*6.1. Single-Index Measure*

### 6.1.1. Single-Index Measure Matrix

Set μijrq = μ xijr ∈ cq to express the degree that xijr belongs to cq, which is the qth evaluation class (rating). μ must meet the conditions as follows:

$$\mathbf{r} \cdot \mathbf{0} \le \mu \left( \mathbf{x}\_{\text{jirq}} \in \mathbf{c}\_{\text{q}} \right) \le \mathbf{1}, \mathbf{i} = 1, 2, \dots, \mathbf{r}, \mathbf{n}; \mathbf{j} = 1, 2, \dots, \mathbf{m}; \mathbf{r} = 1, 2, \dots, \mathbf{k}; \mathbf{q} = 1, 2, \dots, \mathbf{p} \tag{1}$$

$$\mathbf{u}(\mathbf{x}\_{\text{irr}} \in \mathbb{C}) = 1, \mathbf{i} = 1, 2, \cdots, \mathbf{r}, \mathbf{i} \text{; } \mathbf{j} = 1, 2, \cdots, \mathbf{r}, \mathbf{m}; \mathbf{r} = 1, 2, \cdots, \mathbf{k} \tag{2}$$

$$\mu\left(\mathbf{x}\_{\text{ijr}} \in \bigcup\_{l=1}^{\mathsf{q}} \mathbf{c}\_{\text{l}}\right) = \sum\_{l=1}^{\mathsf{q}} \mu\left(\mathbf{x}\_{\text{ijr}} \in \mathsf{c}\_{\text{l}}\right) \neq \mathbf{q} = 1, 2, \dots, \mathsf{p} \tag{3}$$

Define Equation (2) as the normalization and Equation (3) as the additivity. That which meets the three equations above is unascertained measurement. The matrix that follows is a single index measure matrix [50].

$$\mathbf{u} \cdot \begin{pmatrix} \mu\_{\text{ij}\text{n}} \\ \mu\_{\text{i}\text{p}\text{q}} \end{pmatrix}\_{\text{k}\times\text{p}} = \begin{bmatrix} \mu\_{\text{i}\text{j}\text{1}} & \mu\_{\text{i}\text{j}\text{2}} & \cdots & \mu\_{\text{i}\text{j}\text{1}} \\ \mu\_{\text{i}\text{j}\text{2}1} & \mu\_{\text{i}\text{2}2} & \cdots & \mu\_{\text{i}\text{j}\text{2}} \\ \vdots & \ldots & \ddots & \ldots \\ \mu\_{\text{i}\text{jk}\text{1}} & \mu\_{\text{i}\text{jk}\text{2}} & \cdots & \mu\_{\text{i}\text{jk}\text{p}} \end{bmatrix} \mathbf{i} = 1, 2, \cdots, \text{\textquotedblleft m} \mathbf{j} = 1, 2, \cdots \text{\textquotedblright} \mathbf{m} \end{cases} \tag{4}$$

#### 6.1.2. Distinction Weight of Single-Index

Using the concept of information entropy to define the peak of index Iijr.

$$N\_{\rm ijr} = 1 + \frac{1}{\ln \mathbf{p}} \sum\_{\mathbf{q}=1}^{\mathbf{P}} \mu\_{\rm ijr\mathbf{q}} \ln \mu\_{\rm ijr\mathbf{q}} \tag{5}$$

p in Equation (5) represents the number of the evaluate ratings, μijrq is the measure of single index, and the value of Vijr expresses the degree that Iijr different to each evaluation class. The distinction weight is as follows:

$$\omega\_{\rm ilr} = \frac{\text{V}\_{\rm ilr}}{\text{s}} \text{ i} = 1,2,\cdots \text{ },\text{n:}\\\mathbf{j} = 1,2,\cdots \text{ },\text{m:}\\\mathbf{r} = 1,2,\cdots \text{ },\text{k} \tag{6}$$

k ∑ <sup>r</sup>=1 <sup>ω</sup>ijr = 1, 0 ≤ <sup>ω</sup>ijr ≤ 1, <sup>ω</sup>ijr is the classification weights of Ijr. <sup>ω</sup>ij = <sup>ω</sup>ij1, <sup>ω</sup>ij2, ··· , <sup>ω</sup>ijk is the classification weight vector of secondary grade index [51].

#### *6.2. First Grade Index Measure*

Set μiq = μ xi ∈ cq expresses the degree that sample xi belongs to cr, which is the rth evaluation class (rating).

$$\mathbf{u}\_{\text{iq}} = \sum\_{\mathbf{j}=1}^{\text{m}} \omega\_{\text{i}\text{j}} \mu\_{\text{i}\text{q}}; \mathbf{i} = 1, 2, \dots, \mathbf{v}, \mathbf{n}; \mathbf{q} = 1, 2, \dots, \mathbf{v} \text{ } \tag{7}$$

Due to 0 ≤ μiq ≤ 1, and p ∑ q=1 μiq = p ∑ q=1 m ∑ j=1 <sup>ω</sup>ijμijq = m ∑ j=1 <sup>ω</sup>ij p ∑ q=1 μijq = m ∑ j=1 <sup>ω</sup>ij = 1, μiq is the unascertained measure. Define μi1, μi2, ··· , μip as measure evaluation vector of xi's composite indicator. The matrix μiq n×p is measure matrix of comprehensive index [52].

$$\begin{aligned} \left(\mu\_{\text{iq}}\right)\_{\text{n}\times\text{p}} = \begin{bmatrix} \mu\_{11} & \mu\_{12} & \cdots & \mu\_{1p} \\ \mu\_{21} & \mu\_{22} & \cdots & \mu\_{2p} \\ \dots & \dots & \dots & \ddots & \dots \\ \mu\_{\text{n}1} & \mu\_{\text{n}2} & \cdots & \mu\_{\text{np}} \end{bmatrix} \end{aligned} \tag{8}$$

#### *6.3. Determination of First Grade Index Weight by AHP*

AHP was proposed by Saaty, an American operational research expert, in the 1970s [53]. It is a method of combining qualitative and quantitative, systematized and hierarchical qualities. It is a process of modeling and quantifying decision makers' decision thinking processes for complex systems. By using AHP, decision makers decompose the complex problems into several levels and factors, and make simple comparisons and calculations among the factors, so that they can ge<sup>t</sup> the weight of different plans and provide the basis for the best plan selection. In AHP, in order to make the judgment quantified, the key is to quantitatively describe the relative superiority of any two schemes to a certain criterion. For a single criterion, the comparison between the two plans can always demonstrate the advantages and disadvantages. AHP adopts the 1–9 scale method to give the quantitative scale for the evaluation of different situations. This scale is adopted in matrices to look for relative criteria's weights and to compare the alternatives linked to every criterion. Table 3 summarizes the basic ratio scale. All final weighted coefficients are shown in matrices. Alternatives and criteria can be ranked based on the overall aggregated weights in matrices. The alternative with the highest overall weight would be the most preferable [28,54].



Based on this first index's judgment matrix, the weights of every first grade index can be calculated by the geometric calculation method of mean.

$$\overline{\varpi\_{\mathbf{i}}} = \sqrt[n]{\prod\_{j=1}^{n} \mathbf{a}\_{\overline{\mathbf{i}}}} \left( \mathbf{i} = 1, 2, \cdots, n; \mathbf{j} = 1, 2, \cdots, n \right) \tag{9}$$

Then, by employing normalized processing, using the following equation:

$$
\omega\_{\bar{\mathbf{i}}} = \frac{\overline{\varpi\_{\bar{\mathbf{i}}}}}{\sum\_{i=1}^{n} \overline{\varpi\_{\bar{\mathbf{i}}}}} \tag{10}
$$

The weight vector of first index is obtained: ω = (<sup>ω</sup>1, ω2, ··· , <sup>ω</sup>n) T. The largest characteristic roots λmax = 1 n n ∑ i=1 (AW)i Wi can be calculated.

When solving practical problems, due to the complexity of objective phenomena and the mind's cognitive limitations, our understanding of the problem is subjective and involves one-sidedness and fuzzy judgment; the structure of the judgment matrix is, therefore, may not fully meet the requirements of consistency, and often includes some deviation. The constructed judgement matrix does not always meet consistency condition. If the judgement matrix passes the consistency test, the calculated index weight can be adopted; if the consistency test is not passed, the judgement matrix needs to be adjusted. The consistency index is divided into complete consistency (CI) and satisfactory consistency (CR), CI = λmax<sup>−</sup><sup>n</sup> <sup>n</sup>−1.

When CI = 0, the judgment matrix is considered to be completely consistent. CI = 0, it is considered that the judgment matrix is not completely consistent. CR = CI RI , RI is the average random consistency index, and the value of RI are shown in Table 4 for the judgment matrix of n = 1–14.

**Table 4.** The mean random consistency index.

