# BIM Application in Multi-Criteria Assessment of
## 3 Methodology
### Methods for Multi-Criteria Ranking of Alternatives  
Our task of multi-criteria evaluation and ranking of suspended brickwork facades consists of \(m\) alternative facades and \(n\) criteria, describing the performance of the alternatives. The variable \(x_{ij}\) means the performance of alternative \(i\) when it is described by the criterion \(j\), where \(i=1,\,\ldots,\,m\), and \(j=1,\,\ldots,\,n\). The variable \(w_{j}\) means the weight (relative significance) of the criterion \(j\).  
The basis for estimating the weights of the criteria is via the application of an expert survey. Then, the responses of the experts \(t_{jk}\), \(k=1,\,\ldots,\,r\) (where \(r\) is number of experts), as scores obtained according to the scale, are statistically processed.  
First, the mean value of scores \(\overline{t_{j}}\) is calculated as follows:  
\[\overline{t_{j}}=\frac{\sum_{k=1}^{r}t_{jk}}{r} \tag{1}\]  
The weight of every criterion \(w_{j}\) is calculated as follows:  
\[w_{j}=\frac{\overline{t_{j}}}{\sum_{j=1}^{n}\overline{t_{j}}} \tag{2}\]  
The compatibility of expert opinions is evaluated by applying the concordance coefficient \(W\)[48; 49]:  
\[W=\frac{12\Big{(}\sum_{j=1}^{n}\Big{(}\sum_{k=1}^{r}t_{jk}-\frac{1}{n}\sum_{j= 1}^{n}\sum_{k=i}^{r}t_{jk}\Big{)}\Big{)}}{r^{2}(n^{3}-n)}. \tag{3}\]  
The significance of the concordance coefficient \(\chi^{2}\) is calculated by applying the following equation:  
\[\chi^{2}=\frac{12\Big{(}\sum_{j=1}^{n}\Big{(}\sum_{k=1}^{r}t_{jk}-\frac{1}{n} \sum_{j=1}^{n}\sum_{k=i}^{r}t_{jk}\Big{)}\Big{)}}{rn(n+1)}. \tag{4}\]  
The calculated value \(\chi^{2}\) is compared to the normative value \(\chi^{2}_{lbl}\) at a certain level of significance. If the calculated value \(\chi^{2}\) exceeds the normative value \(\chi^{2}_{lbl}\), we can conclude that the hypothesis regarding the compatibility of expert opinions is accepted.  
Next, to evaluate and rank the alternatives, the WASPAS (Weighted Aggregated Sum Product Assessment) method developed by [PERSON] et al. is applied [50]. The steps of calculation are presented below according to the initial source of the method.  
The method involves two classical criteria of optimality; namely, WSM (weighted sum) and WPM (weighted product). A joint generalized criterion involving the two mentioned criteria of optimality (weighted aggregated function, using aggregation coefficient \(\lambda\), \(\lambda=0,\,\ldots,\,1\)) for calculating the relative importance of alternative \(Q_{i}\) is suggested.  
The first criterion of optimality is based on WSM. The relative importance of alternative \(i\) is defined as follows:  
\[Q_{i}^{(1)}=\sum_{j\,=\,1}^{n}\overline{x}_{ij}w_{j}, \tag{5}\]  
where linear normalization of the initial criteria values is applied according to Equation (6) for maximized (benefit) criteria and Equation (7) for minimized (cost) criteria:  
\[\overline{x}_{ij}=\frac{x_{ij}}{max\,x_{ij}}, \tag{6}\]  
\[\overline{x}_{ij}=\frac{min\,x_{ij}}{x_{ij}}. \tag{7}\]  
According to the second criterion of optimality--namely, the multiplicative exponential generalized criterion--the relative importance of alternative \(i\), is defined as follows:  
\[Q_{i}^{(2)}=\prod_{j\,=\,1}^{n}\left(\overline{x}_{ij}\right)^{w_{j}}. \tag{8}\]  
Assuming that there is an increase in ranking accuracy and, subsequently, the effectiveness of decision making, a joint generalized criterion of the WSM and WPM methods and the methodology for optimization of weighted aggregated function is proposed [50]:  
\[Q_{i}=\lambda\,\sum_{j\,=\,1}^{n}\overline{x}_{ij}w_{j}+(1-\lambda)\,\prod_{j \,=\,1}^{n}\left(\overline{x}_{ij}\right)^{w_{j}},\;\lambda=0,\;\ldots,\;1. \tag{9}\]  
In this paper, the authors take the value of the aggregation coefficient \(\lambda=0.5\), given the equal weights of both criteria of optimality. Therefore, the following equation is used for further calculations:  
\[Q_{i}=0.5\,\sum_{j\,=\,1}^{n}\overline{x}_{ij}w_{j}+0.5\,\prod_{j\,=\,1}^{n} \left(\overline{x}_{ij}\right)^{w_{j}}. \tag{10}\]