## 3 HTOPSIS for Solving the HMADM Model with Deviation and Fixation Attributes
### New Normalization Approach for Heterogeneous Attributes
#### 3.1.1 Normalization of the Attribute Described as Real Number  
For a benefit or cost attribute described as real number \(f_{ij}\), it holds  
\[r_{ij}=\frac{f_{ij}}{\sqrt{\sum\limits_{i=1}^{m}f_{ij}^{2}}}. \tag{2}\]  
Supposing that the value of a deviation attribute is described as a real number \(f_{ij}\) and the given constant to be deviated from is \(k_{j}\), the attribute value is \(f_{ij}^{\prime}=f_{ij}-k_{j}\) when the deviation attribute is transformed into the benefit one with zero NIS. Then, the equivalent attribute values for the alternatives with \(f_{ij}^{\prime}<0\) can be obtained as \(f_{ij}^{\prime\prime}=\left|f_{ij}-k_{j}\right|\) by symmetrical transformation. Finally, the normalized attribute value \(r_{ij}\) can be calculated by  
\[r_{ij}=\frac{\left|f_{ij}-k_{j}\right|}{\sqrt{\sum\limits_{i=1}^{m}\left(f_{ ij}-k_{j}\right)^{2}}}. \tag{3}\]  
3.1.2 Normalization of the Attribute Described as Interval Number Based on Possibility Degree Method  
For a benefit or cost attribute described as interval number \(f_{ij}=[f_{ij}^{L},f_{ij}^{L}]\), it holds  
\[r_{ij}=[r_{ij}^{L},r_{ij}^{L}]=\left[\frac{f_{ij}^{L}}{\sqrt{\sum\limits_{i=1} ^{m}\frac{1}{2}(f_{ij}^{L2}+f_{ij}^{U2})}},\frac{f_{ij}^{U}}{\sqrt{\sum\limits _{i=1}^{m}\frac{1}{2}(f_{ij}^{L2}+f_{ij}^{U2})}}\right]. \tag{4}\]  
Supposing that the value of a deviation attribute is described as an interval number \(f_{ij}=[f_{ij}^{L},f_{ij}^{U}]\) and the given interval constant to be deviated from is \([k_{j}^{L},k_{j}^{U}]\), the attribute value is \(f_{ij}^{\prime}=[f_{ij}^{L}-k_{j}^{U},f_{ij}^{U}-k_{j}^{L}]\) when the deviation attribute is transformed into the benefit one with zero NIS. According to the possibility degree method for comparing two interval numbers [25], the probability of interval \(f_{ij}^{\prime}\) being not greater than 0 can be calculated as  
\[P(f_{ij}^{\prime}\leq 0)=\left\{\begin{array}{ll}0,&f_{ij}^{L}-k_{j}^{U}\geq 0 ;\\ \frac{-f_{ij}^{L}+k_{j}^{U}}{f_{ij}^{U}-f_{ij}^{U}-k_{j}^{L}+k_{j}^{U}},&f_{ij }^{L}-k_{j}^{U}<0\leq f_{ij}^{U}-k_{j}^{L};\\ 1,&f_{ij}^{U}-k_{j}^{L}\leq 0.\end{array}\right. \tag{5}\]  
With the attribute values satisfying \(P(f_{ij}^{\prime}\leq 0)>0.5\) symmetrically mapped about 0, the equivalent attribute values of all the alternative configurations can be described as  
\[f_{ij}^{\prime\prime}=\left\{\begin{array}{ll}[k_{j}^{L}-f_{ij}^{U},\ k_{j}^{U}-f_{ij}^{L}],&P(f_{ij}^{\prime}\leq 0 )>0.5;\\ [f_{ij}^{L}-k_{j}^{U},\ f_{ij}^{U}-k_{j}^{L}],&P(f_{ij}^{\prime}\leq 0)\leq 0.5. \end{array}\right. \tag{6}\]  
Finally, the normalized attribute value \(r_{ij}\) can be calculated as  
\[r_{ij}=[r_{ij}^{L},\ r_{ij}^{U}]=\left\{\begin{array}{ll}\left[\frac{k_{j}^{ L}-f_{ij}^{U}}{c_{j}^{L}},\ \frac{k_{j}^{U}-f_{ij}^{L}}{c_{j}^{L}}\right],&P(f_{ij}^{\prime}\leq 0 )>0.5;\\ \left[\frac{f_{ij}^{L}-k_{j}^{U}}{c_{j}^{L}},\ \frac{f_{ij}^{U}-k_{j}^{U}}{c_{j}^{L}} \right],&P(f_{ij}^{\prime}\leq 0)\leq 0.5.\end{array}\right. \tag{7}\]where \(c_{j}^{*}=\sqrt{\sum\limits_{i=1}^{m}\frac{1}{2}\bigg{[}\left(f_{ij}^{L}-k_{j}^{U} \right)^{2}+\left(f_{ij}^{U}-k_{j}^{L}\right)^{2}\bigg{]}}\).