# A Decision Concept to the Historic Pedestrian Bridges Recovery Planning
## 2 Materials and Methods
### The TOPSIS Method  
The TOPSIS method was proposed by [PERSON] and [PERSON] [58], and then lately it was developed by [PERSON] and [PERSON] [59]. It is a technique of preference order by closeness to ideal solution. It is useful and practical multicriteria decision making tool to deal with real-life complex problems [60]. The ideal alternative, according to this method, is the one that is closest to the positive ideal option and farthermost from the negative ideal option [61]. The positive ideal option maximizes the benefit criteria and minimizes the cost criteria, whereas the negative ideal option, called anti-ideal option, maximizes the cost criteria and minimizes the benefit criteria [10]. In short, the positive ideal option is composed of all ideal achievable criteria values, whereas the negative ideal option consists of all the worst achievable criteria values [62]. It is assumed that each criterion is monotonically increasing or decreasing. If each criterion has a monotone increasing (or decreasing) effective function, the positive ideal option, which consists of the best criterion values, and the negative ideal option, which consists of the worst, can be computed [63]. Euclidean distances are used to measure the distance of each alternative from the positive ideal option and anti-ideal option. The preference order of alternatives is achieved through the comparison of Euclidean distances.  
Let a multicriterial decision making problem has \(n\) alternatives, \(A_{1}\),..., \(A_{n}\), and \(m\) decision criteria, \(C_{1}\),..., \(C_{m}\). Alternatives are evaluated with respect to the \(m\) criteria. All the ratings assigned to the alternatives with respect to each criterion form a decision matrix are denoted by \(X=\left(x_{ij}\right)_{m\times n}\). Let \(W=\left(w_{1},\ldots,\ w_{m}\right)\) be the relative weight vector about the criteria, satisfying \(\sum_{j=1}^{m}w_{j}=1\). Then, the TOPSIS process is performed as follows [64]:  
(i) Normalization of the decision matrix \(X=\left(x_{ij}\right)_{n\times m}\):  
\[b_{ij}=\frac{x_{ij}}{\sqrt{\sum_{k=1}^{n}x_{kj}^{2}}},\ \ i=1,\ldots,\ m;\ \ j=1, \ldots,\ n \tag{14}\]  
where \(b_{ij}\) is the normalized criteria rating.  
(ii) Calculation of the weighted normalized decision matrix \(V_{ij}=\left(v_{ij}\right)_{m\times n}\)  
\[v_{ij}=w_{j}r_{ij},\ \ i=1,\ldots,\ m;\ \ j=1,\ldots,\ n \tag{15}\]  
where \(w_{j}\) is the relative weight of the \(j\)th criterion, and \(\sum_{j=1}^{m}w_{j}=1\).  
(iii) Determination of the positive ideal and negative ideal solutions.  
\[A^{*}=v_{1}^{*},\ldots,v_{m}^{*}=\big{\{}\left(max_{j}v_{ij}\big{|}je\ell \Omega_{a}\right),\ \left(min_{j}v_{ij}\big{|}je\ell\Omega_{b}\right) \tag{16}\]  
\[A^{-}=v_{1}^{-},\ldots,v_{m}^{-}=\big{\{}\left(max_{j}v_{ij}\big{|}je\ell \Omega_{a}\right),\ \left(min_{j}v_{ij}\big{|}je\ell\Omega_{b}\right) \tag{17}\]  
where \(\Omega_{a}\) and \(\Omega_{b}\) are the sets of benefit criteria and cost criteria, respectively.  
(iv) Calculation of the Euclidean distances of each alternative from the positive ideal solution and the negative ideal solution, respectively.  
\[d_{i}^{*}=\sqrt{\sum_{j=1}^{n}\left(v_{ij}-v_{j}^{*}\right)^{2}},\ \ \ \ \ \ \ \ i=1,2,\ldots,\ m \tag{18}\]  
\[d_{i}^{-}=\sqrt{\sum_{j=1}^{n}\left(v_{ij}-v_{j}^{-}\right)^{2}},\ \ \ \ \ \ \ \ i=1,2,\ldots,\ m \tag{19}\](v) Calculation of the relative closeness coefficient of each alternative to the ideal solution. The relative closeness coefficient of the alternative \(A_{i}\) with respect to \(A^{*}\) is defined as:  
\[RC_{i}=\frac{d_{i}^{-}}{d_{i}^{*}+d_{i}^{-}},\ \ i=1,\ldots,\ m \tag{20}\]  
(vi) The alternatives ranking comparing the relative closeness coefficient is determined.