Abstract
In the application of uncertainty measure theory, the determination method of index weight mainly includes the subjective weight determination method and the objective weight determination method. The subjective weight determination method has the disadvantages affected by the subjective preference of the decision-maker. The objective weight determination method often ignores the participation degree of the decision-maker, and when using the uncertainty measure evaluation model to perform multi-index classification evaluation, the credible degree recognition criterion is often used as the attribute recognition of the object to be measured, because the credible degree is taken by the subjective people, and the different values of different people have a great influence on the evaluation results. In order to solve the above problems in the uncertainty measure theory, this paper used the combination weighting of game theory to determine the optimal weight. At the same time, the credible degree recognition criterion was improved on the basis of the concept of minimum uncertainty measure distance, and a game theory-improved uncertainty measure optimization model was proposed. Finally, the validity of the model was proven by a case.
1. Introduction
Uncertainty measure theory [1, 2] is a mathematical method for studying uncertainty information. It can quantitatively describe the size of a thing that is in an uncertainty state or has uncertainty. Since the uncertainty measure model was put forward, it has been widely used in various fields. But, in the practical application process, the authors found that there are two defects in the uncertainty measure theory. On one hand, the index weight is determined. So far, the determination of index weight is roughly divided into the subjective weight algorithm and the objective weighting method [3, 4]. The subjective weighting method is the method in which the decision-maker directly gives preference information, such as Analytic Hierarchy Process (AHP) [5], minimum square method [6], and Delphi method [7]. The objective weighting method is a method on the basis of decision matrix information, such as entropy method [8], multiobjective optimization method [9], principal component analysis method [10], and scheme closeness degree [11]. In actual decision-making, due to the complexity of the decision-making problem, it is difficult to agree with the actual situation by relying solely on the subjective judgment of the decision-maker or the weight directly given by the objective algorithm. On the other hand, it is the criterion of attribute recognition. At present, in the process of applying the uncertainty measure theory, the credible degree criterion is mainly used to identify and determine the classification level of the judged object. Since the credible degree is artificially determined, when the credible degree is taken to be of different values, the different discrimination results will be obtained and sometimes even get the opposite judgment or classification results. Therefore, in order to avoid the above problems in the use of the model, based on the existing subjective and objective weight solving models, this paper introduced the game weighting model and put forward an uncertainty measure model to improve the credible degree recognition criteria. The effectiveness of the algorithm is proven by an example.
2. Improved Uncertainty Measure Theory
Uncertainty measure theory is the base of improved uncertainty measure theory. First, uncertainty measure model of the research objects is established based on uncertainty measure theory. Then, the single index measure function of each discriminant index is established, and the uncertainty measure value of each evaluation index is calculated. Moreover, the weight of each discriminant index is determined by the combination weighting model based on game theory, and the multiple index comprehensive measure of the discriminant object is calculated. At last, the category of the object to be measured is determined based on the uncertainty measure distance.
2.1. Establishment of Discriminant Index System for the Research Objects
The influencing factors of the research object are analyzed and the discriminant index set for the research objects is established. are a set of objects to be evaluated, represented as , which is termed as the domain. Each evaluation object has one-way evaluation index spaces, represented as , with being denoted as the observed value of the object under the index .
2.2. Classification of Samples
is set as an evaluation space, wherein represents the evaluation level, and the level is higher than the level; that is, >.
2.3. Uncertainty Measure of Single Index
When the observed value of the index for the object is different, this index makes the level of the evaluation level of different, and the degree of the evaluation level of is a set of . As a measure, it must meet , and
is uncertainty measure; the single index evaluation matrix of the object is
2.4. Combination Weighting Model Based on Game Theory
The combination weighting of game theory is different from the traditional simple linear combination weighting. Its central idea is to “coordinate conflicts and maximize benefits,” that is, to comprehensively consider the relationship between the indexes, balance the subjective and objective weights, and find the optimization of weights [12, 13]. The basic algorithm is as follows.
① Construction of the basic weight vector set. Assuming that the weight values are obtained using the weighting method, the basic weight vector set of the method isAny linear combination of weight vectors is
where is linear combination coefficient; is a comprehensive weight value of weight set.
② The optimal combination weight. In order to find the balance in the different weights, the optimal effect weight vector is obtained. In the calculation process, it can be converted into an optimization of the weight coefficient to minimize the deviation between and differences . The calculation formula is as follows:
From the differential properties of the matrix, the first-order derivative condition for the optimization of (5) is
By solving the equation, the combination coefficient can be obtained, and it can be normalized; is obtained. The final combination weight is
2.5. Integrated Evaluation System
According to the index weightings determined, a comprehensive multiple-index measure for evaluating the object is obtained, where if , then is the uncertainty measure and is the comprehensive multiple-index evaluation uncertainty measure vector of .
2.6. Evaluation Criterion
The classification of the evaluation level is orderly, and of evaluation levels is better than of evaluation levels. Therefore, the maximum measure identification criterion is not suitable, and the confidence recognition criterion is used. is set at a credible degree (, usually a value of 0.6 or 0.7 is taken), , and is considered for the evaluation level of [14–20].
2.7. Calculation of Uncertainty Measure Distance
In order to reduce the influence of artificial subjective factors, the credible degree recognition criteria are optimized on the basis of the uncertainty measure distance, and the category of the object to be measured is determined by the minimum uncertainty measure distance. In the hierarchical space , is set as the uncertainty measure of each classification level, the minimum uncertainty measure distance is the Euclidean distance from the multiple indexes comprehensive measure to the classification level , and
2.8. Identification of Classification Level for the Objects to Be Evaluated
The uncertainty measure distance is compared; if , the objects to be evaluated are considered to be the closest to the classification pattern system, so the classification level of the objects to be evaluated belongs to [20–23].
3. A Case
The paper selects the data of Hanqiao typical abandoned mines in Jiawang district of Xuzhou provided by Ting Li [24] as the research object. Multiple indices related to the groundwater pollution risk of abandoned mines are considered in the following evaluation. Each index is subdivided into three aspects, which are the risk of pollution sources, the risk of pollution channel, and the hazard of the pollution receptors. Individual aspects are determined by multiple parameters. Qualitative indices are evaluated by semiquantitative methods, and quantitative indices are evaluated using measured values. The criteria used for classification and valuation are presented in Table 1. Each evaluation index was classified and valued, and the evaluation set is . Finally, the evaluation indices are assigned levels: I (high risk), II (medium risk), and III (low risk). The basic situations of three typical abandoned mines zones are shown in Table 2.
3.1. Single Index Measure Functions of Uncertainty Measure
In the light of the concept of single index measure function, the classification standard in Table 1, and the monitoring data in Table 2, the single index measure functions are constructed by linear measure functions most widely used in this paper. Figures 1–8 show the specific single index measure functions for all evaluation indexes, respectively.
In the light of single index measure functions mentioned in Figures 1–8 and combined with the monitoring data in Table 2, the single index evaluation matrix of Hanqiao coal mine of Jiawang in Xuzhou is calculated as follows:
3.2. Determination of Index Weights
In the study, according to the established evaluation index system of groundwater pollution risk in abandoned mines, the weights of all indexes are calculated by AHP. Similarly, using the entropy weight method, the weights of all indexes are calculated. Finally, combining the index weights obtained by AHP and the entropy method, the optimal weight coefficients and are calculated according to (6) and (7), and all the index weights are obtained based on the combined weighting of game theory. The calculation results are shown in Table 3.
From Table 3, the weight obtained by AHP fluctuates greatly, because the method is influenced by the expert subjective factors, highlighting the main factors involved in the indexes, ignoring the influence of some minor factors, which also directly affects the accuracy of the evaluation results. The weight obtained by entropy method has less fluctuation, because the method relies too much on the original sample data, but the original sample data is usually not very different, resulting in a relatively small difference in the weight distribution, and the weight obtained by this method also lacks accuracy. The weight obtained by game theory is located between the two, and the weight gap between the indexes has achieved a balance, which has reduced the impact of ignoring some minor factors and expert experience. An optimal weight of balance subjective and objective results is obtained.
3.3. Multiple Index Measure Evaluation Matrix Calculation
In the light of single index measure matrix and multiple index calculation formula, the multiple index vector is .
3.4. Credible Degree Recognition
According to the improved credible degree recognition formula (9), the optimized uncertainty measure distances , , and are obtained. Hence, the grade of Hanqiao coal mine is determined to be Class I (high risk). In order to verify the feasibility and accuracy of the model, the credible degree is taken as 0.6, 0.7, 0.8, and 0.9, respectively, and compared with the attribute recognition optimization model by the distance discriminant. When the credible degree is taken as 0.6 and .65>0.6, the grade of Hanqiao coal mine is determined to be Class I (high risk); when the credible degree is taken as 0.7 and .7.7, the grade of Hanqiao coal mine is determined to be Class II (medium risk); when the credible degree is taken as 0.8 or 0.9 and .8 or 0.9, the grade of Hanqiao coal mine is determined to be Class III (low risk). The results show that when the credible degree is different, it has a great influence on the accuracy of the discriminant results. And the result is compared with the evaluation result of Ting Li [24] who used AHP-fuzzy comprehensive evaluation method. The results are consistent, so it is feasible and accurate to predict the risk level of groundwater pollution in abandoned mines using the game theory and improved uncertainty measure model.
4. Conclusions
Analyzing the weight solving model and confidence recognition criteria in the uncertainty measure theory, based on the existing subjective and objective weight solving models, the optimal weight was determined by the idea of game theory. Based on the idea of the minimum uncertainty measure distance, the credible degree recognition criterion was improved, and a game theory-improved uncertainty measure optimization model was proposed. The example shows that the game theory-improved uncertainty measure evaluation model is more scientific, objective, and reasonable and deserves further study.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was financially supported by the open fund for State Key Laboratory of Water Resources Protection and Utilization in Coal Mining (SHJT-16-30.19) and the Special Fund for Basic Scientific Research of Central Colleges (310827173702).