## 3 Materials and Methods
### Fuzzy Analytic Hierarchy Process (FAHP)  
Fuzzy logic was integrated with the AHP technique to address the lack of clarity arising from human judgments, for example, in paired comparisons [44, 45, 46]. To this end, fuzzy sets provide a solid foundation for solving real-world MCDM problems, which inevitably involve some level of noise in their structure. Under consideration of these aspects, AHP becomes FAHP. The judgments are represented by triangular numbers \(M\) denoted by (\(a.\ b.\ c\)), and the membership function is described as follows:  
\[\mu_{M}^{\sim}(x)=\left\{\begin{array}{ll}\frac{x-a}{b-a}&a\leq x\leq b\\ \frac{c-x}{c-b}&b\leq x\leq c\\ 0.&otherwise\end{array}\right.\]  
Here, \(-\infty<a\leq b\leq c<\infty\).  
The lower and upper limits are \(a\) and \(c\), whereas the strongest degree is symbolized by the parameter \(b\). The fuzzy triangular numbers to be employed in evaluating the importance of operation selection criteria and sub-criteria are described in Table 3. A shorter version of the [PERSON]'s scale (5-point scale) is proposed to accelerate the decision-making process and reduce bias when making the paired judgments [47].  
The FAHP procedure is as follows:  
**Step 1:** The fuzzy triangular numbers referred to in Table 3 are used to make a pairwise comparison between criteria and sub-criteria. In this way, it is possible to obtain the matrix of fuzzy judgments for \(\widetilde{A}^{K}\) (Equation (1)):  
\[\widetilde{A}^{K}=\left[\begin{array}{cccc}\widetilde{a}_{11}^{k}& \widetilde{a}_{12}^{k}&\cdots&\widetilde{a}_{1n}^{k}\\ \widetilde{a}_{21}^{k}&\widetilde{a}_{22}^{k}&\cdots&\widetilde{a}_{2n}^{k}\\ \cdots&\cdots&\cdots&\cdots\\ \widetilde{a}_{n1}^{k}&\widetilde{a}_{n2}^{k}&\cdots&\widetilde{a}_{nn}^{k} \end{array}\right] \tag{1}\]  
\(\widetilde{a}_{ij}^{k}\) represents the \(k\)th decision-maker's prediction of \(i\)th decision element (criterion/sub-criterion) over \(j\)th decision element (criterion/sub-criterion).  
**Step 2:** If the decision is supported by a decision-making group, the comparisons should be aggregated by applying Equation (2). Here, \(K\) represents the number of experts involved in the judgment process. After this, the initial matrix of fuzzy comparisons is updated, as shown in Equation (3).  
\[\widetilde{a}_{ij}=\sqrt[k]{\widetilde{a}_{ij}^{1}*\widetilde{a}_{ij}^{2}* \cdots*\widetilde{a}_{ij}^{k}} \tag{2}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Shorter Version of Saaty’s Scale** & **Importance Degree** & **Fuzzy Triangular Number** \\ \hline
1 & Equally important & [1.1.1] \\
3 & More important & [2.3.4] \\
5 & Much more important & [4.5.6] \\
1/3 & Less important & [1/4.1/3.1/2] \\
1/5 & Much less important & [1/6.1/5.1/4] \\ \hline \hline \end{tabular}
\end{table}
Table 3: Fuzzy triangular numbers employed in FAHP.  
\[\widetilde{A}=\left[\begin{array}{ccc}\widetilde{a_{11}}&\ldots&\widetilde{a_{1n} }\\ \vdots&\ddots&\vdots\\ \widetilde{a}_{n1}&\ldots&\widetilde{a}_{nn}\end{array}\right] \tag{3}\]  
**Step 3:** The geometric mean of the fuzzy comparisons \((\widetilde{r}_{i})\) for each decision element (criterion or sub-criterion) is computed, taking Equation (4) as a reference:  
\[\widetilde{r}_{i}=\left(\prod_{j=1}^{n}\widetilde{a}_{ij}\right)^{1/n}.i=1.2 \ldots n \tag{4}\]  
**Step 4**: The fuzzy relative priorities of all criteria \((\widetilde{w}_{i})\) are estimated by employing Equation (5)  
\[\widetilde{w}_{i}=\widetilde{r}_{i}\otimes(\widetilde{r}_{1}\oplus\widetilde{ r}_{2}\oplus\ldots\oplus\widetilde{r}_{n})^{-1}=(lw_{i}.mw_{i}.uw_{i}) \tag{5}\]  
**Step 5:**\((\widetilde{w}_{i})\) is defuzzified using the center of area method [48] described in Equation (6). Here, \(M_{i}\) denotes a non-fuzzy number.  
\[M_{i}=\frac{lw_{i}+mw_{i}+uw_{i}}{3} \tag{6}\]  
**Step 6:**\(M_{i}\) is normalized by implementing Equation (7)  
\[N_{i}=\frac{M_{i}}{\sum_{i=1}^{n}M_{i}} \tag{7}\]