# Conceptual Design Evaluation Considering Confidence Based on Z-AHP-TOPSIS Method
## 3 Materials and Methods
### Z-Ahp  
When DMs utilize fuzzy AHP [38] to calculate the weight of criteria, the linguistic terms are expressed as (\(L,M,U\)) TFNs. Assuming that there are \(n\) criteria, the matrix \(P\) is used for pairwise comparison of criteria by DMs as follows:  
\[P=\left(P_{ij}\right)=\left(L_{ij},M_{ij},U_{ij}\right)=\] \[\left[\begin{array}{cccc}(1,1,1)&(L_{12},M_{12},U_{12})&\cdots& (L_{1n},M_{1 nr},U_{1n})\\ (1/U_{12},1/M_{12},1/L_{12})&(1,1,1)&\cdots&(L_{2n},M_{2 nr},U_{2n})\\ \vdots&\vdots&\vdots&\vdots\\ (1/U_{1 nr},1/M_{1 nr},1/L_{1n})&(1/U_{2n},1/M_{2 nr},1/L_{2n})&\cdots&(1,1,1)\\ \end{array}\right], \tag{9}\]  
where the symmetric fuzzy number of the main diagonal is \(P_{ij}=P_{ij}^{-1}=\left(1/U_{ij},1/M_{ij},1/L_{ij}\right)\).  
This paper uses the Chang algorithm [39] and improves it based on Z-number as follows:  
Step 1. Build pairwise comparison matrix with linguistic terms.  
DMs utilize Z = (\(A,B\)) to compare the indicators, \(A\) is the evaluation value, \(B\) is the confidence, so the pairwise matrix can be expressed as:  
\[P=\left(P_{ij}\right)=\left((L_{ij}^{A},M_{ij}^{A},U_{ij}^{A}),( L_{ij}^{B},M_{ij}^{B},U_{ij}^{B})\right)=\] \[\left[\begin{array}{cccc}(1,1,1)&((L_{12}^{A},M_{12}^{A},U_{12}^ {A}),(L_{12}^{B},M_{12}^{B},U_{12}^{B}))&\cdots&((L_{1n}^{A},M_{1 nr}^{A},U_{1n} ^{A}),(L_{1 nr}^{B},M_{1 nr}^{B},U_{1 nr}^{B}))\\ -&(1,1,1)&\cdots&((L_{2n}^{A},M_{2 nr}^{A},U_{2n}^{A}),(L_{1 nr}^{B},M_{2 nr}^{B},U_{2 nr}^{B}))\\ \vdots&\vdots&\vdots&\vdots\\ -&-&\cdots&(1,1,1)\\ \end{array}\right], \tag{10}\]  
where the symmetric elements of the main diagonal are temporarily vacated, and the main diagonal elements are (1, 1, 1).  
Step 2. Transform the Z-numbers into regular fuzzy numbers and supplement the elements of the diagonal symmetric position.  
By using the method proposed in Section 3.2, the Z-numbers in the pairwise comparison matrix is transformed into regular fuzzy numbers, and then the symmetric elements of the main diagonal are supplemented to obtain the following matrix:  
Figure 4: TFN of “good” corresponding to different confidence levels.  
\[\left[\begin{array}{cccc}P=\left(P_{ij}\right)=\\ (1,1,1)&\left(\widetilde{L_{12}},\widetilde{M_{12}},\widetilde{U_{12}}\right) &\cdots&\left(\widetilde{L_{1n}},\widetilde{M_{1n}},\widetilde{U_{1n}}\right) \\ \left(1/\widetilde{U_{12}},1/\widetilde{M_{12}},1/\widetilde{L_{12}}\right)&(1,1,1)&\cdots&\left(\widetilde{L_{2n}},\widetilde{M_{2n}},\widetilde{U_{2n}} \right)\\ \vdots&\vdots&\vdots&\vdots\\ \left(1/\widetilde{U_{1n}},1/\widetilde{M_{1n}},1/\widetilde{L_{1n}}\right) &\left(1/\widetilde{U_{2n}},1/\widetilde{M_{2n}},1/\widetilde{L_{2n}}\right) &\cdots&(1,1,1)\end{array}\right], \tag{11}\]  
Step 3. Calculate the extension of the fuzzy synthesis.  
The extension of the fuzzy synthesis \(S_{i}\) is calculated for the \(i\)-th criterion by the following formula:  
\[S_{i}=\sum_{j=1}^{n}P_{ij}\otimes\left\{\sum_{i=1}^{n}\sum_{j=1}^{n}P_{ij} \right\}^{-1};i=1,2,\ldots,n., \tag{12}\]  
Step 4. Calculate the degree of possibility.  
The degree possibility for a convex fuzzy number to be greater than \(i\) convex fuzzy numbers \(V_{j}\) can be defined by Equation (13) [39].  
\[V\big{(}S_{j}\geq S_{i}\big{)}=\left\{\begin{array}{cc}1,&\text{if }\widetilde{M}_{j}\geq \widetilde{M}_{i},\text{ }i=1,2,\ldots,n;j=1,2,\ldots,n.\\ \frac{\widetilde{L_{i}}-\widetilde{U_{j}}}{\left(\widetilde{M_{j}}-\widetilde {U_{j}}\right)-\left(\widetilde{M_{i}}-\widetilde{L_{i}}\right)}&\text{if } \widetilde{U_{j}}\geq\widetilde{L_{i}},i=1,2,\ldots,n;j=1,2,\ldots,n.\\ 0,&\text{in other cases}\end{array}\right., \tag{13}\]  
The theory of the comparison of the fuzzy numbers is applied to the comparison of the values.  
\[V_{j}=V\big{(}S_{j}\geq S_{1},S_{2},\ldots,S_{j-1},S_{j+1},\ldots,S_{n}\big{)} =min\big{\{}V\big{(}S_{j}\geq S_{i}\big{)},i=1,\ldots,n;i\
eq j\big{\}}, \tag{14}\]  
Step 5. Calculate normalized weight of criteria \(w_{j}\).  
\[w_{j}=\frac{V_{j}}{\sum_{j=1}^{n}V_{j}},j=1,2,\ldots,n., \tag{15}\]