A compositional approach to air quality index (AQI) based on a log-contrast of pollutants over non-polluted air fraction was recently developed. This approach let to three open questions about consistency and formalization of such index. First question is to find out the conditions under which the approximation of the log-contrast by the logarithm of the geometric mean of pollutants concentrations, are reliable. Second, is to examine a way to define a reasonable scale for express air quality, taking into account the effects of air pollution on population health and on social or economic activity. The third one is to find out the properties of this kind of index with respect to the addition of a new set of air pollutants in the composition. To deal with these open questions, a decision making tree should be designed in order to select suitable weights on the air pollutants. A preliminary application study of air pollutants composition distribution in some cities is presented. In addition, the approach of the log-contrast by the logarithm of the weighted geometric mean is described for these cities.