# Urban Flood Risk Assessment Based on a Combination of Subjective and Objective Multi-Weight Methods
## 2 Materials and Methods
### Combination of Multi-Weight Methods
#### 2.3.3 Analytic Hierarchy Process to Determine Subjective Weights  
The analytic hierarchy process (AHP) is a combination of qualitative and quantitative methods for weighting analysis and decision-making, as proposed by [PERSON] [49]. The AHP uses mathematical relationships to express human subjective judgment, reducing the influence of subjective factors on the weights to a certain extent and, thus, making the assessment results more scientific. This method is more suitable for decision-making problems that have a target system with hierarchical interlocking evaluation indicators and for which the target values are difficult to describe quantitatively. AHP is an effective risk assessment method based on qualitative and quantitative analyses of multiple indicators and has been widely used for flood risk assessment in recent years [39; 50; 51; 52]. The specific steps for determining the weights are as follows:  
Construct a hierarchical structure model  
When choosing AHP for problem-solving, the problem is first stratified to create a hierarchical structural model. The model is divided into three layers, namely the task layer, the criterion layer, and the indicator layer. The task level represents the final problem to be solved, the criterion level represents the factors to be considered, and the indicator level represents the alternatives in decision-making. In the criterion level of the hierarchical model, the factors at the same level are called target criteria, and the indicators at the indicator level are governed by the influencing factors at the criterion level based on their characteristics, while the lower-level criteria are governed by the criteria at the upper level,so this kind of tree-type progression is regarded as a multilayered structural analytical model [53]. (2) Use 1-9 and its reciprocal scale method to construct an index judgment matrix  
In order to improve the accuracy of the constructed matrix, relative scales were used to compare the assessment factors two by two to minimize the difficulty of comparing factors of different natures with each other. The judgment matrix \(R=\left(r_{ij}\right)_{n\times n}\) is defined using 1 to 9 and its reciprocal as the scale, and the method of constructing the judgment matrix is shown in Table 3 below. (3) Hierarchical ordering of items in a list  
The eigenvector corresponding to the largest eigenroot of the judgment matrix, \(\lambda_{\max}\), is normalized and recorded as \(W\). The elements of \(W\) are the ranking weights of the elements of the same level with respect to the relative importance of a factor in the previous level, a process known as hierarchical single ranking. The synthetic weights of the elements of each level on the system goal are calculated, and the total ranking is performed to determine the importance of each element of the lowest level in the recursive structure diagram in the total goal. (4) Consistency check  
The consistency indicator \(CI\) is calculated using the following formula:  
\[CI=\frac{\lambda_{\max}-n}{n-1} \tag{12}\]  
where n is the dimension of the matrix. If \(CI=0\), there is perfect consistency; if \(CI\) is close to 0, there is satisfactory consistency; the larger \(CI\) is, the more serious the inconsistency is. To measure the magnitude of \(CR\), the random consistency index \(RI\) is introduced, as shown in Table 4 below. The consistency ratio is calculated by the following formula:  
\[CR=\frac{CI}{RI} \tag{13}\]  
When \(CR<0.1\), the consistency of the judgment matrix is reasonable; however, it is necessary to re-test the consistency. Since this study discusses the impact of precipitation intensity on storm flooding levels over different return periods, a total of six experts (the authors of this paper) were invited to participate in the decision-making process when setting up the guideline layer indicators. This involved setting up the weights according to the different return periods, taking