# Vulnerability of Buildings to Meteorological Hazards:
## 2 Materials and Methods
### The AHP Method and the Weighting of Indicators  
The AHP, developed by [PERSON] [53], is one of the most common and popular participatory approaches for the weighting of indicators. The method has many advantages, including the possibility to include qualitative and quantitative data as well as the transparency of the indicators. Nevertheless, the pairwise comparison makes its implementation time-consuming for the participating experts and computationally expensive. Moreover, results based on expert judgment depend on the experts that have contributed, as well as on their knowledge, experience, and interests [54]. The indicators were first weighted pairwise by the authors, and then the MACBETH software (Measuring Attractiveness through a Category Based Evaluation Technique) was used for the analysis of the results. MACBETH is a specially designed computer program for multi-criteria decision analysis. It was used to implement pairwise comparisons of indicators and obtain relative weights. The main difference between MACBETH and other MCDA (multi-criteria decision analysis) methods is that to generate numerical scores for the options in each criterion and to weight the criteria, MACBETH only requires qualitative judgments (e.g., weak, moderate, strong) about the difference of attractiveness between two elements at a time. The weighting approach followed at this point was the same by [PERSON] [10; 55]. The approach of [PERSON] was designed for tsunamis. The present approach is based on the same weighting method but on a different set of indicators associated with the impact of strong wind, heavy rainfall, and hail on the buildings.  
A questionnaire was developed and sent to the experts to diminish the subjectivity and bias of the given weights and to finalize the weighting of the indicators. The questionnaire consisted of two parts. In the first part, the indicators selected for each hazard were introduced and input by the experts was required. In addition, a brief explanation of each indicator and their importance was given to support the experts in the weighting of the indicators. In the second part, a blank area was provided for unstructured comments on the model itself, additional information on the recommended weights, suggestions for indicators that should be also considered, and other feedback.  
The questionnaire was sent to 15 experts from various backgrounds (structural engineering, insurance industry, and natural disasters research) to determine different weights for indicators according to their knowledge of building engineering and experience with past events, respectively. Nevertheless, 8 experts (more than 50%) responded to the survey. The next step was the analysis of the questionnaires by MACBETH.  
Firstly, the indicators (options) were introduced to the software and by using an evaluation matrix, the pairwise comparisons between indicators affecting the structural vulnerability of buildings for each hazard were conducted. The indicators in each row and column were contrasted with each other. When it was determined that an indicator in a row was more significant than an indicator in a column, their qualitative difference was stated in the appropriate cell. The options for the difference between the two factors' relative weights were the semantic categories. Furthermore, as imaginary references, lower and higher factors were introduced. The higher one is equally significant as the most important element (engineering-based state of the building for windstorm), but the lower one makes no contribution to the degree of structural vulnerability.  
The results of the survey and the pairwise comparison (AHP) led to the final weights for each indicator that are shown in Table 5.