## 3 Methodology
### Psychological Behavioral in Decision Making
: _Consider alternative \(X_{i}(i=1\ldots n)\) with choosing consequences are \(x_{i}(i=1\ldots n)\). The overall utility value is obtained by alternative \(X_{i}\) can be defined as Equations (4) and (5) [65]:_  
\[utility(x_{i})=utility^{risk}(x_{i})+utility^{regret}(x_{i},x^{*}) \tag{4}\]  
\[\ \  
\[d_{ik}=\left|x_{ij}-x_{kj}\right|\quad,j=1,\ldots,J,\ i=1,\ldots,I,\ k=1,\ldots,I,\ i\
eq k \tag{13}\]  
Step 8. For each Euclidean distance matrix, the double-centering process is performed. Accordingly, the row means, column means, and matrix means of the Euclidean distance matrix are determined. The row mean and column mean are deducted sequentially from each element. Then, the matrix mean is added to each element of the distance matrix. As the results, the double-centered matrices are constructed for each criterion as shown in Equation (14). \[T_{j}= \begin{bmatrix}t_{11}^{j}&t_{12}^{j}&\cdots&t_{11}^{j}\\ t_{21}^{j}&t_{22}^{j}&\cdots&t_{2I}^{j}\\ \vdots&\vdots&\ddots&\vdots\\ t_{I1}^{j}&t_{I2}^{j}&\cdots&t_{II}^{j}\end{bmatrix} ,j=1,\ldots,J \tag{14}\]  
Step 9. The distance covariance between \(j\)th criterion and \(j^{{}^{\prime}}\)th criterion (\(dCOV_{jj^{{}^{\prime}}}\)) is determined by the following sub-steps:  
* Multiply double-centered matrices elementwise using the Hadamard product, also known as the Schur product or the entry wise product. * Calculate the average value of the elementwise multiplication matrix. * Calculate the square root of the average value. Step 10. The distance variance of the \(j\)th criterion (\(dVAR_{j}=dCOV_{jj}\)) is determined similarly to step 9. Step 11. The distance correlation between the \(j\)th criterion and \(j^{{}^{\prime}}\)th criterion (\(dCOR_{jj^{{}^{\prime}}}\)) is determined according to Equation (15). \[dCOR_{jj^{{}^{\prime}}}=\ \frac{dCOV_{jj}}{\sqrt{dVAR_{j}\times dVAR_{j^{{}^{ \prime}}}}} \tag{15}\]  
Step 12. The information content of the \(j\)th criterion (\(I_{j}\)) is determined according to Equation (16). \[I_{j}=s_{j}\sum\limits_{j=1}^{I}\left(1-dCOR_{jj^{{}^{\prime}}}\right) \tag{16}\]  
Step 13. The absolute weight of the \(j\)th criterion (\(w_{j}\)) is determined according to Equation (17). \[w_{j}=\ \frac{I_{j}}{\Sigma_{j=1}^{I}I_{j}} \tag{17}\]  
Step 14. According to the principle of reference dependence of the Prospect theory, the relative weight of \(j\)th criterion (\(\hat{w}_{j}\)) is estimated based on the absolute weight of reference criterion as Equation (18). \[\hat{w}_{j}=\ \frac{w_{j}}{w_{r}},wherev_{r}=\max\limits_{1\leq i\leq I}\left(w_{ j}\right) \tag{18}\]  
Step 15. The contribution of each \(j\)th criterion (\(\omega_{ik}^{j}\)) to the level of dominance (\(\theta_{ik}\)) of \(i\)th alternative over \(k\)th alternative is shown in Equations (19) and (20). Accordingly, if \(x_{ij}>x_{kj}\), the value of \(\omega_{ik}^{j}\) will be a gain to \(\theta_{ik}\). if \(x_{ij}<x_{kj}\), it will be a loss to \(\theta_{ik}\). Otherwise, the \(j\)th criterion has no contribution to \(\theta_{ik}\). Besides, according to Prospect theory, the loss attenuation coefficient \(\mu\) is used to express the loss attenuation of the decision makers. \[\omega_{ik}^{j}=\left\{ \begin{array}{ll}\sqrt{\frac{\bar{\omega}_{j}\left(s_{ik}^{* -s_{ik}^{*}}\right)}{\Sigma_{j=1}^{j}\bar{\omega}_{j}}}&,if\ x_{ij}>x_{kj}\\ 0&,if\ x_{ij}=x_{kj}\\ -\frac{1}{\mu}\sqrt{\frac{\left(s_{ik}^{*-s_{ik}^{*}}\right)\Sigma_{j=1}^{j} \bar{\omega}_{j}}{\bar{\omega}_{j}}}&,if\ x_{ij}<x_{kj}\end{array} \right. \tag{19}\]  
\[\theta_{ik}=\sum\limits_{j=1}^{I}\omega_{ik}^{j} \tag{20}\]  
Step 16. The total dominance of the \(i\)th alternative is calculated according to Equation (21). \[\theta_{i}^{*}=\sum\limits_{k=1,\ k\
eq i}^{I}\theta_{ik} \tag{21}\]  
Step 17. The overall evaluation score (\(\tau_{i}\)) of the \(i\)th alternative is determined as Equation (22). The higher the value of \(\tau_{i}\), the better the alternative. \[\tau_{i}=\frac{\theta_{i}^{*}-\ min_{i=1}^{I}\theta_{i}^{*}}{max_{i=1}^{I} \theta_{i}^{*}-\ min_{i=1}^{I}\theta_{i}^{*}} \tag{22}\]