## 3 Mathematical Models of MCDMs  
Eight different MCDMs and seven normalization methods were computed for deciding the UHS mitigation intervention. Additionally, the AHP was also applied for calculating the weightage of criteria. The procedure and key equations of methods are given in Table 1. Mathematical formulas of applied normalized methods are presented in Table 2.  
Figure 1: Type of heat resilience interventions. \begin{table}
\begin{tabular}{c c c} \hline
**MCDM** & **Steps** & **Reference** \\ \hline Step 1: make decision matrix & \\ Step 2: normalize decision matrix & \\ Step 3: weighted normalized decision matrix & \(r_{ij}\): normalized decision matrix & \\  & \(\mathcal{V}_{ij}=w_{j}r_{ij}\) & \\  & \(w_{j}\): weight of the Jth criteria (attribute) & \\ Condition \(\sum_{j=1}^{n}w_{j}=1\) & \\ Step 4: ideal best \(\mathcal{V}_{j}^{+}\) and ideal worst \(\mathcal{V}_{j}^{-}\) values & \\  & If beneficial criteria: & \\ TOPSIS: Technique for Order & \(\mathcal{V}_{j}^{+}\ =\ \underset{i}{max}\Big{(}v_{ij}\Big{)}\ =\ \max\Big{\{}v_{ij},i\ =\ 1,\ldots,m\Big{\}}\) & \\  & If cost criteria: & \\  & \(\mathcal{V}_{j}^{+}\ =\ \underset{i}{min}\Big{(}v_{ij}\Big{)}\ =\ \min\Big{\{}v_{ij},i\ =\ 1,\ldots,m\Big{\}}\) & \\ Step 5: Calculate the distances of each alternative from the positive ideal & \\  & \(\mathcal{S}_{i}^{\pm}\ =\ \sqrt{\frac{n}{\sum\limits_{j=1}^{n}\big{(} \mathcal{V}_{ij}\ -\ \mathcal{V}_{j}^{\pm}\big{)}^{2}}}\) & \\ Step 6: Calculate the relative closeness to the ideal solution (performance score) & \\  & \(R_{i}=\frac{\mathcal{S}_{i}}{\mathcal{S}_{i}^{+}\mathcal{S}_{i}^{-}}\) & \\ Step 7: Ranking the best alternative & \\ \hline Step 1: make decision matrix & \\ Step 2: normalize decision matrix & \\ Step 3: weighted normalized decision matrix & \\  & \(v_{ij}=R\times W=\begin{vmatrix}r_{11}\cdot\mathcal{W}_{1}&r_{12}\cdot\mathcal{W} _{2}&\text{K}&r_{1n}\cdot\mathcal{W}_{n}\\ r_{21}\cdot\mathcal{W}_{1}&r_{22}\cdot\mathcal{W}_{2}&\text{K}&r_{2n}\cdot \mathcal{W}_{n}\\ \text{K}&\text{K}&\text{K}&\text{K}\\ r_{m1}\cdot\mathcal{W}_{1}&r_{m2}\cdot\mathcal{W}_{2}&\text{K}&r_{nm}\cdot \mathcal{W}_{n}\\ \end{vmatrix}\) \\ Step 4: concordance and discordance interval sets & \\  & \(C_{ab}=\Big{\{}j\ |\ v_{aj}\geq v_{bj}\Big{\}}\) & \\  & \(D_{ab}=\Big{\{}j\ |\ v_{aj}<v_{bj}\Big{\}}=J-C_{ab}\) & \\ Step 5: calculation of the concordance interval matrix & \\  & \(C_{ab}=\sum\limits_{j\in C_{ab}}\mathcal{W}_{j}\) & \\  & \(C=\begin{vmatrix}-&c(1,2)&\ldots&c(1,m)\\ c(2,1)&-&\ldots&c(2,m)\\ \vdots&\vdots&\vdots&\vdots\\ c(m,1)&c(m,2)&\ldots&-\\ \end{vmatrix}\) & \\ Step 6: Determine the concordance index matrix & \\  & \(\overline{c}=\frac{\sum_{i=1}^{n}\sum_{j=1}^{n}c(a,b)}{m(m-1)}\) & \\  & \(e(a,b)=1\) if \(c(a,b)\ \geq\ \overline{c}\) & \\  & \(e(a,b)=0\) if \(c(a,b)<\overline{c}\) & \\ Step 7: Calculation of the discordance interval matrix & \\  & \(d(a,b)\ =\ \frac{\frac{\max}{\sum\limits_{j\in\mathcal{S}_{ab}}\left| \mathcal{V}_{aj}-\mathcal{V}_{bj}\right|}}{\max\mathcal{V}_{bj}\ -\mathcal{V}_{bj}}\) & \\  & \(D=\begin{vmatrix}-&d(1,2)&\ldots&d(1,m)\\ d(2,1)&-&\ldots&d(2,m)\\ \vdots&\vdots&\vdots&\vdots\\ d(m,1)&d(m,2)&\ldots&-\\ \end{vmatrix}\) & \\ \hline \end{tabular}
\end{table} 
Table 1: Key equations of MCDMs used in this study.