Ranking of non-linear qualitative decision preferences using copulas

In this paper we address the problem of option ranking in qualitative evaluation models. Current approaches make the assumptions that when qualitative data are suitably mapped into discrete quantitative data, they form monotone or closely linear tabular value functions. Although the power of using monotone and linear functions to model decision maker’’s preferences is impressive, there are many cases of non-linear decision preferences that need to be modeled using non-linear functions. In this paper, we present one possibility of how to capture the discrete non-linear decision maker preferences by employing copulas. Copulas are functions that manage to capture the non-linear dependences between random variables. Mainly, they are used for aggregation of two attributes. We extend the concept to multivariate case by introducing a hierarchical copula. That way we capture the non-linear dependences among all uniformly distributed variables. We use the obtained dependence structure for copula-based median regression which results into the required option ranking. The results show that this method may outperform the current approaches for qualitative option ranking of non-monotone decision preferences for a class of non-linear preferences. Furthermore, the mathematics behind copula functions allows extending their usage on preferences expressed with continuous attributes.