# A Decision Concept to the Historic Pedestrian Bridges Recovery Planning
## 2 Materials and Methods
### The Proposed Decision Concept  
Vagueness dominates in real-life decisions due to insecure, unavailable, and uncertain data such as fuzziness, imprecision, incompleteness, and ignorance [65]. Most of the decisions involve the opinions and attitudes of a group of decision makers instead of an individual decision maker. For this reason, in this research focus is on the decision making with uncertainty and multiple decision makers that are gathered in groups for a belief multicriteria decision making problem. Let there be \(M\) alternatives \(A_{i}\) (\(i=1,\ldots,M\)) and \(P\) decision groups \(D_{p}\) (\(p=1,\ldots,P\)). Alternatives are assessed by \(C_{j}\) (\(j=1,\ldots,L\)) criteria. To handle the uncertainty of decision problems, the decision concept is proposed and applied to present the judgments of decision makers. Due to the above context, the decision concept is defined through AHP-evidential reasoning-TOPSIS methodology for the group multicriteria decision making problem demonstrated in the 13 steps and illustrated in Figure 1.  
**Step 1. ** Defining the problem and gathering of the relevant experts and stakeholders. Dividing them into the groups, and determining the research area. **Step 2. ** Mapping and defining the historic pedestrian bridges. **Step 3. ** Modeling a goal hierarchy structure, introducing the main goal and dividing it into the objectives and criteria, and at the lowest level, alternatives. **Step 4. ** Defining the comparison matrix for each group by AHP method using [PERSON]'s nine-grade scale, where 1 = equally important, 3 = slightly important, 5 = very important, 7 = much more important, 9 = extremely more important. The 2, 4, 6, and 8 are values in between. Each expert from each group gave his or her attitude on each criterion. Their opinions are further integrated using average values of criteria weights and are expressed as scenarios. **Step 5. ** After the definition of criteria weights by each group, priority vector, and the maximum eigenvalue (\(\lambda_{max}\)) for each matrix is calculated. **Step 6. ** Checking the validity of comparison matrices using consistency ratio. **Step 7. ** Constructing a belief decision matrix of each group and defining the weights of decision makers' groups. Assessing the bridges by each group according to each defined criterion as follows:  
\[M^{p}= \begin{array}{ccccc}A_{1}&\begin{array}{ccccc}C_{1}&\ldots&C_{j}& \ldots&C_{L}\\ S_{11}^{p}&\ldots&S_{1j}^{p}&\ldots&S_{1L}^{p}\\ \vdots\\ A_{i}&\vdots\\ A_{M}&\end{array}\end{array}  \tag{21}\]  
\[W^{D}=\left[ \begin{array}{ccccc}w_{1}^{D}&\ldots&w_{p}^{D}&\ldots&w_{p}^{D }\\ \end{array} \right] \tag{22}\]  
\[W^{C}=\left[ \begin{array}{ccccc}w_{1}^{C}&\ldots&w_{j}^{C}&\ldots&w_{L}^{C} \\ \end{array} \right] \tag{23}\]  
where \(M^{p}\) is the group belief decision matrix for each group, \(w_{p}^{D}\) is the weight of decision group \(D_{p}\), where \(\sum_{p=1}^{p}w_{p}^{D}=1\). \(w_{j}^{C}\) is weight of criterion \(C_{j}\), and \(\sum_{j=1}^{L}w_{j}^{C}=1\). \(S_{ij}^{p}=\left\{\left(H_{n},\beta_{n,p}\right),\ n=1,\ldots,N\right\}\) is a rating in a belief decision matrix. It means that, in the opinion of decision group \(D_{p}\), alternative \(A_{i}\) is evaluated as rating \(H_{n}\) with the degree of belief of \(\beta_{n,p}\) according to criterion \(C_{j}\). **Step 8. ** Aggregation of group belief decision matrices \(M^{p}\) into the belief decision matrix \(M\) using the evidential reasoning. The aggregated belief decision matrix \(M\) can be expressed as [65]:  
\[M=\big{[}S_{ij}\big{]}_{M\times L}=M^{1}\oplus M^{2}\oplus\ldots\oplus M^{p} \tag{24}\]  
With all its elements, \(S_{ij}\), defining aggregated ratings of alternative \(A_{i}\) according to criterion \(C_{j}\).