A Relative Distance-based Scalarization Scheme using Reference Levels

When variables measured in different units are used to analyze a given phenomenon, it is usually necessary to scale these variables in order to bring all of them down to a common scale. This allows their subsequent aggregation into a single measurement. This is the case, for example, of the processes of constructing composite indicators from a system of simple indicators. One way to perform this scaling is through distance-based schemes, used when reference levels are available for the different variables, which allow defining different performance bands. In these cases, the scaling function is often called the achievement scalarizing function. However, the linear nature of the achievement scalarizing function employed so far implies that the achievement level of each entity is scalarized in absolute terms, that is, considering their absolute distance to the reference levels, without considering the distribution of achievement values across all considered entities. In this paper, we propose a new achievement scalarizing function based on reference levels. This new function encompasses formulations that allow for the inclusion of relative assessments. Thus, we seek to broaden the application scope of distance-based scalarizations to analyze societal aspects where it is crucial to compare the performances of the entities not only with respect to the reference levels, but also with respect to the performances achieved by other entities. Finally, since absolute or relative measures may be required for different values when scaling the same variable, we propose a more general hybrid scheme that allows the combination of both schemes.