Weight-free TOPSIS-based port atmospheric pollution comprehensive evaluation method
By using the unweighted TOPSIS method to optimize weights in the comprehensive evaluation of port air pollution, the problem of improper weight setting was solved, resulting in more stable evaluation results, adapting to the dynamic changes of complex polluted environments, and improving the robustness and scientific nature of the evaluation.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-10-31
- Publication Date
- 2026-03-31
AI Technical Summary
The existing TOPSIS method in the comprehensive assessment of port air pollution suffers from improper weight setting due to subjective or objective reasons, which affects the accuracy and reliability of the assessment results and makes it difficult to adapt to the dynamic changes of complex polluted environments.
The unweighted TOPSIS method is adopted, which treats the weights as optimization variables in the computation stage. The optimal solution is found in the feasible region with reasonable weight configuration through mathematical programming. The weighted distance function is constructed and the relative proximity interval is solved, thus avoiding the inherent limitations of the pre-weights.
It provides more stable and reliable evaluation results, can adapt to changes in pollution structure, reduces sensitivity to outliers, and improves the robustness and scientific rigor of the evaluation.
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Figure CN121767151A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of environmental monitoring and comprehensive evaluation technology, specifically involving a multi-indicator comprehensive evaluation method for air pollution in port areas based on unweighted TOPSIS. Background Technology
[0002] The multi-source and complex nature of air pollution in port areas means that pollutants come from various sources, including ship fuel combustion, dust from loading and unloading operations, bulk material storage in stockpiles, road transportation, and emissions from port-related industries. These emissions are complex in composition and exhibit dramatic spatial and temporal variations. This multi-source nature and concentration fluctuations directly affect the scientific validity and representativeness of the indicator system in the comprehensive evaluation.
[0003] TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) is a classic multi-indicator decision analysis method. Due to its simple principle and convenient calculation, it has been attempted to be applied in the field of environmental assessment. However, the traditional TOPSIS method and its common improved forms, such as the Analytic Hierarchy Process-TOPSIS (AHP-TOPSIS) and the Entropy Weight TOPSIS (EW-TOPSIS), all have a common drawback: the weights of each indicator need to be explicitly set at the beginning of the assessment.
[0004] This pre-assessment method has inherent limitations. On the one hand, subjective weighting methods (such as AHP) rely heavily on expert experience and judgment; once weights are set, they remain fixed and cannot adapt to the dynamic changes in port pollution structures, resulting in highly subjective evaluation results. On the other hand, objective weighting methods (such as entropy weighting) determine weights based on the degree of variation in the data itself, but they are extremely sensitive to outliers or extreme values in the data. Small data fluctuations can lead to significant changes in weights, making the evaluation results unstable. In the complex pollution environment of ports, both subjective bias and data anomalies can seriously affect the accuracy and reliability of the final evaluation results due to inappropriate weight settings.
[0005] Therefore, there is an urgent need to develop a TOPSIS improvement method that can avoid the risks of pre-assignment and improve the robustness of the model. This is of great practical significance for the refined management of air quality in complex environments such as ports. Summary of the Invention
[0006] Purpose of the invention: The purpose of this invention is to overcome the dependence of existing TOPSIS methods on pre-set weights and provide a comprehensive evaluation method for port air pollution based on unweighted TOPSIS. This method treats weights as optimization variables in the calculation stage rather than preset parameters, and uses mathematical programming to find the optimal solution within the feasible domain composed of all reasonable weight configurations, thereby obtaining a more stable and reasonable comprehensive evaluation result.
[0007] Technical solution: The port air pollution comprehensive evaluation method based on unweighted TOPSIS described in this invention includes: Step 1: Data Collection and Standardization. Acquire concentration data of n evaluation objects in the target port area under m air pollutant indicators within a specified time period to construct an initial evaluation matrix; then standardize the initial evaluation matrix to eliminate the influence of dimensions, obtaining a standardized matrix.
[0008] Step Two: Determine the Ideal Solution. Without pre-setting any indicator weights, determine the positive ideal solution (PIS) and negative ideal solution (NIS) from the standardized matrix based on the attributes of each indicator (e.g., cost-based, benefit-based). For cost-based indicators such as air pollutants, the positive ideal solution consists of the minimum value of each indicator, and the negative ideal solution consists of the maximum value of each indicator.
[0009] Step 3: Construct the weighted distance function. Set the weights of each indicator as unknown variables that vary within a pre-defined set of weight constraints, where the set of constraints defines the range (upper and lower limits) of each weight variable and the sum constraint. Based on these weight variables, construct a weighted distance function from each object to be evaluated to the positive and negative ideal solutions.
[0010] Step 4: Solve for the relative proximity interval. Based on the weighted distance function, establish the relative proximity function for each object to be evaluated. This function is also a function of the weight variables. By solving a mathematical programming problem with this function as the objective function, determine the minimum and maximum values of the relative proximity function for each object to be evaluated under the set of weight constraints, thus constructing the relative proximity interval for that object.
[0011] Step 5: Comprehensive Evaluation and Ranking. Based on the relative proximity range of all objects to be evaluated, calculate the final comprehensive evaluation value (such as the mean of the range) for each object, and rank the port air pollution levels of each object accordingly. The higher the evaluation value, the lower the pollution level.
[0012] Beneficial effects: Compared with the prior art, the advantages of the present invention are as follows: This invention, by using weights as post-optimization variables, fundamentally avoids the problem of improper weight setting in the early stages of evaluation caused by subjective or objective reasons in traditional methods.
[0013] The evaluation result obtained by this invention is an interval number, which can reflect the stable range of the evaluation conclusion under reasonable weight fluctuations. It is not sensitive to data outliers and changes in contamination structure, making the evaluation conclusion more reliable and robust.
[0014] The interval evaluation results provided by this invention contain more information than a single numerical value, which helps decision-makers understand the potential range of variation in the evaluation conclusions and supports more robust decision-making. It is particularly suitable for scenarios such as ports where pollution sources are complex and the importance of each pollutant is difficult to determine a priori, providing a more scientific and flexible evaluation tool for environmental management. Attached Figure Description
[0015] Figure 1 This is a flowchart of the port air pollution comprehensive evaluation method based on the unweighted TOPSIS model described in this invention.
[0016] Figure 2 This is a comparison chart of the daily evaluation results of the method of this invention (UW-TOPSIS) and the traditional methods (AHP-TOPSIS, EW-TOPSIS) and AQI.
[0017] Figure 3 This is a comparison chart of the daily relative errors of the evaluation results of the three TOPSIS methods relative to the AQI. Detailed Implementation
[0018] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings, but the scope of protection of the present invention is not limited to the embodiments described.
[0019] Example 1: As Figure 1 The port air pollution comprehensive evaluation method based on unweighted TOPSIS, as shown, includes the following steps: Step 1: Data Collection and Indicator System Establishment. Obtain n monitoring data points for m air quality evaluation indicators in the target port area within a specified time period, and construct a decision matrix. Specifically, the selection of indicators fully considered the composition of port pollution sources and the temporal response of pollutants in the air, choosing six typical pollutants—PM2.5, PM10, SO2, NO2, CO, and O3—as evaluation indicators. PM2.5 and PM10 reflect the characteristics of particulate pollution, while SO2, NO2, CO, and O3 represent the characteristics of gaseous pollution, comprehensively characterizing the overall level and changing trend of complex pollution in the port area. This indicator system takes into account the synergistic effects of particulate matter and gaseous pollutants, reflecting both the complexity of port pollution and the comprehensive impact of concentration fluctuations on air quality, providing a reliable data foundation for the subsequent unweighted TOPSIS comprehensive evaluation.
[0020] Daily average concentration data of six typical pollutants—PM2.5, PM10, SO2, NO2, CO, and O3—were obtained to construct an initial evaluation matrix. .
[0021] Step 2: Data Standardization. The initial data is normalized to eliminate the influence of dimensions.
[0022] Step 3: Determine the Positive Ideal Solution (PIS) and Negative Ideal Solution (NIS). Based on the attributes of each evaluation index, determine the positive and negative ideal solutions from the normalized decision matrix; this step is performed without introducing weights. In the standardized matrix... For cost-related indicators (pollutant concentrations) where smaller values are better, the positive ideal solution consists of the optimal standardized value (minimum value) of each indicator, while the negative ideal solution consists of the worst standardized value (maximum value) of each indicator.
[0023] Step 4: Introduce a set of weight constraints and construct a weighted distance function. In a preferred embodiment of the invention, the upper and lower limits of the weights of each indicator are determined based on the objective weighting results of historical data. Specifically, the average weight of each pollutant indicator is first calculated using the entropy weight method on all monitoring data within the evaluation period. This is used as the center value of the weight interval. While ensuring model stability and flexibility, the upper and lower limits of each indicator weight are set to ±20% of the average value, i.e.: in, and For each of the pre-set numbers j The lower and upper limits of the weight of each indicator.
[0024] Weighting each indicator Considered as being in the constraint set For unknown variables within the range, in order to ensure that the mathematical programming problem has a feasible solution and the evaluation result is reasonable, the weight constraints must satisfy: the sum of all lower limits of weights is not greater than 1 and the sum of all upper limits of weights is not less than 1.
[0025] In applications at different ports or during different time periods, the weight range can be dynamically adjusted based on the characteristics of the pollution structure and the variability of monitoring data. When the coefficient of variation of pollutant concentration is large (indicating significant changes in the pollution structure), the weight fluctuation range can be appropriately widened (e.g., ±30%) to improve the model's adaptability; when the pollution level is stable, the fluctuation range can be narrowed (e.g., ±10%) to enhance the model's stability. Through the above adaptive mechanism, the weight constraint set... It can automatically update according to changes in port pollution characteristics, so that the comprehensive evaluation results of the present invention maintain high robustness and comparability under different environmental conditions.
[0026] Define each evaluation object (the first) i The weighted Euclidean distance function from the ideal solution to the positive and negative ideal solutions: Step 5: Calculate the relative closeness function to the ideal solution. Define the relative closeness function for each evaluation object. Step Six: Solve for the relative proximity of the intervals. By solving the following two mathematical programming problems, determine the upper and lower bounds of the relative proximity of each evaluation object, thus forming an interval number. The mathematical programming problem is a fractional programming problem, which is transformed into an equivalent linear programming problem using the Charnes-Cooper transformation for solution. The specific steps are as follows: 1. Establish the original fractional programming model: For the i For each evaluation object, the relative proximity function is defined as: in, and They represent the first i The weighted Euclidean distance from each evaluation object to the positive and negative ideal solutions: The objective function is in fractional form, and solving it directly will lead to a nonlinear optimization problem.
[0027] 2. Introduce auxiliary variables for transformation: Let the auxiliary variable t > 0, and define a new variable: The original constraints will be: Transform into 3. Substitute into the original expression and linearize: Will Substituting the relative proximity function, t in the denominator is eliminated, and the original fractional form is transformed into a function of the variable. The linear form of . For the minimization problem, it can be expressed as: Among them, coefficient , , All are derived from the numerical differences of the indicators in the standardized matrix. ) was calculated.
[0028] 4. Solving linear programming problems: After the above transformation, the model becomes a standard linear programming form, and the optimal solution can be directly solved using existing linear programming algorithms. ).
[0029] according to Back-substitution is used to calculate the weights of each indicator, and then the corresponding values are obtained. or .
[0030] 5. Forming a relatively close proximity range: Repeat the above minimization and maximization process for each evaluation object to obtain the lower and upper limits of its relative closeness: This constitutes the comprehensive evaluation range of the object under all reasonable weight combinations.
[0031] By employing the Charnes-Cooper transformation and linear programming solution described above, this invention effectively linearizes the original fractional programming problem, significantly reducing the solution complexity and computational uncertainty, and ensuring the robustness and reproducibility of the evaluation results under different weight configurations.
[0032] Step Seven: Comprehensive Evaluation and Ranking. Using the interval number ranking method, determine the relative proximity intervals of all evaluated objects. , , Sort the samples and derive robust relative proximity scores. R * i The higher the value, the lighter the overall pollution level and the better the air quality for that day.
[0033] Example 2: Taking the air pollution assessment of Zhenjiang Port in July 2023 as an example, the specific implementation of the present invention will be described in detail. This month was chosen because its overall pollution concentration was the lowest, which helps to clearly demonstrate the stability and distinguishing ability of the method of the present invention under complex backgrounds.
[0034] This embodiment applies the method of the present invention to comprehensively evaluate the daily air pollution status of Zhenjiang Port in July 2023 (a total of 31 days, i.e., n=31 evaluation objects).
[0035] Step 1: Data Collection and Standardization Daily average concentration monitoring data of six major pollutants (PM2.5, PM10, SO2, NO2, CO, and O3, i.e., m=6 indicators) were collected in Zhenjiang Port from July 1 to 31, 2023, and an initial evaluation matrix was constructed. .
[0036] The initial evaluation matrix D was standardized using vector normalization. Taking PM2.5 as an example, the concentration data for 31 days in July were as follows: Then the first i PM of the day 2.5 The standardized value is: in, Let j be the original concentration value of the j-th pollutant (indicator) on day i (the evaluation object). This is the normalized value.
[0037] Repeat this operation for the remaining five pollutants to obtain the standardized moments: .
[0038] Step 2: Determine the positive and negative ideal solutions in the case of no weights Since all six pollutants are cost-effective indicators where lower concentrations are better, the positive ideal solution (r⁺) and negative ideal solution (r⁻) are directly determined from the standardized matrix R without introducing weights. Positive ideal solution A + The negative ideal solution A is formed by the minimum values of each column (each pollutant) in R. - It consists of the maximum values of each column in R. The calculation formula is as follows: in, and Let represent the positive and negative ideal solution values of the j-th index, respectively. Up to this point, no weights have been introduced into any of the calculations.
[0039] Step 3: Introduce a set of weight constraints and construct a weighted distance function Weight vectors of the six pollutants Treat it as an unknown variable, and define the set of constraints it satisfies as follows: The requirement is that all weight constraints must satisfy the following conditions: the sum of all lower weight limits ≤ 1 and the sum of all upper weight limits ≥ 1.
[0040] Construct a weighted Euclidean distance function from each evaluation object (each day) to the positive and negative ideal solutions, taking July 15th (assumed to be the day) as the starting point. i Taking 15 days as an example, its weighted Euclidean distance function to the positive and negative ideal solutions is... and The calculation formula is as follows: Step 4: Solve for the relative proximity interval Based on the weighted distance function, taking July 15th as an example, the relative proximity function for the 15th day is defined. for: Next, in the weight constraint set The following two mathematical programming problems are used to determine... In the constraint set The minimum value (lower limit) ) and maximum value (upper limit) ): These two optimization problems belong to fractional programming problems, which can be transformed into equivalent linear programming problems through appropriate mathematical transformations (such as the Charnes-Cooper transformation) for efficient solutions. After solving, the relative proximity interval for day 15 is obtained: .
[0041] Repeat this step for all 31 days to obtain the daily comprehensive pollution assessment interval value.
[0042] Step 5: Comprehensive Evaluation and Ranking Obtain the relative closeness intervals R1ᴵ, R2ᴵ, ..., R for all evaluation subjects (31 days). 31 Afterwards, an initial sorting is performed using the interval number sorting method. Simultaneously, to obtain a clear single evaluation value for the final sorting, the robustness relative closeness of each interval is calculated. The formula is: The stability is relatively close to the degree. Defined as the comprehensive pollution index for that day, the higher the Rᵢ value, the lighter the overall pollution level and the better the air quality. Based on this, the air quality for the 31 days in July is ranked in the final order.
[0043] To verify the effectiveness and superiority of the method of this invention, the results obtained in this embodiment (UW-TOPSIS) are compared with the evaluation results of the same dataset using the traditional AHP-TOPSIS and EW-TOPSIS methods. Using the normalized AQI score as a reference benchmark, the relative error of the daily evaluation results of each method is calculated. The smaller the error, the higher the degree of agreement between the method and the actual pollution situation, and the more stable the performance.
[0044] The comparison results are as follows Figure 2 and Figure 3 As shown, the method of this invention (UW-TOPSIS) performed most stably during the 31-day low-pollution month. Specifically, the evaluation error was below the reference line of 0.06 for 22 days (68.75%) of the month, with the flattest error curve and the smallest fluctuation range.
[0045] In contrast, the AHP-TOPSIS method, due to its subjective weighting (excessively high combined weights for PM2.5 and PM10), exhibits poor adaptability to changes in pollution structure, resulting in errors exceeding 0.06 for 14 days (45.16%). While the EW-TOPSIS method uses objective weighting, it is extremely sensitive to data fluctuations, showing significant error peaks on dates with abnormal concentration values (such as July 4th, 6th, and 13th), leading to errors exceeding the standard for 16 days (51.61%) throughout the month, making it the most unstable of the three methods.
[0046] The comparative analysis demonstrates that the post-weighted optimization method proposed in this invention significantly improves the stability and robustness of the evaluation results by avoiding the inherent defects of pre-weighting, and its overall performance is superior to the traditional AHP-TOPSIS and EW-TOPSIS methods.
[0047] As described above, although the invention has been shown and described with reference to specific preferred embodiments, it should not be construed as limiting the invention itself. Various changes in form and detail may be made without departing from the spirit and scope of the invention as defined in the appended claims.
Claims
1. A comprehensive evaluation method for port air pollution based on unweighted TOPSIS, characterized in that, include: Step 1: Obtain the concentration data of n evaluation objects in the target port area under m air pollutant indicators within a specified time period, construct an initial evaluation matrix; and standardize the initial evaluation matrix to obtain a standardized matrix; Step 2: Without pre-setting index weights, determine the positive and negative ideal solutions from the standardized matrix based on the attributes of each index; Step 3: Set the weight of each indicator as an unknown variable that varies within a preset set of weight constraints, and construct a weighted distance function from each object to be evaluated to the positive and negative ideal solutions based on the weight variables; Step 4: Based on the weighted distance function, establish the relative proximity function for each object to be evaluated; and by solving a mathematical programming problem, determine the minimum and maximum values of the relative proximity function for each object to be evaluated under the weight constraint set, thereby constituting the relative proximity interval of the object. Step 5: Based on the relative proximity range of all objects to be evaluated, calculate the final comprehensive evaluation value of each object, and rank the port air pollution levels of each object to be evaluated accordingly.
2. The port air pollution comprehensive evaluation method based on unweighted TOPSIS according to claim 1, characterized in that, The air pollutant indicators in step one include PM2.5, PM10, NO2, SO2, CO, and O3.
3. The port air pollution comprehensive evaluation method based on unweighted TOPSIS according to claim 1, characterized in that, The standardization process in step one uses vector normalization, and its calculation formula is as follows: in, For the first i The first evaluation object j The original values of each indicator This is the normalized value.
4. The port air pollution comprehensive evaluation method based on unweighted TOPSIS according to claim 3, characterized in that, In step two, the optimal standardized value of each indicator is used to construct the positive ideal solution, and the worst standardized value of each indicator is used to construct the negative ideal solution, forming a reference benchmark for comparing the evaluation objects. The calculation formula is as follows: ， ； in, and They represent the first j The positive and negative ideal solutions for each index.
5. The port air pollution comprehensive evaluation method based on unweighted TOPSIS according to claim 3, characterized in that, The weight vectors of each indicator in step three The set of constraints that are satisfied is: ； The weight constraints satisfy that the sum of all lower weight values is not greater than 1 and the sum of all upper weight values is not less than 1. in, and The first j The preset lower and upper limits of the weight of each indicator.
6. The port air pollution comprehensive evaluation method based on unweighted TOPSIS according to claim 5, characterized in that, The weighted distance function used in step three is the weighted Euclidean distance function, and the calculation formula is as follows: in, and Each evaluation object i The weighted Euclidean distance to the positive ideal solution and the negative ideal solution.
7. The port air pollution comprehensive evaluation method based on unweighted TOPSIS according to claim 6, characterized in that, The relative proximity function in step four Defined as: The lower and upper limits of relative approximation are obtained by solving the following mathematical programming problem, which is a fractional programming problem, and is then transformed into an equivalent linear programming problem using the Charnes-Cooper transformation for solution: ； ； Thus, the relative proximity interval of the i-th evaluation object is obtained: .
8. The port air pollution comprehensive evaluation method based on unweighted TOPSIS according to claim 7, characterized in that, In step five, the interval number sorting method is used to sort the relative proximity intervals of all evaluation objects. , , Sort the samples and calculate the robust relative proximity using the following formula. R * i : The stability is relatively close to the degree. Defined as a comprehensive pollution index, The higher the value, the lower the overall pollution level.