# A Decision Concept to the Historic Pedestrian Bridges Recovery Planning
## 2 Materials and Methods
### The Proposed Decision Concept
As \(M=\big{[}S_{ij}\big{]}_{M\times L}\) and \(M^{p}=\big{[}S_{ij}^{p}\big{]}_{M\times L}\), where \(S_{ij}\)=\(\{(H_{n},\beta_{n}),\ n=1,\ldots,N\}\) and \(S_{ij}^{p}=\big{\{}\big{(}H_{n},\beta_{n,p}\big{)},\ n=1,\ldots,N\big{\}}\) are both belief structures, then the relation between \(\beta_{n}\) and \(\beta_{n,p}\) is described as follows [65]:  
\[\beta_{n}=\frac{v\Big{(}\prod_{p=1}^{p}\Big{(}w_{p}^{D}\beta_{n,p}+1-w_{p}^{D} \sum_{j=1}^{N}\beta_{j,p}\Big{)}-\prod_{p=1}^{p}\Big{(}1-w_{p}^{D}\sum_{j=1}^ {N}\beta_{j,p}\Big{)}\Big{)}}{1-v\Big{(}\prod_{p=1}^{p}\Big{(}1-w_{p}^{D}\Big{)} \Big{)}},\ n=1,\ldots,N \tag{25}\]  
where  
\[v=\Bigg{(}\sum_{n=1}^{N}\prod_{p=1}^{P}\Bigg{(}w_{p}^{D}\beta_{n,p}+1-w_{p}^{D }\sum_{j=1}^{N}\beta_{j,p}\Bigg{)}-(N-1)\prod_{p=1}^{P}\big{(}1-w_{p}^{D}\sum _{j=1}^{N}\beta_{j,p}\big{)}\Bigg{)}^{-1} \tag{26}\]  
where \(w_{p}^{D}\) is the weight of \(p\)th decision group. **Step 9. ** Normalization of the aggregated group belief decision matrix using the following expression:  
\[R_{ij}=\frac{\beta_{n,j}}{\sqrt{\sum_{j=1}^{L}\big{(}\beta_{n,j}\big{)}^{2}}} \tag{27}\]  
After that, multiplication of the columns of normalized belief decision matrix with the calculated criteria weights is conducted to obtain the weighted decision matrix:  
\[\mu_{ij}=w_{j}\times R_{ij} \tag{28}\]  
where \(w_{j}\) is the weight of criterion \(C_{j}\). **Step 10. ** Definition of the positive ideal and negative ideal alternatives, denoted as \(A^{+}\) and \(A^{-}\), respectively. These alternatives are introduced as follows [66]:  
\[A^{+} =\Big{\{}\mu_{n,1}^{+},\mu_{n,2}^{+},\ldots,\mu_{n,l}^{+}\Big{\}},\ n=1,\ldots,N\] \[\mu_{n,i}^{+} =\bigg{\{}\bigg{(}\max_{i}\mu_{ij}\Big{|}jeK_{b}\bigg{)}\bigg{(} \min_{i}\mu_{ij}\Big{|}jeK_{c}\bigg{)}\bigg{\}} \tag{29}\]  
\[A^{-} =\Big{\{}\mu_{n,1}^{-},\mu_{n,2}^{-},\ldots,\mu_{n,l}^{-}\Big{\}},\ n=1,\ldots,N\] \[\mu_{n,i}^{-} =\bigg{\{}\bigg{(}\min_{i}\mu_{ij}\Big{|}jeK_{b}\bigg{)}\bigg{(} \max_{i}\mu_{ij}\Big{|}jeK_{c}\bigg{)}\bigg{\}} \tag{30}\]  
where \(K_{b}\) is the set of benefit criteria and \(K_{c}\) is the set of cost criteria. **Step 11. ** Calculation of the distance of the existing alternatives from the positive ideal and negative ideal alternatives using Euclidean distance, \(E_{i}^{+}\) and \(E_{i}^{-}\). The positive-ideal and negative-ideal alternatives, respectively, are obtained by following expression:  
\[E_{i}^{+}=\sqrt{\sum_{n=1}^{N}\sum_{j=1}^{l}\Big{(}\mu_{n,i}^{+}-\mu_{n,i} \Big{)}^{2}} \tag{31}\]  
**Step 12. ** Determination of the relative closeness to the ideal alternative as follows:  
\[\beta_{n}=\frac{v\Big{(}\prod_{p=1}^{p}\Big{(}w_{p}^{D}\beta_{n,p}+1-w_{p}^{D} \sum_{j=1}^{N}\beta_{j,p}\Big{)}-\prod_{p=1}^{p}\Big{(}1-w_{p}^{D}\sum_{j=1}^ {N}\beta_{j,p}\Big{)}\Big{)}}{1-v\Big{(}\prod_{p=1}^{p}\Big{(}1-w_{p}^{D}\Big{)} \Big{)}},\ n=1,\ldots,N \tag{25}\]0  
**Step 13. ** Priority ranking of the alternatives according to \(RC_{i}\). \[\beta_{n}=\frac{v\Big{(}\prod_{p=1}^{p}\Big{(}w_{p}^{D}\beta_{n,p}+1-w_{p}^{D} \sum_{j=1}^{N}\beta_{j,p}\Big{)}-\prod_{p=1}^{p}\Big{(}1-w_{p}^{D}\sum_{j=1}^ {N}\beta_{j,p}\Big{)}\Big{)}}{1-v\Big{(}\prod_{p=1}^{p}\Big{(}1-w_{p}^{D}\Big{)} \Big{)}},\ n=1,\ldots,N \tag{25}\]1 The decision concept to the recovery planning of historic pedestrian bridges.