## 2 Aggregation Frameworks  
As climate change has widespread impacts on almost every facet of natural and social systems, the vulnerability assessments in various fields to climate change have been conducted based on the conceptual framework in IPCC TAR [1]. Vulnerability to complex phenomenon such as floods and other multi-dimensional issues has been generally quantified using a single composite indicator [46], which can be estimated using IPCC's standard components of exposure, sensitivity, and adaptability. The adaptability component functions as a resilient force against the vulnerability while the exposure and sensitivity components represent a positive elasticity to the vulnerability. The composite indicators are usually derived by additive or multiplicative multi-attribute utility functions, and these two types of utility functions will produce composite indicators with utterly different attributes [46]. Some compensation between assessment components is allowed in the composite indicators aggregated by additive utility functions, such that the contribution of lower values in some components can be compensated for by much higher values in the other components. On the contrary, no compensation occurs between assessment components in the composite indicators aggregated by multiplicative utility functions, such that a low value in one of the components can dramatically reduce the composite indicator outcomes. It is preferable that the flood vulnerability assessment is measured by a non-compensatory composite indicator for which no trade-off occurs in the constituent components [13; 34], and also a non-compensatory composite indicator needs to be aggregated from all the assessment components with the same directional elasticity to vulnerability for robustness and confidence in composite indicator outcomes [35]. Thus, this study investigates various construction frameworks for non-compensatory composite indicators aggregated from the assessment components with all the same positive elasticity to the vulnerability, such as exposure, sensitivity, lack of adaptability, and combinations of the two of them as below. Given the above considerations of the aggregation schemes for composite indicators, the first flood vulnerability indicator \(FVI_{1}\) can be expressed in Equation (1) by compiling the three primary assessment components, exposure \(E\), sensitivity \(S\), and lack of adaptability \(A\) independently, as presented in previous studies [13; 14; 15; 16; 17]:  
\[FVI_{1}=E\ \times\ S\ \times\ A. \tag{1}\]  
As the vulnerability is also defined in terms of adaptability to cope with potential impacts of other global changes as well as climate change [1; 36; 37], potential impact \(PI\) is undertaken as a function of exposure and sensitivity [18]. Hence, the second flood vulnerability indicator \(FVI_{2}\) can be expressed in Equation (2) by the potential impact \(PI\) linearly aggregated from the exposure \(E\) and sensitivity \(S\)[19; 38; 39; 40] along with a lack of adaptability \(A\), following the framework in previous studies [20; 21; 41; 42; 43]:  
\[FVI_{2}=\ PI\ \times\ A=f(E,S)\ \times\ A. \tag{2}\]Other approaches for vulnerability assessment integrating climate exposures and adaptation practices are developed in the Republic of Korea to evaluate the potential flood damage used in the comprehensive national water resource plan for water resources [23], and also established in Mozambique for comprehensively assessing coastal communities' livelihood vulnerability to climate change [22]. Since households with limited assets are at an increased exposure to climate-induced variability and their ability to cope is restricted [48], livelihood vulnerability can be expressed as the sum of exposure and adaptability.