# Risk and Complexity Assessment of Autonomous Vehicle Testing Scenarios
## 3 Materials and Methods
### Fuzzy ANP Method  
The ANP framework offers a notable advantage compared to the Analytical Hierarchy Process (AHP) method [40; 41; 42]. ANP allows for the characterization of intricate relationships between decision levels and elements without the need for establishing distinct levels for the elements. Furthermore, ANP enables the representation of interdependence and interaction among elements. The conventional ANP comprises two layers: the control layer, which includes a structure controlling the interaction between criteria, typically composed of objectives and indicators; and the network layer, which consists of elements or individual elements that depend on the objectives or indicators. The unique structure of the ANP allows individual objects to exist both as indicators and as elements. Interdependencies within the model are illustrated through bidirectional arrows, while circular arcs denote interdependencies within the same element, with the direction indicating the influence exerted by one sub-element or element on the other sub-element or element it affects. ANP quantifies the relationship of influence between elements based on the decision maker's subjective evaluation using a pairwise comparison matrix under specific criteria, and it is crucial to consider the consistency of the comparison matrix. In ANP, the influence between elements is quantified through a pairwise comparison matrix, which is based on the subjective evaluations of the decision-maker under certain criteria. Maintaining consistency in this matrix is crucial for accurate decision-making. Traditional ANP treats these evaluations as precise values, but this approach can be limited due to factors like ambiguity in human judgment, uncertainty in decision contexts, and incomplete information. To overcome these challenges, fuzzy methods are employed. Fuzzy sets help capture the inherent vagueness in human preferences by using fuzzy numbers to simulate the decision-making process more realistically. They provide a suitable approach \begin{table}
\begin{tabular}{l l} \hline \hline
**Layer** & **Category** \\ \hline \multirow{3}{*}{Ego Vehicle} & Motion \\  & DDT \\  & Types \\ \hline \multirow{4}{*}{Road} & Road Segment Type \\  & Road Surface Conditions \\  & Road Slope \\  & The Number of Lanes \\  & Road Leveling \\ \hline \multirow{2}{*}{Traffic Infrastructure} & Signal Control \\  & Obstacle \\ \hline \multirow{4}{*}{Objects} & Type \\  & Behavior \\ \cline{1-1}  & Number of Objects \\ \cline{1-1}  & Relative Motion Direction to Ego Vehicle \\ \cline{1-1}  & Position Relative to Ego Vehicle \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Basic elements of testing scenario. for handling the complex network of relationships in ANP and offer a logical foundation for the comparison process. SFs enhance this approach by combining the strengths of Pythagorean fuzzy sets and neutrosophic sets while reducing their limitations. SFs extend traditional fuzzy sets by transferring the membership function to a spherical surface, which allows for three levels of representation: membership, non-membership, and hesitation. This representation of fuzzy sets across the entire domain provides a more nuanced way to model uncertainty and indecision, as described by (1) and (2). \[\bar{A}_{S}=\Big{[}\Big{\langle}x,\Big{(}\ \mu_{\bar{A}_{S}}(x),\ v_{\bar{A}_{S}}( x),\pi_{\bar{A}_{S}}(x)\Big{)}\big{|}\ x\in X\Big{\rangle}\Big{]} \tag{1}\]  
\[0\leq\mu_{\bar{A}_{S}}^{2}(x)+v_{\bar{A}_{S}}^{2}(x)+\pi_{\bar{A}_{S}}^{2}(x) \leq 1\ \ \ \forall x\in X \tag{2}\]  
where \(\mu_{\bar{A}_{S}}:X\rightarrow[0,1]\), \(v_{\bar{A}_{S}}:X\rightarrow[0,1]\), \(\pi_{\bar{A}_{S}}:X\rightarrow[0,1]\). For each \(x\in X\), \(\mu_{\bar{A}_{S}}(x)\), \(v_{\bar{A}_{S}}(x)\), and \(\pi_{\bar{A}_{S}}(x)\) are the degree of membership, non-membership, and hesitancy of \(x\) to \(\bar{A}_{S}\), respectively.