# Resilience Assessment for the Northern Sea Route Based on a Fuzzy Bayesian Network
## 2 Proposed Resilience Measurement Methodology
### Fuzzy Theory  
#### 2.1.1 Fuzzy Number Selection to Design a Questionnaire  
Since first proposed by [39], fuzzy set theory has been widely introduced in the analytic hierarchy process (AHP) because experts prefer to express fuzzy judgements rather than crisp comparisons. According to fuzzy set theory, we can define a fuzzy number, \(M\), in the set of real numbers, \(R\), where \(F(R)\) represents the fuzzy sets; then, \(M\in F(R)\). The concept of a fuzzy number can be expressed as [40]:  
\[ \begin{array}{l}\mu_{M}(x_{0})\ =\ 1\ \ \ \ \ x_{0}\in R\\ A_{\alpha}\ =\ \left[x,\mu_{A_{\alpha}}(x)\geq a\right]\ \ \ \alpha\in[0,1]\end{array} \right\} \tag{1}\]  
where \(\mu\) is the membership function and \(\alpha\) is a random real number. Generally, triangular and trapezoidal fuzzy numbers have been widely applied for the fuzzy judgements of experts in varied case studies. According to the definition of a fuzzy number, the membership function of the triangular fuzzy number can be expressed as:  
\[\mu_{M}(x)\ =\ \left\{ \begin{array}{ll}\frac{x}{a_{2}-a_{1}}-\frac{a_{1}}{a-2a _{1}},&x\in[a_{1},a_{2}]\\ \frac{a_{3}}{a_{3}-a_{2}}-\frac{x}{a-2a_{1}},&x\in[a_{2},a_{3}]\\ 0&\text{otherwise}\end{array} \right\} \tag{2}\]  
where the triangular fuzzy number is defined as \(M(a_{1},a_{2},a_{3})\) for \(a_{1}\leq a_{2}\leq a_{3}\); \(a_{1}\) and \(a_{3}\) are the lower and upper values of fuzzy number, \(M\), respectively; and \(a_{2}\) is the modal value. Figure 2: Proposed resilience measurement model for the northern sea route (NSR). For the limit case of \(a_{1}~{}=~{}a_{2}~{}=~{}a_{3}\), \(M\) is a nonfuzzy number. Similarly, for a trapezoidal fuzzy number, \(N(a_{1},a_{2},a_{3},a_{4})\), the membership function is expressed as:  
\[\mu_{N}(x)~{}=~{}\left\{ \begin{array}{ll}\frac{x}{a_{2}-a_{1}}-\frac{a_{1}}{a- 2a_{1}},&x\in[a_{1},a_{2}]\\ 1&x\in[a_{2},a_{3}]\\ \frac{a_{4}}{a_{4}-a_{3}}-\frac{x}{a-4a_{3}},&x\in[a_{3},a_{4}]\\ 0&\textit{otherwise}\end{array} \right\} \tag{3}\]  
To illustrate the operational rules of fuzzy numbers, two different triangular fuzzy numbers are taken as examples, \(M_{1}(a_{1},a_{2},a_{3})\) and \(M_{2}(b_{1},b_{2},b_{3})\), and the algebraic operation rules are:  
\[M_{1}\oplus M_{2}~{}=~{}(a_{1},a_{2},a_{3})\oplus(b_{1},b_{2},b_{3})~{}=~{}(a_ {1}+b_{1},a_{2}+b_{2},a_{3}+b_{3}) \tag{4}\]  
\[M_{1}\otimes M_{2}~{}=~{}(a_{1},a_{2},a_{3})\otimes(b_{1},b_{2},b_{3})~{}=~{}( a_{1}b_{1},a_{2}b_{2},a_{3}b_{3}) \tag{5}\]  
\[\lambda\odot M_{1}~{}=~{}\lambda\odot(a_{1},a_{2},a_{3})~{}=~{}(\lambda a_{1}, a_{2},\lambda a_{3})\qquad\lambda>0,~{}\lambda\in R \tag{6}\]  
\[M_{1}^{-1}~{}=~{}(a_{1},a_{2},a_{3})^{-1}~{}=~{}(\frac{1}{a_{1}},\frac{1}{a_{2} },\frac{1}{a_{3}}) \tag{7}\]  
Generally, the linguistic expressions of experts are meaningful for handling complex circumstances and eliciting meaningful advice. However, it is difficult to directly quantify the qualitative expressions of experts; therefore, the application of fuzzy numbers can solve this problem. Several attempts have been made to transfer qualitative linguistic expressions into corresponding fuzzy numbers [41; 42; 43], and the technique proposed by [43] is widely accepted; this approach includes eight scales associated with verbal expressions. Later, [44] suggested that the estimation of human memory aptitude should include seven terms, plus or minus two patches, indicating that the number of verbal terms should be between five and nine for good expert elicitation. In the present study, transformation scale six, which includes five verbal expressions [45], was selected to design the expert questionnaire, and fuzzy trapezoidal numbers were selected to express expert judgements. The reason for selecting scale six was mainly that the experts preferred to give their judgements through five verbal expressions. In addition, the reliability of scale six has been verified by its successful application in various fields, such as the oil and gas [46], medicine [47] and marine [48] fields.