# Resilience Assessment for the Northern Sea Route Based on a Fuzzy Bayesian Network
## 2 Proposed Resilience Measurement Methodology
### Fuzzy Theory
#### 2.1.3 Expert Viewpoint Aggregation  
To eliminate any cognitive biases, it is necessary to aggregate expert opinions. Suppose that each expert, \(E_{i}(i~{}=~{}1,2,3,\ldots,n)\), states his or her personal viewpoints about a certain issue by utilizing a predefined set of linguistic expressions. These linguistic variables can then be transferred into corresponding triangular or trapezoidal fuzzy numbers, which can be processed further until defuzzification is achieved. (1) Calculation of the degree of similarity. \(S_{ww}(\widetilde{E}_{u},\widetilde{E}_{v})\) is defined as the degree of agreement for different opinions among experts. Suppose \(\widetilde{E}_{u}(a_{1},a_{2},a_{3})\) and \(\widetilde{E}_{v}(b_{1},b_{2},b_{3})\) \begin{table}
\begin{tabular}{c c} \hline \hline
**Linguistic Expressions** & **Fuzzy Trapezoidal Numbers** \\ \hline Very Low (VL) & (0,0,0,1,0,2) \\ Low (L) & (0,1,0,25,0,25,0,4) \\ Medium (M) & (0,3,0.5,0.5,0.7) \\ High (H) & (0,6,0.75,0.75,0.9) \\ Very High (VH) & (0.8,0.9,1,1) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Linguistic expressions and their corresponding fuzzy numbers for scale six. represent two standard triangular fuzzy numbers (\(u\
eq v\)); then, the degree of agreement between \(\widetilde{E}_{u}\) and \(\widetilde{E}_{v}\) can be obtained by:  
\[S_{uv}(\widetilde{E}_{u},\widetilde{E}_{v})\ =\ 1-\frac{1}{J}\sum\limits_{i\ =\ 1}^{J}|a_{i}-b_{i}|\ \ \ i\ =\ 1,2,3 \tag{12}\]  
where \(J\) is the number of fuzzy set members; e.g., \(J\ =\ 3\) is for standard triangular fuzzy numbers and \(J\ =\ 4\) is for standard trapezoidal fuzzy numbers. Additionally, the greater the values of \(S_{uv}(\widetilde{E}_{u},\widetilde{E}_{v})\) are, the greater the similarity between experts \(E_{u}\) and \(E_{v}\). (2) Calculation of the average agreement (AA) degree for each expert viewpoint. \[AA(E_{u})\ =\ \frac{1}{U-1}\sum\limits_{u\
eq\wp,\ =\ 1}^{U}S_{uv}(\widetilde{E}_{u },\widetilde{E}_{v}) \tag{13}\]  
where \(U\) is the total number of experts. (3) Calculation of the relative agreement (RA) degree between two kinds of experts. The value of \(RA(E_{u})\) for the \(u\)thexpert can be obtained by:  
\[RA(E_{u})\ =\ \frac{AA(E_{u})}{\sum\limits_{u\ =\ 1}^{U}AA(E_{u})} \tag{14}\]  
(4) Estimation of the consensus coefficient (CC) for each expert. The value of \(CC(E_{u})\) for the \(u\)thexpert can be obtained by:  
\[CC(E_{u})\ =\ \beta\times P(E_{u})+(1-\beta)\times RA(E_{u}) \tag{15}\]  
where the coefficient \(\beta(0\leq\beta\leq 1)\) is introduced to represent the importance of \(P(E_{u})\) over \(RA(E_{u})\), namely, the greater \(\beta\) is, the more important \(P(E_{u})\) is. Actually, when \(\beta\ =\ 0\), no weight is distributed to \(P(E_{u})\), indicating that a homogenous group of experts is employed; for another limit case, \(\beta\ =\ 1\), the consensus degree among the various expert viewpoints is adequately high. (5) Calculation of the aggregated results of the expert viewpoints. The aggregated results denoted by \(\widetilde{R}_{A}\) can be computed by:  
\[\widetilde{R}_{A}\ =\ CC(E_{1})\otimes\widetilde{E}_{1}\oplus CC(E_{2})\otimes \widetilde{E}_{2}\oplus\cdots\oplus CC(E_{U})\otimes\widetilde{E}_{U} \tag{16}\]  
(6) Defuzzification of the aggregated results. The defuzzification of fuzzy numbers is critically important for the application of fuzzy set theory. The center of area (CoA) method is widely used for the defuzzification operation, and it is expressed as:  
\[X\ =\ \frac{\int\mu_{M}(x)xdx}{\int\mu_{M}(x)dx} \tag{17}\]  
where \(X\) represents the defuzzification result and \(\mu_{M}(x)\) indicates the aggregated membership functions defined in (1) and (2) for fuzzy triangular and trapezoidal numbers, respectively. Therefore, the fuzzy numbers of the aggregated results, denoted as \(\widetilde{R}_{A}(c_{1},c_{2},c_{3})\) for fuzzy triangular numbers and \(\widetilde{R}_{A}(c_{1},c_{2},c_{3},c_{4})\) for fuzzy trapezoidal numbers, can be defined by (18) and (19), respectively.