# A Three-Way Acceleration Approach for Interval-Valued Multi-Attribute Decision-Making Problems
## 3 TWD Model with Degree Measure
### Interval Fuzzy Dominance Relationship
Therefore, we define \(T\) as an interval-valued decision matrix mentioned above. \(T=\left[ \begin{array}{cccc}[a_{1k_{1}}^{l},a_{1k_{1}}^{u}]&[a_{1k_{2}}^{l}, a_{1k_{2}}^{u}]&\cdots&[a_{1k_{x}}^{l},a_{1k_{x}}^{u}]\\ [a_{2k_{1}}^{l},a_{2k_{1}}^{u}]&[a_{2k_{2}}^{l},a_{2k_{2}}^{u}]&\cdots&[a_{2k_{ x}}^{l},a_{2k_{x}}^{u}]\\ \vdots&\vdots&[a_{ik_{p}}^{l},a_{ik_{p}}^{u}]&\vdots\\ [a_{mk_{1}}^{l},a_{mk_{1}}^{u}]&[a_{mk_{2}}^{l},a_{mk_{2}}^{u}]&\cdots&[a_{mk_{ x}}^{l},a_{mk_{x}}^{u}]\end{array} \right].\)  
For \(x_{i}\in U\), attributes \(k_{p}\in AT\), \(1\leq i\leq m\), \(1\leq p\leq n\), of the evaluation value of the \(i\)th alternatives are denoted by \(a_{ikp}\). Thus, \(a_{ik_{p}}=f(x_{i},k_{p})=[a_{ik_{p}}^{l},a_{ik_{p}}^{u}]\). That is, \(k_{p}\in K\), the corresponding fuzzy dominance probability matrix is denoted by \(X_{k_{p}}=(x_{ij}^{k_{p}})\), and \(x_{ij}^{k_{p}}\) denotes the degree of \(x_{i}\) superior to \(x_{j}\) under the attribute \(k_{p}\). According to the actual situation, there are two ways to build \(x_{ij}^{k_{p}}\):  
1. If \(\sim[f(x_{i},k_{p})]^{\succcurlyeq}\cup[f(x_{j},k_{p})]^{\succcurlyeq}=U\), the dominance degree of \(x_{i}\) superior to \(x_{j}\) is: \[x_{ij}=\frac{|U|}{|U|+|[f(x_{i},k_{p})]^{\succcurlyeq}\cup\sim[f(x_{j},k_{p}) ]^{\succcurlyeq}|}.\] (9)
2. If \(\sim[f(x_{i},k_{p})]^{\succcurlyeq}\cup[f(x_{j},k_{p})]^{\succcurlyeq}\
eq U\), the dominance degree of \(x_{i}\) superior to \(x_{j}\) is: \[x_{ij}=\frac{|\sim[f(x_{i},k_{p})]^{\succcurlyeq}\cup[f(x_{j},k_{p})]^{\succcurlyeq }|}{|U|+|\sim[f(x_{i},k_{p})]^{\succcurlyeq}\cup[f(x_{j},k_{p})]^{\succcurlyeq}|}.\] (10)  
Here, \([f(x_{i},k_{p})]^{\succcurlyeq}\) denotes the dominance class of \(x_{i}\) under attribute \(k_{p}\), \(\sim[f(x_{j},k_{p})]^{\succcurlyeq}\) denotes the non-dominance class of \(x_{j}\) under attribute \(k_{p}\). The matrix composed of \(x_{ij}\) is called an interval fuzzy dominance probability matrix, which is denoted as \(X_{k_{p}}\). According to the Definition 6, the fuzzy dominance probability matrix \(\widetilde{X_{k_{p}}}\) is as follows:  
\(\widetilde{X_{k_{p}}}=\left[ \begin{array}{cccc}x_{11}^{k_{p}}&x_{12}^{k_{p} }&\cdots&x_{1m}^{k_{p}}\\ x_{21}^{k_{p}}&x_{22}^{k_{p}}&\cdots&x_{2m}^{k_{p}}\\ \vdots&\vdots&x_{ij}^{k_{p}}&\vdots\\ x_{m1}^{k_{p}}&x_{m2}^{k_{p}}&\cdots&x_{mm}^{k_{p}}\end{array} \right].\)  
**Definition 7**. : _Given that \(S=(U,AT,V,f)\) and \(x_{i}\) and \(x_{j}\) are any two objects under same attributes, \(R^{\succeq}_{B}\) and \(R^{\prec}_{B}\) can be described separately in the following_  
\[ \begin{split} R^{\succeq}=\{(x_{i},x_{j})\in U^{2}\mid f(k_{p},x_{ i})\geq f(k_{p},x_{j})(k_{p}\in K)\},\\ R^{\prec}=\{(x_{i},x_{j})\in U^{2}\mid f(k_{p},x_{i})<f(k_{p},x_{j})(k_{ p}\in K)\},\end{split}  \tag{11}\]  
_where \(R^{\succeq}\) and \(R^{\prec}\) are called dominating and dominated relationships, respectively, and also \(f(k_{p},x_{i})\geq f(k_{p},x_{j})\iff[a^{l}_{kp},a^{u}_{ik_{p}}]\geq[a^{l}_{ jk_{p}},a^{u}_{jk_{p}}](k_{p}\in K)\)._