Multi-objective optimization under positivity constraints, with a meteorological example

In many practical situations, we need to optimize several objectives under the positivity constraints. For example, in meteorological and environmental studies, it is important to collect various types of data, such as temperature and wind speed and direction, from weather stations. For maintenance purposes, it is convenient to place instruments that collect different weather data on the same weather station. Thus, we need to find the “best” location for a weather station. The “best” means, for example, that the external influences, such as flux of cars passing on nearby road, have a minimal impact on the measurement results. There are several such criteria, so we face a multi-objective optimization problem. In this paper, we show that traditional approaches for solving such problems — such as the weighted sum approach — are not fully adequate for solving our problem. We show that fuzzy heuristics lead to a more adequate approach — of using a generalized form of Nash bargaining solution. We then prove that under reasonable assumptions of scale-invariance, the generalized Nash bargaining solution is the only adequate solution for the general problem of multi-objective optimization under positivity constraints — and, in particular, for the problem of selecting an optimal location for a weather station.