## 2 Materials and Methods
### Development of a Multiattribute Similarity Measure  
The Dice similarity measure is a kind of similarity evaluation method that is used to calculate the similarity degree between two objectives [41]. In regard to statistics, this similarity degree indicates the amount of association between two given objectives compared to the amount of association between them expected by chance [42]. Compared with other similarity measures, it can evaluate the similarity when one vector is equal to zero and its calculation process is relatively convenient. Here, the Dice similarity measure was incorporated into the neutrosophic number to develop a method for slope stability assessment, since slope is a complex system that consists of a series of variables with different ranges.  
Considering two vectors \(A=\left(a_{1},a_{2},\cdots,a_{n}\right)\) and \(B=\left(b_{1},b_{2},\cdots,b_{n}\right)\) that feature length \(n\) and positive coordinates. The Dice similarity measure can be defined as  
\[D(A,B)=\frac{2A\times B}{\|A\|_{2}^{2}+\|B\|_{2}^{2}}=\frac{2\sum_{i=1}^{n}a_{i }b_{i}}{\sum_{i}^{n}a_{i}^{2}+\sum_{i}^{n}b_{i}^{2}} \tag{7}\]Presume that there are two neutrosophic numbers \(N_{Aj}=a_{Aj}+b_{Aj}I\) and \(N_{Bj}=a_{Bj}+b_{Bj}I\). For convenience, those numbers can be written as two sets \(A=\{N_{A1},N_{A2},\cdots,N_{An}\}\) and \(B=\{N_{B1},N_{B2},\cdots,N_{Bn}\}\). Then, the neutrosophic Dice similarity measure is defined as  
\[D(A,B)=\frac{1}{n}\sum\limits_{j=1}^{n}\frac{2\big{[}\big{(}a_{Aj}+\inf\big{(} b_{Aj}I\big{)}\big{)}\times\big{(}a_{Bj}+\inf\big{(}b_{Bj}I\big{)}\big{)}+ \big{(}a_{Aj}+\sup\big{(}b_{Aj}I\big{)}\big{)}\times\big{(}a_{Bj}+\sup\big{(}b _{Bj}I\big{)}\big{)}\big{]}}{\big{[}\big{(}a_{Aj}+\inf\big{(}b_{Aj}I\big{)} \big{)}^{2}+\big{(}a_{Bj}+\inf\big{(}b_{Bj}I\big{)}\big{)}^{2}\big{]}+\big{[} \big{(}a_{Aj}+\sup\big{(}b_{Aj}I\big{)}\big{)}^{2}+\big{(}a_{Bj}+\sup\big{(}b _{Bj}I\big{)}\big{)}^{2}\big{]}} \tag{8}\]  
Obviously, a single value can be denoted as a special interval between the lower and upper bounds. For example, a single value 6 can be transformed into an interval value [6; 6] that is computable in Equation (8). This measure \(D(A,B)\) has the following properties.  
\[1\geq D(A,B)\geq 0) \tag{9}\]  
\[D(A,B)=D(B,A) \tag{10}\]  
\[D(A,B)=1,\text{ if }A=B \tag{11}\]  
The measure with a single attribute does not satisfy the requirement of slope stability assessment because a slope possesses attributes of various types and levels. Multiple attribute method has to be put forward to solve this problem.  
The neutrosophic numbers \(N_{Aj}\) and \(N_{Bj}\) were further divided into \(N_{Ajk}\) and \(N_{Bjk}\), which restructured the evaluation system to two levels. By doing so, both attribute and subattributes of slopes can be considered in the developed method. Let \(N_{An}=\big{\{}N_{Aj1},N_{Aj2},\cdots,N_{Ajr}\big{\}}\) and \(N_{Bn}=\big{\{}N_{Bj1},N_{Bj2},\cdots,N_{Bjr}\big{\}}\) be two sets of subattributes that can be expressed as \(N_{Ajk}=a_{Ajk}+b_{Ajk}I\) and \(N_{Bjk}=a_{Bjk}+b_{Bjk}I\). Then, the weight of elements \(N_{Aj}\) and \(N_{Bj}\) are given as \(w_{j}\in[0,1]\) with \(\sum\limits_{j=1}^{n}w_{j}=1\). The weight of subelements is evaluated as \(\sum\limits_{k=1}^{r}w_{k}=1\). and \(w_{k}\in[0,1]\). Hence, the neutrosophic Dice similarity measure of multiple elements is denoted as  
\[D_{W}(A,B) =\sum\limits_{j=1}^{n}w_{j}\sum\limits_{k=1}^{r}w_{k}\frac{2\big{[} \big{(}a_{Ajk}+\inf\big{(}b_{[PERSON]}I\big{)}\big{)}\times\big{(}a_{[PERSON]}+\inf\big{(} b_{[PERSON]}I\big{)}\big{)}+\big{(}a_{[PERSON]}+\sup\big{(}b_{[PERSON]}I\big{)}\big{)}\times \big{(}a_{[PERSON]}+\sup\big{(}b_{[PERSON]}I\big{)}\big{)}\big{]}}{\big{[}\big{(}a_{[PERSON]}+ \inf\big{(}b_{[PERSON]}I\big{)}^{2}+\big{(}a_{[PERSON]}+\inf\big{(}b_{[PERSON]}I\big{)}\big{)}^ {2}\big{]}+\big{[}\big{(}a_{[PERSON]}+\sup\big{(}b_{[PERSON]}I\big{)}^{2}+\big{(}a_{[PERSON]}+ \sup\big{(}b_{[PERSON]}I\big{)}\big{)}^{2}\big{]} \tag{12}\]  
\(D_{W}\) (\(A\),\(B\)) also satisfies the properties denoted in Equations (9)-(11).