## 3 Methodology
### Fuzzy Evaluation Based on the Distance from the Average Solution (Fuzzy EDAS)  
EDAS is an effective MCDM methodology that was developed in 2015 based on measuring the deviation from the average solution. The efficiency of this MCDM methodology has been evaluated in comparison to prior valid methodologies, such as Vise Kriterijumska Optimizacija Kompromisno Resenje (VIKOR), Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), Simple Additive Weighting (SAW), and COmplex Proportional Assessment (COPRAS), with consistent results [16]. As ambiguity is an important condition in most problems, a combination of fuzzy set theory and EDAS was developed one year later [17]. Recently, fuzzy EDAS has been developed in new studies, and some new ones are mentioned here (as follows):  
* In supplier selection [17];
* In solid waste disposal site selection [18];
* For subcontractor evaluation [19];
* For supplier evaluation and order allocation with environmental considerations [20];
* For the evaluation of construction equipment with sustainability considerations [21];
* In hydrogen mobility roll-up site selection [22];
* For evaluating suppliers [23]. In this study, fuzzy EDAS methodology and its calculation procedure were mainly based on previous studies [17], as follows. An assessment methodology was proposed through the following five steps. Step 1 (determining the average distance): We assumed \(n\) alternatives were defined as a screened building facade and \(m\) criteria were considered in the assessment process. In addition, the number of respondents was set as \(k\). Thus, the average distant matrix was calculated as follows:  
\[\widetilde{x}_{ij}=\frac{1}{k}\left( \begin{array}{cc}k\\ \oplus&\widetilde{x}_{ij}^{p}\end{array} \right)\Rightarrow X=\left[\widetilde{ x}_{ij}\right]_{n\times m}, \tag{6}\]  
where \(\widetilde{x}_{ij}^{p}\) denotes the \(p\)th respondent opinion about alternative \(i\) with respect to the \(j\)th criterion. Step 2 (calculating criteria weight): The criteria average weight based on the respondents' opinions was calculated as follows:  
\[\widetilde{w}_{j}=\frac{1}{k}\left( \begin{array}{cc}k\\ \oplus&\widetilde{w}_{j}^{p}\end{array} \right)\Rightarrow W=\left[\widetilde{ w}_{j}\right]_{1\times m}, \tag{7}\]  
where \(\widetilde{w}_{j}^{p}\) denotes the weight of the \(p\)th respondent on the \(j\)th criteria weight. Step 3 (negative and positive distances from the average): Average responses with regard to every criterion were calculated as follows:  
\[\widetilde{w}_{j}=\frac{1}{n}\left( \begin{array}{cc}n&\\ \oplus&\widetilde{x}_{ij}\\ i=1&\end{array} \right)\Rightarrow AV=\left[\widetilde{a}\widetilde{v}_{j} \right]_{1\times m}. \tag{8}\]  
Then, the positive distance from the average (pda), as well as the negative (nda), were calculated as follows:  
\[\widetilde{pda}_{ij}=\frac{\Psi\big{(}\widetilde{x}_{ij}\odot\widetilde{a} \widetilde{v}_{j}\big{)}}{\kappa\big{(}\widetilde{a}\widetilde{v}_{j}\big{)} }\Rightarrow PDA=\left[\widetilde{pda}_{ij}\right]_{m\times m}, \tag{9}\]  
\[\widetilde{nda}_{ij}=\frac{\Psi\big{(}\widetilde{a}v_{j}\odot\widetilde{x}_{ ij}\big{)}}{\kappa\big{(}\widetilde{a}\widetilde{v}_{j}\big{)}}\Rightarrow NDA=\left[ \widetilde{nda}_{ij}\right]_{m\times m}, \tag{10}\]  
where \(\Psi(\overline{x})=\max(\overline{x})\), and \(\kappa(\overline{x})\) is the defuzzification function. Consequently, the calculated distances from the average were weighted as follows:  
\[\widetilde{sp}_{i}= \begin{array}{cc}m&\\ \oplus&\Big{(}\widetilde{w}_{j}\otimes\widetilde{pda}_{ij}\Big{)},\\ j=1&\end{array}  \tag{11}\]  
\[\widetilde{sn}_{i}= \begin{array}{cc}m&\\ \oplus&\Big{(}\widetilde{w}_{j}\otimes\widetilde{nda}_{ij}\Big{)}.\\ j=1&\end{array}  \tag{12}\]  
Step 4 (calculating the normal value): In order to calculate the normal value and remove the units, all of the values had to be divided by a maximum value, as follows:  
\[\widetilde{nsp}_{i}=\frac{\widetilde{sp}_{i}}{\max\limits_{i}\!\!\!\!\!\!\!\! \!\!