limit sub-model of the objective function and the upper limit sub-model of the objective function for solving.

[0092] This step employs a two-step method to reduce the dimensionality of the intervals. Based on the positive and negative attributes of the coefficients of the decision variables in the objective function, all decision variables are divided into positive and negative sets. When constructing the lower bound sub-model of the objective function, variables in the positive set are set to the lower bound of the interval, and variables in the negative set are set to the upper bound of the interval. The constraints are then aligned with the most favorable boundary, and linear programming operations are performed. After obtaining the lower bound solution, this solution is used as known conditions to reverse the direction of variable selection and tighten the constraints, constructing and solving the upper bound sub-model of the objective function.

[0093] Step 4: Extract the optimal decision variables and optimal target evaluation values ​​for the corresponding levels from the solution results of each sub-model.

[0094] The real scalar solutions output by the lower and upper bound sub-models of the objective function are merged and recombined. The upper and lower bounds of the corresponding variables are extracted and concatenated to restore the original data structure into an interval array. This yields the optimal decision variables and optimal objective evaluation values ​​necessary for subsequent sensitivity benchmark verification and spatial mapping.

[0095] Example 3

[0096] Based on the above embodiment 2, this embodiment further describes the decision variable sensitivity extraction mechanism in the solution process of the two-layer interval multi-objective programming model, and focuses on the bidirectional perturbation evaluation mechanism of constraint perception and endpoint alignment.

[0097] In one possible implementation, obtaining the comprehensive sensitivity index of each decision variable based on the optimal decision variable may include the following steps:

[0098] Step 301: For each decision variable, the optimal decision variable of the upper-level interval model is used as the benchmark point, and the perturbation amount is determined based on the interval solution width of the decision variable; when the interval solution width degenerates to below the preset minimum perturbation threshold, a fixed proportion of the pre-calculated feasible region width jointly determined by all constraints of the decision variable is used as the replacement perturbation amount.

[0099] Specifically, sensitivity analysis quantifies the impact of input changes on output by introducing small perturbations. In interval programming, if the optimal solution interval for a decision variable is extremely narrow, or even degenerates into a single point, directly truncating the interval width as the perturbation step size at a fixed ratio will cause numerical differentiation to fail. To prevent this degradation, a global physical scale is introduced for fallback calculations.

[0100] When x jU + -x jU - >β0·R j When, Δx j =α·(x jU + -x jU - );

[0101] When x jU + -x jU - ≤β0·R j When, Δx j =α·R j ;

[0102] Where, Δx j Let x be the perturbation of the j-th decision variable, α be the preset perturbation scaling factor, and x be the perturbation scaling factor. jU + x represents the upper limit of the optimal decision variables in the upper-level interval model. jU - R represents the lower bound of the optimal decision variables in the upper-level interval model, β0 is the minimum perturbation threshold coefficient, and R0 is the lower bound of the optimal decision variables in the upper-level interval model. j Let be the width of the feasible region for the j-th decision variable before calculation.

[0103] In this embodiment, the perturbation scaling factor α is set to 0.03, and the minimum perturbation threshold factor β0 is set to 0.005. This decision branch ensures that the system can obtain an effective perturbation amplitude sufficient to drive the numerical difference under extreme output conditions.

[0104] Step 302: Apply a perturbation at the reference point to construct a perturbation point. Substitute the perturbation point into the upper-level evaluation function and the lower-level evaluation function respectively to calculate the rate of change of the objective function and obtain the upper-level sensitivity coefficient and lower-level sensitivity coefficient of each decision variable for each level of objective.

[0105] In one possible embodiment, the process of applying a perturbation at a reference point to construct a perturbation point specifically includes:

[0106] Step ①: After initially constructing the perturbation points, perform a constraint feasibility check.

[0107] Specifically, after determining the perturbation quantity of the theory, test coordinate points are constructed in multidimensional space along the dimension of a single variable.

[0108] z new =z base +d·ρ·Δx j ·e j ;

[0109] Among them, z new For the initially constructed perturbation point vector, z base Let d be the reference point vector, ρ be the perturbation direction coefficient, and ρ be the scaling factor used to control the perturbation amplitude. j e is the corresponding perturbation quantity. j Let j be a unit vector whose j-th component is 1 and all other components are 0.

[0110] Because the original system contains numerous rigid boundaries defined by resource quotas and environmental capacity, the coordinates after perturbation are prone to cross these boundaries and enter infeasible regions. If the evaluation function is calculated within the infeasible region, the resulting gradient information will be a distorted virtual influence. Therefore, the initially constructed perturbation points must be substituted into all preset upper and lower constraints for logical judgment to verify whether they violate the upper or lower limits of physical capacity.

[0111] Step ②: If the initially constructed perturbation point does not meet the upper or lower constraints, the scaling factor used to control the perturbation amplitude is reduced proportionally. The perturbation point is reconstructed using the reduced scaling factor, and the constraint feasibility check is performed again until the perturbation point meets all constraints or the scaling factor is lower than the preset minimum limit. When the scaling factor is lower than the preset minimum limit, the sensitivity value of the decision variable in the current perturbation direction is recorded as zero.

[0112] For coordinate points that fail the constraint feasibility check, an iterative attenuation mechanism is triggered. The initial value of the scaling factor is set to 1, and after each failed check, the scaling factor is reduced according to a preset attenuation ratio. In this embodiment, the attenuation ratio is set to 0.5, i.e., ρ = ρ × 0.5. The updated scaling factor is then substituted back into the perturbation point construction formula to obtain new coordinates, which are then used in the next round of checks.

[0113] The termination condition of this iterative loop includes two triggering branches. The first is that the new coordinates successfully pass all determined physical inequality checks. The second is that the scaling factor, after multiple halvings, falls below a preset minimum limit. In this embodiment, the minimum limit of the scaling factor is set to 0.01. If the second branch is triggered, it indicates that the decision variable has approached the rigid physical boundary in the current test direction, leaving almost no feasible adjustment margin.

[0114] One possible implementation involves obtaining the upper-level sensitivity coefficients and lower-level sensitivity coefficients of each decision variable for each level of objective, specifically including:

[0115] Step (1): Implement bidirectional perturbations at the lower limit and upper limit of the interval of the reference point.

[0116] Furthermore, to comprehensively capture the asymmetric sensitivity of the variables, tests are conducted at the two extreme stationary points in space for each decision variable. Specifically, the lower bound of the baseline point vector is extracted to form the lower endpoint vector, and the upper bound is extracted to form the upper endpoint vector. At each endpoint vector, the perturbation direction coefficient is assigned a positive +1 and a negative -1, respectively. Through the orthogonal combination of the two endpoints and two directions, the system generates four independent test paths for each variable.

[0117] Step (2): When calculating the rate of change of the objective function at the lower limit of the interval, only the perturbation points that satisfy the constraints are substituted into the evaluation function corresponding to the lower limit sub-model of the objective function; when calculating the rate of change of the objective function at the upper limit of the interval, only the perturbation points that satisfy the constraints are substituted into the evaluation function corresponding to the upper limit sub-model of the objective function.

[0118] This step is used to eliminate the mathematical ambiguity of numerical differentiation of interval functions at discrete endpoints. In interval fuzzy programming, since the model has been decomposed into lower bound sub-models and upper bound sub-models, conventional methods are prone to sub-model mismatch when substituting the evaluation function, resulting in discontinuous partial derivatives.

[0119] The endpoint alignment matching principle is implemented. When the perturbation operation occurs at the lower bound of the interval of the reference point, the lower bound sub-model of the objective function generated in the previous decomposition stage is invoked, and its corresponding evaluation function is extracted to calculate the rate of change. Similarly, when the perturbation occurs at the upper bound of the interval, the evaluation function corresponding to the upper bound sub-model of the objective function is strictly bound and invoked. This matching mechanism ensures at the underlying logic that the numerical differentiation operation is continuous and well-defined in the local function space.

[0120] Step (3) takes the maximum value of the sensitivity values ​​obtained for the same decision variable at each perturbation direction and at each interval endpoint as the conservatively estimated upper-level sensitivity coefficient and lower-level sensitivity coefficient.

[0121] Substitute the actual perturbation points verified by constraints and the output values ​​of the evaluation function into the difference formula to calculate the absolute value of sensitivity on a single path.

[0122] S path =|f(z new )-f(z base )| / (ρ·Δx j );

[0123] Among them, S path f(z) is the absolute value of sensitivity on a single path. new ) represents the output of the evaluation function when the perturbation point is substituted into it, f(z) base The output of the same evaluation function is substituted into the reference point, and the denominator is the actual effective perturbation step size.

[0124] After calculating the four test paths, a control set containing four values ​​is constructed, and the maximum value is extracted as the final sensitivity of the decision variable to a single-level target. Extracting the maximum value is to adopt a conservative defensive strategy, ensuring that drastic oscillations near the feasible region boundary are not overlooked, thereby generating reliable upper-level and lower-level sensitivity coefficients.

[0125] Step 303: Calculate the comprehensive sensitivity index based on the upper-layer sensitivity coefficient and the lower-layer sensitivity coefficient.

[0126] Since both the upper-level and lower-level evaluation functions are standardized to dimensionless quantities through ideal point standardization, and the normalization operation maps the sensitivity coefficients of each variable to the dimensionless interval of 0 to 1, the subsequent cross-variable comparisons and weighted synthesis are carried out in the dimensionless space, eliminating the influence of the differences in the physical dimensions of the original variables.

[0127] After obtaining the single-layer coefficients, the coefficients of the two different dimensions are algebraically synthesized to output a comprehensive sensitivity index of a single dimension, providing a unified data interface for tolerance allocation in subsequent steps.

[0128] In some alternative implementations, if computational power is limited or the non-convex features of the current model are predicted to be weak, a simplified one-way perturbation scheme can be used instead of the aforementioned two-way perturbation scheme. Specifically, a positive perturbation is implemented only at the lower limit of the interval, and a negative perturbation is implemented at the upper limit of the interval, thereby reducing the test paths from four to two, reducing the computational load of numerical differentiation while ensuring the acquisition of basic boundary gradients.

[0129] To address the problem of easy overshooting or distortion in low-level feature extraction under multiple uncertain boundaries, this invention adopts a two-way perturbation evaluation mechanism of constraint perception and endpoint alignment, which can eliminate numerical calculation ambiguities at interval boundaries, so that the obtained sensitivity information can truly reflect the dynamic characteristics within the feasible domain, and transform gradient evaluation from fuzzy theoretical deduction to controlled physical boundary mapping.

[0130] Example 4

[0131] Based on the above embodiment 3, this embodiment further details the refined allocation mechanism of adaptive tolerance and the construction process of the membership function of decision variables.

[0132] In one possible implementation, the comprehensive sensitivity index is calculated based on the upper-layer sensitivity coefficient and the lower-layer sensitivity coefficient, which may include the following steps:

[0133] Normalize the upper-level and lower-level sensitivity coefficients respectively; determine the number of objective functions in the upper-level and lower-level interval models; determine the hierarchical weights by the inverse ratio of the number of objective functions in the upper-level and lower-level interval models; and use the hierarchical weights to perform weighted synthesis of the normalized upper-level and lower-level sensitivity coefficients to obtain the comprehensive sensitivity index.

[0134] Specifically, to eliminate the magnitude interference caused by differences in physical meaning among different evaluation dimensions, all decision variables are divided by the maximum value in the sensitivity coefficient set of each level, mapping them to a dimensionless interval of 0 to 1. When merging levels, since the number of upper-level and lower-level targets in a two-level model is often unequal, directly adding them with equal weights will dilute the influence weight of the level with more targets.

[0135] The weights are calculated using the single-objective contribution equalization rule:

[0136] w1 = m2 / (m1 + m2);

[0137] Where w1 is the hierarchical weight assigned to the upper-level sensitivity coefficient, m1 is the number of objective functions in the upper-level interval model, and m2 is the number of objective functions in the lower-level interval model.

[0138] w2 = m1 / (m1 + m2);

[0139] Where w2 is the hierarchical weight assigned to the sensitivity coefficient of the lower layer.

[0140] S j =w1·S j_upper_norm +w2·S j_lower_norm ;

[0141] Among them, S j S is the comprehensive sensitivity index for the j-th decision variable. j_upper_norm S represents the normalized upper-layer sensitivity coefficient. j_lower_norm This represents the lower-level sensitivity coefficient after normalization.

[0142] This mechanism compensates for the contribution dilution effect caused by the difference in the number of objectives at each level by assigning relatively large level weights to levels with fewer objective functions, thus maintaining the structural balance of different levels in the comprehensive sensitivity assessment in the bi-level programming.

[0143] One possible implementation involves determining the adaptive tolerance of each decision variable based on a comprehensive sensitivity index, such as... Figure 3 As shown, it includes:

[0144] Step 401: Calculate the difference between the optimal decision variables of the upper-level interval model and the optimal decision variables of the lower-level interval model, and extract the baseline tolerance for each decision variable. Extracting the baseline tolerance for each decision variable specifically includes:

[0145] Based on the feasible region jointly determined by the upper and lower constraints, the feasible region width of each decision variable is pre-calculated. A fixed proportion of the pre-calculated feasible region width of each decision variable, jointly determined by the constraints, is obtained as the minimum tolerance guarantee. The absolute value of the difference between the optimal decision variable in the upper interval model and the optimal decision variable in the lower interval model is compared with the minimum tolerance guarantee, and the larger value is taken as the benchmark tolerance of each decision variable. The feasible region width is obtained by performing extreme value optimization on each decision variable in the joint constraint space beforehand.

[0146] To provide an initial adjustment span for fuzzy programming, information is extracted from the solution divergence in a two-level game. If the optimal solutions at the upper and lower levels are identical, directly taking the difference will cause the subsequent tolerance to degenerate to 0, thus locking up the optimization space. A pre-computed feasible region width is introduced to provide a physical benchmark baseline.

[0147] t base_j =max(|mid(x jU )-mid(x jL )|,δ t ·R j );

[0148] Among them, t base_j Let x be the baseline tolerance for the j-th decision variable. jU x is the optimal decision variable for the upper-level interval model. jL For the optimal decision variables of the lower-level interval model, mid() is the operation of taking the midpoint of the interval, |mid(x jU )-mid(x jL | represents the absolute value of the difference between the center values ​​of the two optimal interval solutions, i.e., the absolute value of the difference between the midpoints of the two intervals, δ. t R is the ratio coefficient for the minimum tolerance guarantee amount. j Let be the width of the feasible region for the j-th decision variable before calculation.

[0149] In this embodiment, the proportional coefficient δ of the minimum tolerance guarantee amount t The value is set to 0.03. By introducing a minimum tolerance guarantee, it is ensured that the decision variables have a minimum degree of adjustment freedom in the degenerate case where the optimal solutions highly overlap.

[0150] Step 402: Using a preset piecewise mapping function, the comprehensive sensitivity index of each decision variable is converted into an adjustment coefficient. Specifically, this includes:

[0151] Statistical features of the comprehensive sensitivity index corresponding to all decision variables are extracted. The first and third quartiles of the comprehensive sensitivity index set are used as the lower and upper thresholds of the piecewise mapping function, respectively. The coefficient of variation of the comprehensive sensitivity index set is calculated, and the lower limit of the adjustment coefficient is determined using the coefficient of variation. Based on the lower threshold, the upper threshold, and the lower limit of the adjustment coefficient, a piecewise mapping function is constructed, and the comprehensive sensitivity indexes in the corresponding numerical intervals are mapped to the adjustment coefficients.

[0152] In this embodiment, the statistical distribution characteristics of objective data are used instead of manual experience-based settings. The first quartile corresponds to the 25th percentile value in the dataset, and the third quartile corresponds to the 75th percentile value. Variables with sensitivity below the lower threshold are automatically classified as low-sensitivity variables, variables above the upper threshold are classified as high-sensitivity variables, and the rest are classified as transitional variables. The coefficient of variation is calculated as the ratio of the standard deviation to the mean of the comprehensive sensitivity index set. The coefficient of variation reflects the dispersion of the influence of each variable on the target in the current system; the higher the dispersion, the greater the required tolerance and differentiated control intensity.

[0153] One possible implementation involves using the coefficient of variation to determine the lower limit of the adjustment coefficient, specifically including the following process:

[0154] Obtain the preset minimum constant; determine whether the coefficient of variation exceeds the preset cutoff threshold. If it does, limit the coefficient of variation to the cutoff threshold and calculate the attenuation margin corresponding to the cutoff threshold. That is, calculate the corresponding attenuation margin based on the limited coefficient of variation or the original coefficient of variation; compare the attenuation margin with the minimum constant and take the larger value as the lower limit of the adjustment coefficient in the piecewise mapping function to prevent the tolerance of highly sensitive variables from completely degrading under a high dispersion distribution.

[0155] To prevent the coefficient of variation from becoming too large under extremely skewed distributions, causing the adjustment coefficient to approach 0, a lower limit determination rule with a cutoff mechanism is constructed, as shown in the following formula:

[0156] h min =max(δ0,1-min(CV,1));

[0157] Among them, h min δ0 is the lower limit of the adjustment coefficient, CV is the preset minimum constant, CV is the coefficient of variation of the comprehensive sensitivity index set, and the cutoff threshold is 1.

[0158] In this embodiment, the minimum constant δ0 is set to 0.1. The control logic of this formula is that when the coefficient of variation is greater than 1, the attenuation margin is calculated based on 1, and the minimum constant takes over the lower limit. The upper limit of the adjustment coefficient is fixed at 1. The piecewise mapping function outputs the lower limit of the adjustment coefficient for the high-sensitivity group and the upper limit of the adjustment coefficient for the low-sensitivity group. The transition group obtains the adjustment coefficient value for a smooth transition through linear interpolation.

[0159] Step 403: Multiply the adjustment coefficient by the baseline tolerance to obtain the adaptive tolerance of each decision variable.

[0160] t j =t base_j ·h j ;

[0161] Among them, t j Let t be the adaptive tolerance of the j-th decision variable. base_j h represents the corresponding baseline tolerance. j This is the corresponding adjustment coefficient obtained through the piecewise mapping function.

[0162] This achieves accurate constraint allocation with high-sensitivity locking and low-sensitivity relaxation. The tolerance of high-sensitivity variables is compressed, while low-sensitivity variables retain the complete baseline game space.

[0163] One possible implementation involves constructing a membership function for the decision variables, such as... Figure 5 As shown, the details are as follows:

[0164] The optimal decision variable of the upper-level interval model is used as the peak center of the membership function;

[0165] The adaptive tolerance of the corresponding decision variable is used as the half-width of the membership function;

[0166] Construct a linear function with the peak center as the maximum value, which decreases as the distance from the peak center increases, and drops to zero when it exceeds half the width, as the membership function of the decision variable.

[0167] Specifically, a piecewise linear geometric form representing the satisfaction variable is constructed in the real number field:

[0168] η j (x j )=1-|x j -x jU | / t j ;

[0169] Where, η j (x j Let x be the membership output value of the j-th decision variable. j Let x be the current value of the decision variable. jU For the peak center, |x j -x jU | represents the Euclidean distance (i.e., the difference in ordinary absolute value) between the current value and the peak center, t j This corresponds to the adaptive tolerance.

[0170] During the calculation, when the absolute value of the decision variable offset is greater than the adaptive tolerance, the forced membership output value is 0.

[0171] In some optional implementations, bypass pass-through logic is set up to address extreme degradation of the input data structure. When the total number of decision variables is 1, the quartile partitioning loses its ranking basis, and the adjustment coefficient is directly set to 1. When the total number of decision variables does not exceed 3, the statistical significance of the quartile partitioning degrades, and approximate quartile values ​​can be obtained using conventional small-sample quartile interpolation methods, or the adjustment coefficient can be uniformly set to 1 to simplify processing. When the mean of the comprehensive sensitivity index set is 0 or the coefficient of variation is 0, it indicates that all variables have no difference in their impact on the system, and the adjustment coefficient is also uniformly degraded and set to 1. In this case, the adaptive tolerance of each decision variable fully retains the numerical range of the baseline tolerance.

[0172] An adaptive tolerance configuration method based on coefficient of variation truncation and robust benchmark guarantee is introduced. Driven by objective data distribution characteristics, it realizes differentiated configuration of high sensitivity tight constraints and low sensitivity wide tolerance, adapting to various extreme data forms such as high discreteness and strong concentration, and avoiding the blindness of manual experience setting.

[0173] Example 5

[0174] Based on the above embodiments, this embodiment further describes the sequential constraint satisfaction aggregation solution module in the compromise process of the two-layer model, as well as the hierarchical orderly concession mechanism.

[0175] In one possible implementation, satisfaction constraints are constructed using the membership functions of decision variables and the optimal objective evaluation value, and sequential optimization is performed in conjunction with a comprehensive sensitivity index, including the following steps:

[0176] Step 501: Construct the upper-level target membership function and the lower-level target membership function based on the optimal target evaluation value.

[0177] Furthermore, based on the optimal target evaluation value and the optimal decision variables, we construct the upper-level target membership function and the lower-level target membership function.

[0178] Specifically, the satisfaction level of each objective function needs to be quantitatively evaluated. For the upper-level interval model, the optimal objective evaluation value is used as the optimal boundary, and the satisfaction level is set to a maximum of 1. The calculated value obtained by substituting the optimal decision variables of the lower level into the upper-level evaluation function is used as the worst-case boundary, and the satisfaction level is set to a minimum of 0. By constructing a linear interpolation ratio, the current evaluation function value is mapped to a continuous membership degree value. The membership function of the lower-level objective is constructed using symmetric logic, and the calculated value obtained by substituting the optimal decision variables of the upper level into the lower-level evaluation function is used as the worst-case boundary. Specifically, for the lower-level objective, the optimal objective evaluation value is used as the optimal boundary, and the satisfaction level is set to a maximum of 1; the calculated value obtained by substituting the optimal decision variables of the upper level into the lower-level evaluation function is used as the worst-case boundary, and the satisfaction level is set to a minimum of 0. A linear interpolation mapping relationship of the same form as the upper-level boundary is constructed between the optimal boundary and the worst-case boundary.

[0179] In one possible implementation, the upper-level target membership function and the lower-level target membership function are constructed based on the optimal target evaluation value, as follows:

[0180] Step 5011: Determine whether there is a conflict of objectives between the optimal solutions of the upper and lower level interval models in a single evaluation dimension.

[0181] When extracting the worst-case boundary, the difference between the worst-case boundary value and the optimal boundary value is compared. If the value obtained by substituting the optimal decision variable of the lower level into the evaluation function of the upper level is equal to the optimal evaluation value of the upper level itself, it means that under the current physical model parameters, the process of the lower level pursuing the optimal has not harmed the interests of the upper level. At this time, there is no objective conflict on the predetermined evaluation dimension, and the denominator of the membership function will degenerate to 0.

[0182] Step 5012: When it is determined that the target is in a degenerate state without conflict, stop constructing the continuous membership function in fractional form, and directly perform Boolean branch assignment on the upper-level target membership function or the lower-level target membership function based on the matching state between the current evaluation function value obtained by substituting the decision variables into the corresponding level evaluation function and the optimal target evaluation value.

[0183] To prevent system crashes caused by division by zero, discrete Boolean assignment logic is used instead of continuous mapping logic. If the current evaluation function value is exactly equal to the optimal target evaluation value, the corresponding membership function value is directly assigned a value of 1. If the current evaluation function value deviates from the optimal target evaluation value, since no further inter-layer concessions are needed for this dimension, the corresponding membership function value is directly assigned a value of 0.

[0184] Step 502: Perform the first stage of solution. Within the tolerance range of all constraints and the membership functions of the decision variables, independently optimize the membership function of the upper-level objective to obtain the maximum satisfaction of the upper level.

[0185] In this step, the first stage of the three-stage sequential optimization is initiated. This stage performs optimization operations within the original physical constraint set set of the model and the various adaptive tolerance boundaries output in Example 4. The optimization objective is uniquely locked to maximizing the membership function of the upper-level objective, without considering the interests of the lower-level objectives. By calling linear programming solvers such as the simplex method or interior point method, the highest satisfaction point that can be reached in the first stage is extracted, and this value is stored as the maximum satisfaction of the upper level.

[0186] One possible implementation, such as Figure 4 As shown, the upper layer relaxation amount and the lower layer relaxation amount are obtained in the following ways:

[0187] Step a: Extract the average normalized sensitivity of the corresponding level based on the sensitivity of the decision variables, that is, extract the average normalized sensitivity of the corresponding level based on the normalized sensitivity coefficient of each decision variable at the corresponding level.

[0188] After determining the maximum satisfaction level, a reasonable compromise margin, or slack, needs to be established. For all decision variables involved in the calculation, their corresponding normalized upper-level sensitivity coefficients are extracted, and all coefficients are arithmetically averaged to obtain the average normalized sensitivity. This parameter objectively reflects the overall sensitivity characteristics of all variables within the current system to the upper-level objective.

[0189] Step b: Calculate the product of the maximum satisfaction level and the preset maximum relaxation ratio for the corresponding level, and use it as the upper bound for the safety cutoff of the corresponding level.

[0190] To prevent the leadership's defenses from collapsing completely during subsequent compromises, a mandatory stop-loss mechanism is introduced.

[0191] ε max =γ max ·τ * ;

[0192] Where, ε max To safely truncate the upper bound, γ max τ is the preset maximum relaxation ratio. * To achieve maximum satisfaction at the upper level. In this embodiment, the maximum relaxation ratio γ max Set it to 0.4.

[0193] Step c: Calculate the product of the maximum satisfaction and the average normalized sensitivity derived value as the theoretical relaxation amount. Compare the theoretical relaxation amount with the upper bound of the safety cutoff, and take the smaller value as the final upper-level relaxation amount or lower-level relaxation amount applied to the corresponding level.

[0194] Specifically, the difference between 1 and the average normalized sensitivity is calculated as the sensitivity compensation factor, which is a derived value of the average normalized sensitivity and is used to characterize the margin space of the current level's allowable relaxation adjustment. The product of the maximum satisfaction of the corresponding level and the sensitivity compensation factor is calculated as the theoretical relaxation amount. The theoretical relaxation amount is compared with the upper bound of the safety cutoff, and the smaller value of the two is taken as the final upper-level relaxation amount or lower-level relaxation amount applied to the corresponding level.

[0195] The compromise space allowed by the inverse derivation of sensitivity.

[0196] ε1=min(τ * ·(1-S upper ),ε max );

[0197] Where ε1 is the relaxation amount finally applied to the corresponding layer, and τ * To maximize the satisfaction of the upper management, S upper For average normalized sensitivity, ε max The upper limit is cut off for safety.

[0198] This mechanism exhibits significant adaptive adjustment characteristics. When system variables are generally highly sensitive to the upper-level objective, the average normalized sensitivity approaches 1, at which point the theoretical relaxation amount approaches 0, providing relatively strict protection for upper-level satisfaction. Conversely, when sensitivity is generally low, a larger adjustment margin is released, but the maximum release magnitude is strictly controlled within the upper bound of the safety cutoff. The calculation of the lower-level relaxation amount employs symmetrical logic execution.

[0199] Step 503: Perform the second stage of solution. Under the constraint that the membership function of the upper-level objective is not lower than the lower limit of the upper-level retreat, optimize the membership function of the lower-level objective to obtain the maximum satisfaction of the lower level. The lower limit of the upper level retreat is the maximum satisfaction of the upper level minus the predetermined relaxation amount of the upper level.

[0200] In the second stage, a new algebraic inequality constraint is added to the existing set of constraints, requiring that the value of the membership function of the upper-level objective must be greater than or equal to the lower yield limit of the upper-level objective. Under the protection of this new constraint, the optimization objective is switched to maximizing the membership function of the lower-level objective. After optimization by the solver, the maximum satisfaction level of the lower-level objective is obtained and recorded.

[0201] Step 504: Perform the third stage of solution. Under the constraints of adding upper-level and lower-level concession limits, perform weighted aggregation optimization on the membership functions of all decision variables with the comprehensive sensitivity index as the weight to obtain the final decision variable interval solution. The lower-level concession limit is the lower-level maximum satisfaction minus the predetermined lower-level relaxation amount.

[0202] In the third stage, a double-lockdown boundary is constructed to protect the interests of both the upper and lower layers. A second algebraic inequality constraint is added, requiring the membership function of the lower-level objective to be greater than or equal to the lower-level concession limit. Within this highly constrained joint feasible region, an aggregate objective function is constructed.

[0203] maxZ=∑(S j ·η j (x j_± ));

[0204] Where Z is the weighted aggregation optimization objective function value, S j Let η be the comprehensive sensitivity index for the j-th decision variable. j (x _j_± Let x be the membership function of the j-th decision variable. j_± Let be the interval variable to be solved, and ∑() be the summation function across variables.

[0205] By using sensitivity as a weighting factor in the overall calculation, highly sensitive variables are guided to preferentially approach their ideal values. The final combination of variables derived from solving this objective function represents the final decision variable interval solution after compromise and balance.

[0206] Optionally, if no feasible solution is found during the third stage of the solution process, a multi-level ordered relaxation strategy is triggered: the protection of the lower-level target is relaxed first, and the lower-level yield limit is relaxed by gradually increasing the lower-level relaxation amount, and the solution is tried again.

[0207] In engineering applications, sharp zero-sum game conflicts often exist between upper and lower layers, causing the feasible region of the third stage constraints to be empty, meaning the optimizer returns a state with no feasible solution. In this case, an adaptive solution is implemented strictly following the hierarchical priority principle of the two-layer model. The relaxation amount of the lower layer is increased in fixed steps, for example, by 10% each time, thereby lowering the lower yield limit and driving the solver to execute the third stage solution again.

[0208] If no feasible solution is found after the lower-level relaxation amount has been accumulated to the preset limit, the protection of the upper-level objective is further relaxed by gradually increasing the upper-level relaxation amount until a feasible solution is obtained or the preset termination condition is reached. When the preset termination condition is reached, the optimization solution process is terminated and a conflict flag is output.

[0209] If the relaxation of the lower-level concession limit reaches a preset limit, such as the cumulative increment equaling the initial lower-level relaxation amount, and the model still has no solution, it indicates that the unilateral sacrifice of the lower level cannot bridge the conflict. In this case, the protection of the upper-level objective is relaxed, and the upper-level relaxation amount is gradually increased in a similar proportion to reduce the upper-level concession limit.

[0210] Optionally, in the process of relaxing the lower limit of the upper-level concession by gradually increasing the upper-level relaxation amount, the following may also be included:

[0211] S1 sets the global maximum relaxation threshold and accumulates the actual relaxation count at each level.

[0212] To prevent the algorithm from getting stuck in an infinite loop and deadlocking, a constant variable is injected into the configuration module to preset a global maximum relaxation count threshold. In this embodiment, the global maximum relaxation count threshold is set to 15. Each time the incrementing operation in the aforementioned steps is executed, the counter is automatically incremented by 1 to generate the actual relaxation count.

[0213] S2. If a feasible solution cannot be obtained even when the actual number of relaxations reaches the upper limit of the global maximum number of relaxations threshold, it is determined that there is an irreconcilable underlying conflict between the upper-level objective and the lower-level objective.

[0214] When the counter records a value equal to the set upper limit, and the solution engine still reports an optimization failure, the sequential optimization process is interrupted. This logical judgment, in a physical sense, indicates that there is a severe structural disconnect between micro-level production and environmental demands and macro-level population carrying capacity benefits in the current input resource and environmental configuration data, making coexistence difficult within acceptable compromise limits.

[0215] S3 terminates the optimization process and calculates and outputs the inter-layer conflict degree diagnostic index based on the ratio between the actual number of relaxations and the upper limit of the global maximum relaxation threshold.

[0216] Shut down the solver thread and switch to diagnostic report generation mode.

[0217] ConflictIndex=N actual / N max ;

[0218] Wherein, ConflictIndex is the calculated inter-layer conflict level diagnostic index, N actual N represents the actual number of relaxations. max This represents the upper limit of the global maximum number of relaxations thresholds.

[0219] The value of this indicator is normalized and limited to between 0 and 1. This indicator will be passed to the external business judgment application module as an abnormal exit code and conflict quantification parameter, providing managers with a basis for adjusting subsequent resource allocation strategies.

[0220] This application proposes a multi-level ordered concession sequential solution strategy with a safe truncation upper bound. This strategy strictly maintains the priority of macro-level hierarchy while preventing the collapse of the leadership target defense line due to extreme dilution effects, ensuring orderly convergence of the system under sharp conflict environments, and effectively improving the reliability of boundary capacity determination for complex systems.

[0221] Example 6

[0222] Based on the above embodiment 5, the post-processing judgment logic reflecting the water resource carrying capacity and system-level commercial applications are further described in detail.

[0223] In one possible implementation, after obtaining the final decision variable interval solution reflecting the water resource carrying capacity of the specified area, the following steps are also included:

[0224] Step 601: Extract the carrying capacity interval solution representing the regional carrying target scale from the final decision variable interval solution.

[0225] By analyzing the final decision variable interval solution output from the third stage, we identify and separate the variable dimensions directly related to the macro-carrying capacity target. In the water resource carrying capacity assessment scenario, the carrying capacity interval solution specifically represents the population carrying capacity range or industrial economic scale range for each calculation unit. This interval solution includes a lower limit and an upper limit, which physically define the minimum and maximum safe scale that can be supported under the current water resource constraints.

[0226] Step 602: Obtain the current status index value corresponding to each calculation unit within the specified area.

[0227] By accessing external socioeconomic statistical databases or real-time sensor monitoring networks, actual statistical data occurring within the assessment baseline period is obtained. The current status index values ​​must correspond to the carrying capacity interval solution in terms of data type and dimension. When the carrying capacity target is population size, the current status index values ​​are the actual population statistics for the corresponding calculation unit.

[0228] Step 603: Compare the current status index value with the lower limit and upper limit of the bearing capacity interval solution.

[0229] Execute the numerical magnitude determination logic. Using the obtained deterministic current status index value as a comparison benchmark, calculate the numerical difference between it and the lower limit and upper limit of the interval, respectively, to provide a mathematical basis for subsequent state classification.

[0230] Step 604: If the current index value is lower than the lower limit of the interval, the carrying capacity is determined to be in a surplus state; if the current index value is between the lower limit and the upper limit of the interval, the carrying capacity is determined to be in a critical state; if the current index value is higher than the upper limit of the interval, the carrying capacity is determined to be in an overload state.

[0231] Output discrete classification states based on defined comparison rules:

[0232] P actual <x pop_min A state of surplus;

[0233] x pop_min ≤P actual ≤x pop_max Critical state;

[0234] P actual >x pop_max Overloaded state.

[0235] Among them, P actual x represents the obtained current status indicator value. pop_min Let x be the lower limit of the solution interval for bearing capacity. pop_max This represents the upper limit of the interval solution for the bearing capacity interval.

[0236] This decision-making mechanism transforms abstract mathematical interval solutions into discrete state parameters that can be directly utilized by the business side.

[0237] Optionally, after determining the load-bearing status, the following may also be included:

[0238] Step 605: Output the carrying status to the water resources scheduling and management terminal.

[0239] Using a preset communication protocol, the generated discrete state identifiers and the corresponding underlying carrying capacity interval solution data are packaged and sent to the water resource scheduling and management terminal used for decision-making configuration, thus completing the data flow from the evaluation algorithm layer to the business execution layer.

[0240] Step 606: When the carrying capacity of the designated area is determined to be overloaded, an overload warning command is triggered on the water resource scheduling and management terminal.

[0241] Upon receiving an overload status indicator, the water resource dispatch and management terminal activates its internal event monitoring mechanism, generating an overload warning command containing data on the location and magnitude of the overloaded units. This command then activates external alarm modules or alarm pop-ups on the system interface, thus alerting administrators that the current resource allocation has exceeded the system's physical capacity limit.

[0242] Step 607: Automatically generate a resource reduction strategy based on the upper limit of the interval solution of the final decision variable interval, and distribute the resource reduction strategy to the water supply control node of the corresponding computing unit to limit the actual water quota.

[0243] Furthermore, a resource reduction strategy is automatically generated based on the difference between the upper limit of the interval solution of the final decision variable and the current indicator value, and the resource reduction strategy is distributed to the water supply control node of the corresponding computing unit to limit the actual water quota.

[0244] After an alarm is triggered, a closed-loop reverse control action is executed. The upper limit of the carrying capacity interval solution is extracted as the upper limit of the safe threshold for resource consumption in that area. The redundancy difference between the current indicator value and the upper limit of the interval is calculated, and a resource reduction strategy is generated based on the reverse mapping of this redundancy difference. The strategy parameters are written into the execution registers of each underlying water supply control node through the control network, and the water allocation is forcibly reduced at the level of physical valves or quota approval, prompting the system state to return to a critical or surplus state.

[0245] Optional, further including:

[0246] Step 608: After outputting the inter-layer conflict degree diagnostic index, extract the target boundary values ​​of the upper-layer target membership function and the lower-layer target membership function corresponding to the triggering of irreconcilable lower-layer conflicts.

[0247] In the event of an abnormal situation where a deadlock is encountered during the sequential optimization phase and a diagnostic index of the degree of inter-layer conflict is output, the diagnostic tracing mechanism is activated. An environment snapshot stored by the optimizer in the last failed relaxation iteration is obtained, and the upper-layer and lower-layer yield limits applied to the model at that time are accurately extracted. These two parameters are the target boundary values ​​that cause the feasible region to become empty at this point.

[0248] Step 609: Generate a conflict diagnosis visualization report based on the inter-layer conflict degree diagnostic index and the target boundary value, which is used to prompt the data incompatibility status of the current two-layer interval multi-objective programming model in the user interface.

[0249] The severity of conflict is characterized by the acquired inter-layer conflict degree diagnostic indicators, and the specific numerical dimensions of the fracture are represented by the target boundary values. These multidimensional data are then integrated into a formatted conflict diagnosis visualization report. This report is pushed to the user interface to help engineering planners intuitively identify the core variables causing the conflict or overly stringent constraints. Based on this, planners decide to modify the model's basic resource constraint boundaries, thereby initiating a new round of evaluation.

[0250] This invention's method, through sensitivity-driven adaptive tolerance configuration and a multi-level ordered concession mechanism with safety truncation, can still stably output effective interval solutions even under extremely degraded data conditions, such as when the decision variable interval width is zero. Compared to traditional methods that use a uniform fixed tolerance and a simple minimum aggregation operator, this method exhibits stronger solution robustness and more refined resource allocation rationality when dealing with sharp conflicts between hierarchical objectives and highly differentiated variable sensitivities. Those skilled in the art will understand that the specific improvement may vary depending on the actual watershed parameter configuration and constraints.

[0251] Example 7

[0252] Based on the same inventive concept as the above-described method embodiments, this embodiment provides a water resource suitability carrying capacity assessment device based on two-layer interval planning. This device can be used to execute the methods provided in the above embodiments, and its implementation principle and technical effects are similar; therefore, the same parts will not be repeated here. The following focuses on describing the module division structure.

[0253] In this embodiment, the device specifically includes a data acquisition module, a model building module, an independent solution module, a sensitivity evaluation module, a tolerance configuration module, and a sequential optimization module.

[0254] Specifically, the data acquisition module is used to acquire a set of resource and environmental configuration interval data for a specified area. This module is equipped with a network communication interface to obtain historical hydrological statistical parameters and socio-economic indicator parameters from external distributed storage nodes, and converts each parameter into an interval array format, which is then loaded into system memory for downstream modules to use.

[0255] The model building module is used to construct a two-level interval multi-objective programming model based on a set of resource and environmental configuration interval data. This module includes a constraint compiler, which maps the available water quantity and environmental pollution carrying capacity in interval form to mathematical inequality boundaries, and constructs a hierarchical nested data structure containing upper-level objective functions and lower-level objective functions.

[0256] The independent solver module is used to solve a two-level interval multi-objective programming model, extracting the optimal decision variables and optimal objective evaluation values. Specifically, it solves the upper and lower interval models independently, extracting the optimal decision variables and optimal objective evaluation values ​​for the corresponding levels. This module integrates a linear programming solver to perform dimensionality reduction decomposition and matrix inversion operations, outputting local optima at the interval endpoints.

[0257] The sensitivity evaluation module is used to obtain the comprehensive sensitivity index of each decision variable based on the optimal decision variable. Specifically, it performs perturbation calculations based on the optimal decision variables of the upper-level interval model to obtain the comprehensive sensitivity index of each decision variable for the two-level interval multi-objective programming model. This module includes a perturbation step size calculation unit and a constraint feasibility verification unit, used to perform bidirectional perturbations in the multi-dimensional feature space and intercept out-of-bounds coordinate points to ensure the logical validity of the numerical partial derivative calculation.

[0258] The tolerance configuration module is used to determine the adaptive tolerance of each decision variable based on a comprehensive sensitivity index and to construct the membership function of the decision variables. Specifically, it maps and controls the baseline adjustment space determined by the differences in optimal decision variables between layers according to the comprehensive sensitivity index, determines the adaptive tolerance of each decision variable, and constructs the membership function of the decision variables based on the adaptive tolerance. This module is equipped with a statistical function processor to calculate quartiles and coefficients of variation, and outputs a sequence of anti-degradation adjustment coefficients.

[0259] The sequential optimization module constructs satisfaction constraints using the membership functions of decision variables and the optimal target evaluation value. It then performs sequential optimization based on a comprehensive sensitivity index to obtain the final interval solution of the decision variables reflecting the water resource carrying capacity of a specified area. Specifically, while maintaining the hierarchical priority sequence, it constructs satisfaction constraints using the membership functions of decision variables and the optimal target evaluation value. Based on the comprehensive sensitivity index, it adjusts the inter-level concession magnitude and performs multi-stage sequential optimization to obtain the final interval solution of the decision variables reflecting the water resource carrying capacity of the region. This module includes a logic relaxation controller, which, when encountering an infeasible constraint state, incrementally increases the relaxation amount in a lower-level-first-upper-level order until a preset maximum relaxation threshold is reached.

[0260] Based on the same inventive concept as the above-described method embodiments, this embodiment also provides an electronic device. The electronic device includes a processor and a memory.

[0261] The memory is used to store computer program code. It is implemented using non-volatile storage elements and is used to persistently store basic model data and intermediate evaluation and diagnostic indicators.

[0262] The processor is used to acquire and execute computer program code stored in memory to implement the water resource suitability carrying capacity assessment method based on two-level interval planning described in this application. The processor can be implemented using a central processing unit or an application-specific integrated circuit.

[0263] Based on the same inventive concept as the above-described method embodiments, this embodiment also provides a computer-readable storage medium. This computer-readable storage medium stores computer program instructions. When the computer program instructions are executed by a processor, they implement the method described in the embodiments of this application.

[0264] In some optional implementations, the aforementioned electronic device may also be configured with a display interface and a control output interface. The display interface is used to render the output final decision variable interval solution and inter-layer conflict degree diagnostic indicators into a conflict diagnosis visualization report. The control output interface is used to send triggered overload warning commands or generated resource reduction strategies to designated physical water supply control nodes, thereby realizing full-link closed-loop reverse control of the evaluation system.

[0265] The preferred embodiments of the present invention have been described in detail above. However, the present invention is not limited to the specific details in the above embodiments. Within the scope of the technical concept of the present invention, various equivalent transformations can be made to the technical solutions of the present invention, and these equivalent transformations all fall within the protection scope of the present invention.

Claims

1. A method for assessing the suitability carrying capacity of water resources based on two-level interval planning, characterized in that, include: Acquire and construct a two-layer interval multi-objective programming model based on the resource and environmental configuration interval data set of a specified area; The two-level interval multi-objective programming model is solved to extract the optimal decision variables and the optimal objective evaluation values, and the comprehensive sensitivity index of each decision variable is obtained based on the optimal decision variables. The adaptive tolerance of each decision variable is determined based on the comprehensive sensitivity index, and the membership function of the decision variable is constructed. Satisfaction constraints are constructed using the membership function of decision variables and the optimal target evaluation value. Sequential optimization is then performed in conjunction with the comprehensive sensitivity index to obtain the final decision variable interval solution reflecting the water resource carrying capacity of the specified area. Based on the resource and environmental allocation interval data set, a two-level interval multi-objective programming model is constructed, including: A higher-level interval model is constructed using the data set of resource and environmental allocation intervals. The higher-level interval model represents the comprehensive benefits of the region with the upper-level objective function and includes the corresponding upper-level constraints. A lower-level interval model is constructed using a set of data on resource and environmental allocation intervals. The lower-level interval model represents production efficiency and environmental load with a lower-level objective function and includes corresponding lower-level constraints. Solve the two-level interval multi-objective programming model, extract the optimal decision variables and optimal objective evaluation values, including: For the upper-level interval model and the lower-level interval model, regularization factors are introduced to eliminate the difference in dimensions, and dimensionless upper-level evaluation functions and lower-level evaluation functions are constructed respectively. The two-level interval multi-objective programming model is decomposed into an upper-level interval programming model and a lower-level interval programming model according to the hierarchy. The upper-level interval programming model and the lower-level interval programming model are respectively decomposed into a lower limit sub-model of the objective function and an upper limit sub-model of the objective function for solving; Extract the optimal decision variables and optimal target evaluation values ​​for the corresponding levels from the solution results of each sub-model; The adaptive tolerance of each decision variable is determined based on a comprehensive sensitivity index, including: Calculate the difference between the optimal decision variables of the upper interval model and the optimal decision variables of the lower interval model, and extract the baseline tolerance for each decision variable; By using a pre-defined piecewise mapping function, the comprehensive sensitivity index of each decision variable is transformed into an adjustment coefficient. Multiply the adjustment coefficient by the baseline tolerance to obtain the adaptive tolerance of each decision variable; Extract the baseline tolerance for each decision variable, including: Obtain a fixed proportion of the pre-calculated feasible region width for each decision variable under the constraints, as the minimum tolerance guarantee. The absolute value of the difference between the optimal decision variables of the upper interval model and the optimal decision variables of the lower interval model is compared with the minimum tolerance guarantee, and the larger value of the two is taken as the benchmark tolerance of each decision variable.

2. The method according to claim 1, characterized in that, After obtaining the final decision variable interval solution reflecting the water resource carrying capacity of the specified area, the following steps are also included: Extract the carrying capacity interval solution representing the scale of the regional carrying target from the final decision variable interval solution; Obtain the current status index values ​​for each computing unit within a specified area; The current index values ​​are compared with the lower limit and upper limit of the interval solution of the bearing capacity interval, respectively; If the current index value is lower than the lower limit of the interval, the carrying capacity is determined to be in a surplus state; if the current index value is between the lower limit and the upper limit of the interval, the carrying capacity is determined to be in a critical state; if the current index value is higher than the upper limit of the interval, the carrying capacity is determined to be in an overload state.

3. The method according to claim 2, characterized in that, After determining the load-bearing status, the following is also included: The carrying capacity status will be output to the water resources dispatch and management terminal; When the carrying capacity of a designated area is determined to be overloaded, an overload warning command is triggered on the water resource scheduling and management terminal.

4. A water resource suitability carrying capacity assessment device based on two-layer interval planning, characterized in that, include: The data acquisition module is used to acquire a set of resource and environmental configuration data for a specified area. The model building module is used to construct a two-level interval multi-objective programming model based on the resource and environment configuration interval data set; An independent solution module is used to solve a two-level interval multi-objective programming model and extract the optimal decision variables and the optimal objective evaluation value. The sensitivity assessment module is used to obtain the comprehensive sensitivity index of each decision variable based on the optimal decision variable; The tolerance configuration module is used to determine the adaptive tolerance of each decision variable based on the comprehensive sensitivity index and to construct the membership function of the decision variable. The sequential optimization module is used to construct satisfaction constraints using the membership function of decision variables and the optimal target evaluation value, and to perform sequential optimization solution in combination with the comprehensive sensitivity index to obtain the final decision variable interval solution reflecting the water resource carrying capacity of the specified area. The apparatus is used to perform the method according to any one of claims 1 to 3.

5. An electronic device, characterized in that, include: Memory, which stores executable programs; A processor for running the program, wherein the program, when running, performs the method according to any one of claims 1 to 3.

6. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores computer program instructions that, when executed by a processor, implement the method of any one of claims 1 to 3.

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