Optimization of multiple satisfaction levels in portfolio decision analysis

We consider a portfolio decision problem in which a set of projects forming a portfolio has to be selected taking into account multiple evaluation criteria and some constraints related to the limited resources (e.g., available budget). Traditionally, such a problem has been approached by Multiple Attribute Value Theory (MAVT) with the aim of maximizing the sum of values associated with the projects included in the selected portfolio. Using MAVT, one represents preferences on the individual projects, and a value of a portfolio is just an aggregate of values of the component projects. This linear value approach does not explicitly account for portfolio balance requirements, raising the risk of selecting a portfolio which is, e.g., composed of projects with good evaluations on the same criterion or on the same small subset of criteria. Thus, we propose a different approach that enables the Decision Maker (DM) to control the distribution of good evaluations on different criteria over the projects composing a portfolio. With this aim, for each criterion we fix a certain number of reference levels corresponding to the qualitative satisfaction degrees. The number of projects entering a portfolio and attaining each of these levels becomes an objective to be maximized. To solve thus formulated multi-objective optimization problem, we use Dominance-based Rough Set Approach (DRSA). The DM is expected to point out some prospective portfolios in a current sample of non-dominated portfolios. DRSA represents the DM's preferences with a set of decision rules induced from such indirect preference information. Their use permits to progressively focus the search on the part of the non-dominated portfolios that satisfy the DM's preferences in the best way