## 2 Hybrid MCDM Model
At the beginning, the fuzzy matrix \(\widetilde{F}\) is defined as follows:  
\[\widetilde{F}=\left[f_{ij}\right]_{m\times n^{\prime}} \tag{4}\]  
where \(f_{ij}\) denotes value of alternatives \(i\) (\(i=1,\ldots,m\)) according to criterion \(j\) (\(i=1,\ldots,n\)). Step 9: Defining the sorted matrix \(P\)  
The elements of the matrix \(P\) are the values of the matrix \(\widetilde{F}\), which are ordered by importance (criteria weight) in descending order. The mathematical formula for the sorted matrix is given below. \[P=\left[p_{ij}\right]_{m\times n^{\prime}} \tag{5}\]  
Step 10: Defining the normalized sorted matrix \(\phi\)  
The normalized values of the sorted matrix for the sets of benefit (\(B\)) and cost (C) criteria are calculated as follows:  
\[\varphi_{kj=} \begin{cases}\frac{p_{ij}}{m_{xx}\;p_{i}},\;for\;j\;e\;B\\ \frac{m_{ij}\;p_{ij}}{p_{ij}},\;for\;j\;e\;C^{\prime}\end{cases}  \tag{6}\]  
Step 11: Determination of coordinates \(a\), \(b\), and \(c\)  
In this step, it is necessary to determine the coordinates \(a\), \(b\), \(c\) of the reference point \(\left(R_{ij}\right)\) and the weighted reference point \(\left(N_{ij}\right)\), which define the complex polyhedron. The coordinates are calculated as follows:  
\[a_{ij}=n_{ij}\times sin\theta_{j},\;\forall j=1,\ldots,n;\;\forall i=1,\ldots,m, \tag{7}\]  
\[b_{ij}=n_{ij}\times cos\theta_{j},\;\forall j=1,\ldots,n;\;\forall i=1,\ldots,m, \tag{8}\]  
\[c_{ij}= \begin{cases}0,forR_{ij}\\ w_{j},\;forN_{ij}\end{cases} ,\;\forall j=1,\ldots,n;\;\forall i=1,\ldots,m, \tag{9}\]  
The angle, denoted as \(\theta_{j}\), plays a crucial role in determining the direction of the vector that defines the value of the alternative, and this angle is obtained as follows:  
\[\theta_{j}=(j-1)\frac{90^{\circ}}{n-1},\;\forall j=1,\ldots,n, \tag{10}\]  
Step 12: Calculation of the volume of a polyhedron \(V_{i}^{C}\)  
In this step, the volume of the polyhedron \(V_{i}^{C}\) is calculated, which consists of the sum of the individual volumes of the pyramids \(V_{k}\). The volume of the polyhedron is calculated as follows:  
\[V_{i}^{C}=\sum_{k=1}^{n-1}V_{k},\;\forall i=1,\ldots,m, \tag{11}\]  
where \(V_{k}\) is calculated as follows:  
\[V_{k}=\frac{1}{3}B_{k}\times h_{k},\;\forall k=1,\ldots,n-1, \tag{12}\]  
where \(B_{k}\) is the area of the base of the pyramid defined by the reference and weighted reference points, which is calculated as follows:  
\[B_{k}=y_{k}\times x_{k}+\frac{x_{k}\times\left(e_{k}-y_{k}\right)}{2}, \tag{13}\]  
where \(x_{k}\) is the Euclidean distance between the reference points of two consecutive criteria, calculated as follows:  
\[x_{k}=\sqrt{\left(a_{j+1}-a_{j}\right)^{2}+\left(b_{j+1}-b_{j}\right)^{2}}, \tag{14}\]\(y_{k},\ e_{k}\) are vector magnitudes corresponding to the weights of two consecutive criteria, that is:  
\[y_{k}=b_{j}, \tag{15}\]  
\[e_{k}=b_{j+1}, \tag{16}\]  
\(h\) is the height of the pyramid from the defined base to the top of the pyramid located in the coordinate origin (O) and is calculated as follows:  
\[h_{k}=\frac{2\sqrt{r_{k}(r_{k}-x_{k})(r_{k}-t_{k})(r_{k}-g_{k})}}{x_{k}}, \tag{17}\]  
where \(r_{k}\) is half of the perimeter of the triangle defined by coordinates \(a\) and \(b\) of two consecutive criteria and the coordinate origin, which is calculated as follows:  
\[r_{k}=\frac{x_{k}+t_{k}+g_{k}}{2}, \tag{18}\]  
where \(t_{k}\) and \(g_{k}\) are the Euclidean distances of two consecutive criteria from the coordinate origin and are calculated as follows:  
\[t_{k}=\sqrt{a_{j}^{2}+b_{j}^{2}}, \tag{19}\]  
\[V_{i}^{C}=\sum_{k=1}^{n-1}V_{k},\;\forall i=1,\ldots,m, \tag{11}\]0  
Step 13: Ranking of alternatives  
The ranking of alternatives in descending order is carried out according to the value of the total volume \(V_{i}^{C}(i=1,\dots,m)\). The best alternative is the one with the highest \(V_{i}^{C}\) value.