# A Three-Way Acceleration Approach for Interval-Valued Multi-Attribute Decision-Making Problems
## 3 TWD Model with Degree Measure
### Interval Fuzzy Dominance Relationship  
This section proposes the concepts of intersecting and non-intersecting interval numbers. We use connection numbers to express the corresponding interval numbers and then establish decision models to compare any two interval numbers. **Definition 3**. : _Let \(y_{i}=[a_{i}^{l},a_{i}^{u}],y_{j}=[a_{j}^{l},a_{j}^{u}]\) be any two interval numbers. The comparison between \(y_{i}=[a_{i}^{l},a_{i}^{u}]\) and \(y_{j}=[a_{j}^{l},a_{j}^{u}]\) is presented as follows:_  
1. _If_ \(a_{i}^{l}<a_{j}^{l}<a_{i}^{u}<a_{j}^{u}\)_,_ \(y_{i}\) _and_ \(y_{j}\) _are called the intersecting interval numbers;_
2. _If_ \((a_{i}^{u}<a_{j}^{l})\vee(a_{j}^{u}<a_{i}^{l})\)_,_ \(y_{j}\) _and_ \(y_{j}\) _are called the non-intersecting interval numbers;_
3. \(a_{j}^{l}<a_{i}^{l}<a_{i}^{u}<a_{j}^{u}\) _or_ \(a_{i}^{l}<a_{j}^{l}<a_{j}^{u}<a_{i}^{u}\)_, they are special examples of intersecting interval numbers._  
_Here, \(e\in[-1,1]\), and it reflects uncertainty of the connection number._  
**Definition 4**. : _Suppose that \(y_{i}=[a_{i}^{l},a_{i}^{u}]\) and \(y_{j}=[a_{j}^{l},a_{j}^{u}]\) are any intersecting interval numbers. \(y_{i}\) and \(y_{j}\) can be defined in the following_  
\[ \begin{split} y_{i}^{\prime}&=\frac{a_{i}^{l}+a_{i} ^{u}}{2}+\frac{a_{i}^{u}-a_{j}^{l}}{a_{i}^{u}-a_{i}^{l}}e,\\ y_{j}^{\prime}&=\frac{a_{j}^{l}+a_{j}^{u}}{2}+\frac {a_{i}^{u}-a_{j}^{l}}{a_{j}^{u}-a_{j}^{l}}e,\end{split}  \tag{6}\]  
_where \(\frac{a_{i}^{l}+a_{i}^{u}}{2}\) and \(\frac{a_{j}^{l}+a_{j}^{u}}{2}\) represent the midpoints of \(y_{i}\) and \(y_{j}\), respectively. \(\frac{a_{i}^{u}-a_{j}^{l}}{a_{i}^{u}-a_{i}^{l}}\) and \(\frac{a_{i}^{u}-a_{j}^{l}}{a_{j}^{u}-a_{j}^{l}}\) represent the proportion of the determined distances on \(y_{i}\) and \(y_{j}\), respectively. \(y_{i}^{\prime}\) and \(y_{j}^{\prime}\) represent the connection numbers derived from \(y_{i}\) and \(y_{j}\), respectively._  
We can interpret that the new \(y_{i}\) is composed of two parts, one is the midpoint of a certain interval number, and another is an uncertain measurement of the intersecting part in the previous \(y_{i}\). Since it is difficult to compare any intersecting interval numbers \(y_{i}\) and \(y_{j}\), they are first represented by the connection number and then compared. In particular, when comparing the interval number \(y_{i}=[a_{i}^{l},a_{i}^{u}]\) and \(y_{j}=[a_{j}^{l},a_{j}^{u}]\), we have \(y_{j}>y_{i}\) if, and only if, \(a_{j}^{l}<a_{i}^{l}<a_{i}^{u}<a_{j}^{u}\). **Definition 5**. : _Let \(a+be\) be a connection number. Based on the principle of connection number plural operation, the modulus of \(a+be\) is given by_  
\[r=\sqrt{a^{2}+b^{2}}, \tag{7}\]  
_where \(r=\sqrt{a^{2}+b^{2}}\) is an interval multi-attribute decision vector projection model based on the connection number whose type is \(a+be\)._  
According to Definition 5, we may quickly obtain the decision vector projection model of the interval number and compare any two-interval-number. The comparison model is as follows:\(y_{i}=[a_{i}^{l},a_{i}^{u}]\) and \(y_{j}=[a_{j}^{l},a_{j}^{u}]\) are any two intersecting interval numbers and the decision vector projection models of \(y_{i}\), and \(y_{j}\) are: \[ \begin{split} r_{i}&=\sqrt{(\frac{a_{i}^{l}+a_{i}^{ u}}{2})^{2}+(\frac{a_{i}^{u}-a_{j}^{l}}{a_{i}^{u}-a_{i}^{l}})^{2}},\\ r_{j}&=\sqrt{(\frac{a_{j}^{l}+a_{j}^{u}}{2})^{2}+( \frac{a_{i}^{u}-a_{j}^{l}}{a_{j}^{u}-a_{j}^{l}})^{2}},\end{split} \] (8) if \(r_{i}>r_{j}\), then \(y_{i}\succcurlyeq y_{j}\), else then \(y_{i}\preccurlyeq y_{j}\). 2. \(y_{i}=[a_{i}^{l},a_{i}^{u}]\) and \(y_{j}=[a_{j}^{l},a_{j}^{u}]\) are any two non-intersecting interval numbers, which can be expressed and compared according to Definitions 4 and 5. Definition 6: Let \(S=(U,AT,V,f)\) be an interval-valued information system [40; 41]. Under the same attribute, we use the interval fuzzy dominance probability to express the relative degree between objects. The interval fuzzy dominance probability is \(x_{ij}\).