## 2 Methodology
### Street-Lighting System Appraisal Using Fuzzy Synthetic Evaluation  
The following assumptions were made to assess the efficacy of the street-lighting system of Buraydah City: (i) there are two drivers (road users) in each household, and (ii) all the drivers hold a driving license and have the same level of understanding about road infrastructure, traffic signs, lane marking, and street-lighting, irrespective of the number of years of driving experience. Statistical analysis in the previous section identifies the important criteria affecting USLS performance. The hierarchical approach to assess the urban street-lighting system assessment presented in Figure 3 requires a weighting scheme and an aggregation scheme, which turns it into a multicriteria analysis (MCA) problem. Fuzzy synthetic evaluation (FSE) was found appropriate to aggregate the data on PIs gathered through survey responses. Fuzzy set theory has been used in the past to solve MCA problems with uncertain data, subjective information, and imprecise judgment [38]. FSE is an application of the fuzzy set theory, which solves a multicriteria decision-making problem by synthetically evaluating an object relative to an objective in a fuzzy environment [39]. FSE has recently been used in smart city development and traffic congestion assessment projects [40; 41]. The FSE method was adopted to assess the urban street-lighting system, which has essentially been defined as an MCA problem in the present study [42]. The hierarchical structure of the problem presented in Figure 3 was solved by FSE using the following steps. The drivers' perception of different aspects of the street-lighting systems was secured through the questionnaire survey presented in Table 1. The universe of discourse (UoD) linguistically defining the five-level rating scheme is presented in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Response Level**} & \multirow{2}{*}{\(S_{j}\)} & **Criteria and Sub-Criteria Performance** & \multirow{2}{*}{**Color Coding Scheme**} \\ \cline{3-3}  & & & \\ \cline{1-1} \cline{3-3}  & & & \\ \cline{1-1} \cline{3-3}  & & & \\ \cline{1-1} \cline{3-3}  & & & \\ \cline{1-1} \cline{3-3}  & & & \\ \cline{1-1} \cline{3-3}  & & & \\ \cline{1-1} \cline{3-3}  & & & \\ \cline{1-1} \cline{3-3}  & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Subjective response levels and street-lighting system effectiveness rating. Figure 3: Hierarchical performance assessment approach for the urban street-lighting system. Details of sub-criteria at level 1 can be seen in Table 1.  
The term \(x_{ij}^{SC}\) denotes the degree of association of each sub-criterion to the first level, where \(S_{j}\) = 1, 2, 3, 4, and 5. Additionally, see Table 2 for \(S_{j}\) values. Equation (7) describes it in the matrix form:  
\[\left(X_{i}^{SC}\right)_{1\times 5}=\left(x_{i1}^{SC},x_{i2}^{SC},x_{i3}^{SC},x_{ i4}^{SC},x_{i5}^{SC}\right) \tag{7}\]  
where \(X_{i}^{SC}\) represents the sub-criteria \(i\) (where \(i\) = 1, 2, 3) under each criteria \(k\), where \(k\) = 1, 2, 3. Equation (8) calculated the impact of each sub-criterion (\(SC_{ki}\)):  
\[SC_{ki}=\sum_{i=1}^{5}\left(S_{j}*x_{ij}^{SC}\right) \tag{8}\]  
Equation (9) estimated the relative weights for the sub-criteria for assessing the criteria performance at level 2 of Figure 1:  
\[w_{i}^{SC}=SC_{ki}/\sum_{i=1}^{n}SC_{ki}\text{, where}n=3 \tag{9}\]  
The weighted matrix developed by Equation (8) and the evaluation matrix by Equation (6) were aggregated using FSE.