## 4 Case Study  
### Result Analysis  
To illustrate the effectiveness of the developed method, Table 4 shows the emergency evacuation evaluation matrices. There are four HSRSs that need to be evaluated, \(X=\{x_{1},x_{2},x_{3},x_{4}\}\). The attributes are guidance system (\(C_{(1)}\), emergency exit (\(C_{(2)}\), emergency route (\(C_{(3)}\), emergency stairway (\(C_{(4)}\) and emergency management (\(C_{(5)}\). Otherwise, the decision attribute (\(C_{d}\)) represents the acceptability level of attributes. The values of each attribute are given by the experts in the form of PFNs based on their professional knowledge. Assume that there are three actions in the two states (HSRS risk unacceptable (\(P\)) and acceptable (\(N\))): taking measures to eliminate risks (\(a_{P}\)), tracking attribute levels continuously (\(a_{B}\)) and taking no measures (\(a_{N}\)). The risk loss of the three actions is represented by PFNs and is shown in Table 5.  
Assume that the thresholds \(\alpha=0.7\), \(\beta=0.3\) and the factor \(\omega=0.5\), the steps of the emergency evacuation evaluation process are as follows:  
**Step 1:** calculate the Pythagorean fuzzy closeness measures \(\vartheta(x_{i},x_{k})(i,k=1,2,3,4)\) between the two stations \(x_{i}\) and \(x_{k}\) related to the condition attribute set \(B=\{c_{1},c_{2},c_{3},c_{4},c_{5}\}\) by Equation (20), and the results are  
\(\vartheta(x_{1},x_{1})=(1,0)\), \(\vartheta(x_{1},x_{2})=(0.8388,0.4976)\), \(\vartheta(x_{1},x_{3})=(0.8926,0.4486)\), \(\vartheta(x_{1},x_{4})=(0.7785,0.5444)\), \(\vartheta(x_{2},x_{2})=(1,0)\), \(\vartheta(x_{2},x_{3})=(0.7750,0.5879)\), \(\vartheta(x_{2},x_{4})=(0.8949,0.3855)\), \(\vartheta(x_{3},x_{3})=(1,0)\), \(\vartheta(x_{3},x_{4})=(0.7094,0.6280)\), \(\vartheta(x_{4},x_{4})=(1,0)\). **Step 2:** calculate the Pythagorean fuzzy similar category \(\vartheta_{\alpha}^{\beta}(x_{i})(i=1,2,3,4)\). The \((\alpha,\beta)\)-level cut set \(X_{\alpha}^{\beta}\) under the Pythagorean fuzzy concept is calculated by Equation (8). Meanwhile, \(\delta(x_{i})=I\left(X_{\alpha}^{\beta},\,\theta_{\alpha}^{\beta}(x)\right)\). \(\vartheta_{\alpha}^{\beta}(x_{1})=\{x_{1},x_{2},x_{3}\}\), \(\vartheta_{\alpha}^{\beta}(x_{2})=\{x_{1},x_{2},x_{4}\}\), \(\vartheta_{\alpha}^{\beta}(x_{3})=\{x_{1},x_{3}\}\), \(\vartheta_{\alpha}^{\beta}(x_Therefore, the stations \(x_{1}\), \(x_{2}\) and \(x_{3}\) have high risks and must take measures to eliminate risks, and \(x_{4}\) is at risk and needs to be continuously observed. Then, the relationships among the decision action, factor and threshold are shown in Figure 1, and we can draw the following conclusions:  
(1) In Figure 1a, the threshold value \(\overline{\pi}\) increases and \(\overline{\beta}\) decreases with the increase of factor \(\omega\), indicating that the more optimistic the decision-maker is about the risks, the higher the requirement to take measures to eliminate the risks; and the more pessimistic the decision-maker is, the lower the requirement to take no measures, which is consistent with people's cognition or intuition. (2) From Figure 1b-e, with the increase of \(\omega\), their decision-making actions will change stations \(x_{1}\) and \(x_{2}\) from tracking risks to taking some measures to eliminate risks. The risk critical point of change is 0.4090, while the decision-making actions of \(x_{3}\) and \(x_{4}\) remain unchanged. (3) Considering that the performance is in two states of acceptance and unacceptability, assuming that \(\alpha+\beta=1\), with the increase of \(\alpha\), the decision-making actions of the stations  
Figure 1: The relationship between decision-making action, factor \(\omega\) and the threshold \(\alpha\). (**a**) Changes of three thresholds; (**b**) Status change of station \(x_{1}\); (**c**) Status change of station \(x_{2}\); (**d**) Status change of station \(x_{3}\); (**e**) Status change of station \(x_{4}\); (**f**) Decision action changes. \(x_{1}\), \(x_{2}\), \(x_{3}\) and \(x_{4}\) will change, as shown in Figure 1f.