# A Methodology for Determining the Rehabilitation Needs of Buildings
## 2 Materials and Methods
### Mathematical Model of Method for Determining the Rehabilitation Needs of Buildings DRBN  
Decision criteria for rehabilitation needs c\({}_{1}\), c\({}_{2}\),..., c\({}_{7}\) determined by meters of decision criteria of t\({}_{ij}\), s\({}_{ij}\), d\({}_{ij}\), o\({}_{ij}\), r\({}_{ij}\), l\({}_{ij}\), h\({}_{ij}\) and the importance of these criteria w\({}_{1}\), w\({}_{2}\),..., w\({}_{7}\) are the output for determining the matrix of indicators of the order of rehabilitation needs k\({}_{ij}\). The sequence of importance of rehabilitation needs of building elements can be determined by prioritizing the order indicators k\({}_{ij}\). Order indicators for n elements in the j-th object can be obtained by solving the matrix equation. \[K_{j}=B_{j}W_{p} \tag{15}\]  
\[[k_{ip,j}]_{mx1}=[b_{ip,j}]_{mxu}\times[w_{p}]_{mx1} \tag{16}\]  
where:  
\(K_{j},[k_{ip,j}]_{mx1}\) matrix of indicators determining the need of rehabilitation of elements in the j-th building \(B_{j},[b_{ip,j}]_{mxu}\) matrix of criteria measures for elements in the j-th building;  
\(W_{p},[w_{p}]_{mx1}\) matrix of importance of criteria;  
\(i\) --denotes an ordinal number of an element in a building, i = 1, 2, 3,..., m;  
\(j\) --denotes an ordinal number of the building; j = 1, 2, 3,..., n;  
\(P\) --denotes an ordinal number of the criteria; p = 1, 2, 3,..., u. The matrix of criteria measures B\({}_{j}\) (for the j-th building) is a rectangular finite matrix with dimensions m \(\times\) u. \[B_{j}=\left[ \begin{array}{cccccc}t_{1j}&s_{1j}&d_{1j}&o_{1j}&r_{1j}&l_{1j}& h_{1j}\\ t_{2j}&s_{2j}&d_{2j}&o_{2j}&r_{2j}&l_{2j}&h_{2j}\\ t_{3j}&s_{3j}&d_{3j}&o_{3j}&r_{3j}&l_{3j}&h_{3j}\\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\ t_{m,j}&s_{m,j}&d_{m,j}&o_{m,j}&r_{m,j}&l_{m,j}&h_{m,j}\\ \end{array} \right] \tag{17}\]  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & **c1** & **c2** & **c3** & **c4** & **c5** & **c6** & **c7** & **Total** & \ \begin{tabular}{c} **Relative Importance** \\ **of Criteria** \\ \end{tabular}  & \begin{tabular}{c} **Importance** \\ **of Criteria** \\ \end{tabular}  \\ \hline c\(c_{1}\) & 1.00 & 5.00 & 3.00 & 5.00 & 0.14 & 9.00 & 7.00 & 30.14 & 0.300 & 0.30 \\ c\(c_{2}\) & 0.20 & 1.00 & 7.00 & 5.00 & 3.00 & 3.00 & 1.00 & 20.20 & 0.201 & 0.20 \\ c\(c_{3}\) & 0.33 & 0.14 & 1.00 & 0.14 The W\({}_{\rm p}\) matrix with u-importance criteria:  
\[\rm W_{p}=\left[ \begin{array}{c}\rm w_{t}\\ \rm w_{s}\\ \rm w_{d}\\ \rm w_{o}\\ \rm w_{r}\\ \rm w_{l}\\ \rm w_{h}\end{array} \right] \tag{18}\]  
The result of Equation (14) is the matrix K\({}_{\rm j}\). Matrix K\({}_{\rm j}\) contains indicators determine the order of needs for rehabilitation of m-elements in the j-th building. \[\rm K_{j}=\left[ \begin{array}{c}\rm k_{i,j}\\ \rm k_{2j}\\ \rm k_{3j}\\ \rm\ldots\\ \rm k_{m,j}\end{array} \right] \tag{19}\]  
The task is to solve the matrix Equation (14). By multiplying the numerical meters of the criteria by the importance of these criteria, we obtain the numerical values assigned to each examined element in a given building that was included in the study. The numerical values resulting from research are indicators of the order of rehabilitation needs--k\({}_{\rm ij}\). The higher the index, the more it is necessary to renovate the i-th element in the j-th object. The indicator is not, however, any physical rehabilitation value of the elements, it is only used to rank the building elements due to the proposed order of rehabilitation needs. The result for a large number of buildings is a set of matrixes corresponding to the number of buildings. Specific words of the matrix give the possibility of ordering the analyzed objects and their elements in any number of buildings, as well as determining the relation of the order of any two objects and their elements to each other. The proposed method can be used in planning rehabilitation works. Analyzing the results of the method of using indicators, one can give the order of rehabilitation of individual elements in a damaged building.