## 3 Problem Statement and Proposed Integrated Methodology
### Proposed Integrated Procedure Under Uncertainty  
This section presents an integrated procedure to evaluate the sustainability and risk pillars under uncertainty in logistics network analysis. The first part of the procedure is related to specifying the main criteria and sub-criteria depended upon the official reports, experts' opinions and academic papers. The second part of the procedure is to obtain the weights when applying the fuzzy-based AHP. The third part of the procedure ranks the alternatives using the TOPSIS method. Finally, comparison studies are conducted with the different scenario analyses. The steps of the proposed integrated procedure are provided as follows:  
Step 1: Specify a group of experts. Step 2: State the main criteria and sub-criteria depended upon the afore-mentioned stakeholders for the problem. Step 3: Start the procedure of the fuzzy-based AHP method, and then go to step 4. Note that the fuzzy extension is necessary due to the vagueness and the lack of exact numerical descriptions of the experts' judgments. Step 4: Pairwise comparison matrices (PCMs) are generated associated with the linguistic scale, and experts make selections to fill the PCMs with the fuzzy number scales in Table 1. Triangular fuzzy numbers (TFNs) are employed in Table 1, and \(\tilde{x}_{ijk}=\left(a_{ijk},b_{ijk},c_{ijk}\right)\) denotes the lower, middle, or upper values of the TFN, respectively. In addition, TFNs are selected in this study due to computational efficiency, widespread use in MCDM literature and proven adequacy in representing expert uncertainty with minimal cognitive burden on stakeholders. Although alternative fuzzy numbers such as trapezoidal or Gaussian could be employed, TFNs offer a favorable trade-off between expressiveness and mathematical tractability, especially when multiple expert evaluations and consistency checks are involved. Step 5: Aggregate the experts' selections using the fuzzy numbers, \(\bar{x}_{ijk}=\left(a_{ijk},b_{ijk},c_{ijk}\right)\), as follows:  
\[\bar{x}_{ij}=\left(a_{ij},b_{ij},c_{ij}\right)\left(k=1,2,\ldots,n\right)\text{ is the aggregated fuzzy number} \tag{1}\]  
where \(a_{ij}=\left(\prod\limits_{k=1}^{n}a_{ijk}\right)^{\frac{1}{k}},b_{ij}=\left( \prod\limits_{k=1}^{n}b_{ijk}\right)^{\frac{1}{k}},c_{ij}=\left(\prod\limits_ {k=1}^{n}c_{ijk}\right)^{\frac{1}{k}}\)  
Step 6: Apply the defuzzification of the aggregated fuzzy number as follows:  
\[Deff\left(\bar{x}_{ij}\right)=\left(a_{ij}\times b_{ij}\times c_{ij}\right)^{ \frac{1}{3}} \tag{2}\]  
where \(Deff\left(\bar{x}_{ij}\right)\) denotes the defuzzification function. Step 7: Apply the normalization of defuzzified aggregated pairwise comparison matrices for the following formula. \[d_{ij}{}^{\prime}=\frac{Deff\left(\bar{x}_{ij}\right)}{\sum\limits_{i=1}^{N} Deff\left(\bar{x}_{ij}\right)}\left(i,j=1,2,\ldots,N\right) \tag{3}\]  
Step 8: Obtain the weights for the specified criteria as follows:  
\[nw_{j}=\left(\frac{1}{N}\right)\sum\limits_{i=1}^{N}d_{ij}{}^{\prime}(i,j=1,2,\ldots,N) \tag{4}\]  
where \(nw_{j}\) denotes the \(j\)th weight. Step 9: Calculate a consistency ratio (CR) to check the credibility of the preferences using the following formula. \[ \begin{array}{l}CR=\frac{CI}{\text{RCI}}\\ \text{where }CI=\frac{\left(\lambda_{\text{max}}-N\right)}{\left(N-1\right)}, \lambda_{\text{max}}\text{ is the principal eigenvalue,}\\ \text{and }RI\text{ is the random consistency index (RCI).}\end{array}  \tag{5}\]  
Step 10: The threshold of CR is 0.10. If the threshold is greater than 0.10, the preferences of the experts should be revised until they obtain a CR value that is smaller than 0.10. Step 11: Repeat Steps 3-10 for the main and all sub-criteria. Then, go to Step 12  
Step 12: Start the procedure of the TOPSIS method, and then go to Step 13. Step 13: The normalized multi-attribute decision-making matrix is found using the following formula:  
\[fnor_{ij}=f_{ij}\Bigg{/}\sqrt{\sum\limits_{i=1}^{n}f_{ij}^{2}} \tag{6}\]