# Robust Simulation of Cyber-Physical Systems for Environmental Monitoring on Construction Sites
## 5 Uncertainty Detection and Processing
### Conflict Detection and Processing of Evaluation Decision  
In order to accurately measure the conflict between pieces of evidence in Scenario 3, this study combines the fuzzy set theory and D-S evidence theory according to the combination approach of conflict evidence proposed by [PERSON] and [PERSON] [49]. The similarity \(Sim\) is defined by combining the two features of fuzzy closeness \(F_{n}\) and correlation coefficient \(Coc\), as well as adjusting the evidence credibility. The high-conflict data fusion based on the improved evidence theory is presented in the sequel. Step 1: Calculation of transition probability  
Assuming that \(\Theta\)\(=\{\theta_{1},\theta_{2},\cdots,\theta_{n}\}\), there are \(k\) bodies of evidence, and the probability distribution of evidence is \(m_{i}(i=1,2,\cdots,k)\), which can be obtained by the probability transition formula \(k\) probability vectors, where the transition probability formula is in Equation (6). In Equation (6), \(BEL=\sum Bel(\theta_{i})\), where Bel represents the value of the belief function, and BEL represents the total value of belief functions. \[P(\theta_{i})=Bel(\theta_{i})+\frac{BEL\cdot Bel(\theta_{i})+(1-BEL)pl(\theta_ {i})}{\sum_{\theta_{i}\in\Theta BEL\cdot Bel(\theta_{i})+(1-BEL)pl(\theta_{i} )}}\big{(}1-BEL\big{)} \tag{6}\]  
Step 2: Combination of the fuzzy closeness and correlation coefficient  
Then, the fuzzy closeness of \(k\) probability vectors is shown in Equation (7). In Equation (7), \(\wedge\) and \(\vee\) are the minimum and maximum operators separately. \(F_{n}\big{(}m_{i},m_{j}\big{)}\in[0,1]\), when \(P_{i}\) and \(P_{j}\) are more similar, the fuzzy closeness \(F_{n}\big{(}m_{i},m_{j}\big{)}\) is closer to 1. \[F_{n}\big{(}m_{i},m_{j}\big{)}=\frac{\sum_{s=1}^{n}\big{(}P_{i}(\theta_{s}) \wedge P_{j}(\theta_{s})\big{)}}{\sum_{s=1}^{n}\big{(}P_{i}(\theta_{s})\lor P _{j}(\theta_{s})\big{)}}\quad i,j=1,2,3\cdots,k \tag{7}\]  
However, it is unreliable to measure the similarity between the types of evidence based on the fuzzy closeness \(F_{n}\), and the robustness stands at an undesirable level [58]. Hence, the correlation coefficient \(Coc\) can be analyzed to measure the similarity between the types of evidence. In this study, the definition of the correlation coefficient is shown in Equation (8). \[Coc\big{(}m_{i},m_{j}\big{)}= \begin{cases}\frac{m_{i}\big{(}\theta_{\max}^{n_ {i}}\big{)}+m_{j}\big{(}\theta_{\max}^{n_{j}}\big{)}}{2},if\theta_{\max}^{m_{ i}}=\theta_{\max}^{m_{j}}\\ \frac{m_{i}\big{(}\theta_{\min}^{n_{i}}\big{)}+m_{j}\big{(}\theta_{\min}^{n_{j} }\big{)}}{if\theta_{\min}^{m_{j}}}\
eq\theta_{\min}^{m_{j}}\end{cases}  \tag{8}\]In Equation (8), \(Coc\) is used to measure the similarity of the propositions supported by the maximum trust of two pieces of evidence. If the propositions of two pieces of evidence are the same, the two pieces of evidence are considered to be similar, and \(Coc\) is equal to the average of the maximum trust values of the two pieces of evidence. On the contrary, if the propositions are different, the two pieces of evidence are considered to be in conflict, and \(Coc\) is equal to the average of the minimum trust values of the two pieces of evidence. \(Fn\) measures the similarity from the basic probability distribution structure of evidence, while \(Coc\) measures the similarity of the evidence from the perspective of evidence reliability. In order to comprehensively measure the similarity between the types of evidence, \(Fn\) and \(Coc\) are combined through \(Sim\) in Equation (9). This shows that the degree of belief of evidence is associated with the degree of similarity. The greater the degree of similarity between the types of evidence, the higher the credibility is.