2.1 Analytic hierarchy process (AHP)
Analytic Hierarchy Process can effectively quantify the qualitative problem, and use the maximum eigenvalue and feature vector of judgment matrix to calculate the weight value of the index or factor of a layer relative to each index or factor of the upper layer. Where the judgment matrix \(B = (b_{ij} )_{m \times m}\) is constructed using the Arabic numerals 1 to 9 and their reciprocals as scales.
The weight is calculated by the feature vector, and the calculation equation is:
$$BW = \lambda_{\max } W$$
(1)
where \(\lambda_{\max }\) is the largest eigenvalue of the judgment matrix, B is hierarchical judgment matrix, W is weight vector.
The weight value of each level index relative to the previous level index or factor can be obtained by normalizing the weight vector. When establishing a judgment matrix, due to the complexity and diversity of objective things and the limitations of people's understanding of objective things, the maximum eigenvalue usually obtained is not unique. In order to avoid the deviation of the weight vector and ensure that the judgment matrix meets the requirements, it is necessary to perform a consistency check on the judgment matrix. Two consistency indexes are introduced: the measure judgment matrix deviation consistency index CI and the average random consistency index RI. Where the average random consistency index RI can be obtained by looking up the table.
CI is an indicator to measure the deviation consistency of the judgment matrix:
$$CI = \frac{{\lambda_{\max } - m}}{m - 1}$$
(2)
$$CR = \frac{CI}{{RI}}$$
(3)
CR is the proportion of consistency, which is the ratio of CI to RI. If CR < 0.1, the judgment matrix consistency check is qualified; Otherwise, the scale value of the judgment matrix needs to be corrected appropriately until the matrix consistency check meets the requirements.
2.2 Entropy weight method (EWM)
Entropy Weighting Method is used to determine the weights, which means that the weights of the indicators are determined based on the objective information contained in the data itself. Suppose the decision matrix is \(Y = \left\{ {y_{ij} } \right\}_{m \times n}\). In the equation, \(y_{ij}\) is the evaluation value of the jth index of the ith evaluation scheme. Then the entropy value of the jth index is calculated by the following equation:
$$e_{j} = - \frac{{\sum\nolimits_{i = 1}^{m} {p_{ij} \ln p_{ij} } }}{\ln m}.$$
(4)
$$p_{ij} = - \frac{{y_{ij} }}{{\sum\nolimits_{i = 1}^{m} {y_{ij} } }}.$$
(5)
where \(p_{ij}\) is the feature ratio of the jth index of the ith evaluation scheme, and the entropy weight of the jth index can be calculated by the following equation:
$$\beta_{j} = - \frac{{1 - e_{j} }}{{\sum\nolimits_{j = 1}^{n} {(1 - e_{k} )} }}$$
(6)
2.3 Comprehensive weight determination
The obtained subjective weight and objective weight are respectively weighted by Equal Weight method, Multiplication Weight Method, Difference Coefficient Method and Game theory to calculate the comprehensive weight W. Difference Coefficient Method (DCM) is shown in Eq. (7).
$$W = a\alpha + b\beta$$
(7)
where a, b are the undetermined coefficients of subjective and objective weighting; a + b = 1, which represents the importance of the evaluation model to the subjective and objective influences. Multiplicative Weighting (MW) is shown in Eq. (8).
$$W_{j} = \frac{{\sqrt {\alpha_{j} \beta_{j} } }}{{\sum\nolimits_{j = 1}^{n} {\sqrt {\alpha_{j} \beta_{j} } } }}$$
(8)
where \(\alpha_{j}\) is the weight calculated by the analytic hierarchy process, \(\beta_{j}\) is the weight calculated by the entropy method.
2.4 Game theory combinatorial empowerment (GTCE)
Combining the game theory idea with the AHP and Entropy method for optimization, the game theory combination weighting is to take the conflict between different weights as the coordination goal, compare and coordinate, so as to find the optimal result that takes both subjective and objective weights into consideration. The weighting steps for merging are as follows:
$$W = \sum\limits_{i = 1}^{m} {\alpha_{i} } w_{i}^{T}$$
(9)
$$\left( {\begin{array}{*{20}c} {w_{1} \cdot w_{1}^{T} } & { \cdot \cdot \cdot } & {w_{1} \cdot w_{m}^{T} } \\ { \cdot \cdot \cdot } & { \cdot \cdot \cdot } & { \cdot \cdot \cdot } \\ {w_{m} \cdot w_{1}^{T} } & { \cdot \cdot \cdot } & {w_{m} \cdot w_{m}^{T} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\alpha_{1} } \\ {\alpha_{2} } \\ {\alpha_{3} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {w_{1} \cdot w_{1}^{T} } \\ { \cdot \cdot \cdot } \\ {w_{m} \cdot w_{m}^{T} } \\ \end{array} } \right)$$
(10)
where \(w_{i}\) is the weight vector determined by the ith method, \(w_{i} = \{ w_{i1} ,w_{i2} ,\ldots,w_{im} \} (i = 1,2,\ldots,m)\), \(\alpha_{i}\) is the linear combination coefficient, (αi > 0).
Since αi and W in Eq. (9) are assumed values, the sum of αi calculated by Eq. (10) may not be 1. According to the idea of game theory, the goal is to minimize the dispersion to find the similarities and differences of different weights. The values need to be normalized to obtain \(\alpha_{i}^{*}\), and the weight \(W^{*}\) obtained by \(\alpha_{i}^{*}\) is:
$$W_{3} = \sum\limits_{i = 1}^{n} {\alpha_{i}^{*} } w_{i}^{T}$$
(11)
2.5 Grey relational analysis (GRA)
Grey relational analysis is a method to quantitatively describe and compare the development and change of the system. The basic idea is to judge whether they are closely connected by constructing the reference data column and the comparison data column, and calculating the geometric similarity of the two. It reflects the degree of association between variables. The specific calculation steps are as follows:
2.5.1 Construct a reference sequence
Suppose the evaluation value of the index j of the kth group of data is \(h_{j} (k)\), and its value is calculated according to the following equation:
$$h_{j} (0) = \left\{ \begin{gathered} \min_{1 \le k \le n} \{ h_{j} (k)\} ,j \in {\text{cost index}} \hfill \\ \max_{1 \le k \le n} \{ h_{j} (k)\} ,j \in {\text{benefit index}} \hfill \\ h(0),{\text{j}} \in {\text{moderate indicator}} \hfill \\ \end{gathered} \right..$$
(12)
The final constructed optimal index set is:\(H(0) = [h_{1} (0),h_{2} (0),...,h_{n} (0)]\). Where \(h_{j} (0)\) represents the optimal value of the jth index, \(j = 1,2,...,n\).
2.5.2 Construct a comparison sequence
Since the dimensions of different indicators will be different and cannot be directly calculated, the indicators should be normalized, and the normalized indicator value \(h^{^{\prime}}_{j}\) is obtained by using the efficacy coefficient method. The calculation equation is as follows:
$$h^{^{\prime}}_{j} (k) = c + \frac{{h_{j} (k) - \min \{ h_{j} (1), \cdots ,h_{j} (k)\} }}{{\max \{ h_{j} (1), \cdots ,h_{j} (k)\} - \min \{ h_{j} (1), \cdots ,h_{j} (k)\} }} \times d$$
(13)
where \(k = 1,2,...,m;j = 1,2,...,n\) and c, d are constants DETERMINED by the requirements of the data gap, c represents the amount of translation, and d represents the amount of zoom; This paper takes c = 0.85, d = 0.15.
2.5.3 Calculate the gray correlation coefficient matrix
After normalizing the indicators, the optimal indicator set \(H^{^{\prime}} (0) = [h_{1}^{^{\prime}} (0),h_{2}^{^{\prime}} (0),...,h_{n}^{^{\prime}} (0)]\) is used as the reference sequence. According to the grey relational analysis, the grey relational coefficient \(\delta_{kj}\) of the jth index of the kth scheme is calculated respectively. The equation is as follows:
$$\delta_{kj} = \frac{{\min_{k} \min_{j} \left| {h^{^{\prime}}_{j} (0) - h^{^{\prime}}_{j} (k)} \right| + \rho \max_{k} \max_{j} \left| {h^{^{\prime}}_{j} (0) - h^{^{\prime}}_{j} (k)} \right|}}{{\left| {h^{^{\prime}}_{j} (0) - h^{^{\prime}}_{j} (k)} \right| + \rho \max_{k} \max_{j} \left| {h^{^{\prime}}_{j} (0) - h^{^{\prime}}_{j} (k)} \right|}}$$
(14)
where \(\rho\) is the resolution coefficient. The smaller the \(\rho\) is, the greater the resolution. Generally, the value range of \(\rho\) is (0, 1). When \(\rho \le 0.5463\), the resolution is the best, usually \(\rho\) = 0.5.
After all the gray correlation coefficients are calculated, the gray correlation coefficient matrix G is further obtained:
$$G = \left( {\begin{array}{*{20}c} {\delta_{11} } & \ldots & {\delta_{1n} } \\ \vdots & \ddots & \vdots \\ {\delta_{m1} } & \cdots & {\delta_{mn} } \\ \end{array} } \right)$$
(15)
2.5.4 Calculate relevance
The correlation degree \(r_{k}\) of each scheme is calculated, which is the score of each scheme. The higher the score, the closer the evaluation plan is to ideal optimal plan, and the one with the highest score is recorded as optimal plan. The specific equation is as follows:
$$r_{k} = \frac{1}{n}\sum\limits_{j = 1}^{n} {\delta_{kj} }$$
(16)
2.6 Fuzzy comprehension evaluation method (FCEM)
Fuzzy comprehensive evaluation is a method to comprehensively evaluate the subordinate level of evaluation objects by using fuzzy mathematical tools. Based on the comprehensive weight vector W and the correlation coefficient evaluation matrix G calculated above, the comprehensive evaluation mathematical model is constructed as follows:
where C is the final decision vector of the m evaluation schemes, \(C = [c(1),c(2),...c(m)]\) and \(c(i)\) are the gray correlation degree of the ith scheme; G is the evaluation matrix of each index, there is \(G = \{ g_{i} (j)\}\); W is the vector weight of the n evaluation indicators, there is \(W = [w_{1} ,w_{2} ,...w_{n} ]\); "0" indicates a fuzzy operator, and here the weighted average type synthetic operator is chosen, which is \(C = W^{0} G = W \times G\).
Therefore, the grey relational degree is finally expressed as:
$$c(i) = \sum\nolimits_{j = 1}^{n} {w_{j} g_{i} (j)}.$$
(18)
The final calculated correlation degree is sorted by size. If the evaluation scheme is close to the ideal optimal scheme, the higher the correlation degree is. The solution with the greatest correlation is the optimal solution for comparison.
