Robust Ordinal Regression

Making any type of decision, from buying a car to siting a nuclear plant, from choosing the best student deserving a scholarship to ranking the cities of the world according to their liveability, involves the evaluation of several alternatives with respect to different aspects, technically called evaluation criteria. Multiple Criteria Decision Aiding (MCDA) (see [13, 14]) provides methodologies to recommend the Decision Maker (DM) a decision that best fits the DM’s preferences. Formally, in MCDA, a set of n alternatives A = {a1, . . . , an} is evaluated with respect to a consistent family of m criteria G = {g1, . . . , gm} [50]. In general, each criterion gj ∈ G can be considered as a realvalued function gj : A → Ij ⊆ R, where the elements of Ij are real numbers having either the meaning of quantities for quantitative criteria, or the meaning of ordered identifiers for qualitative criteria, e.g., 1=“bad”, 2=“medium”, 3=“good”. Each criterion gj can have an increasing or a decreasing direction of preference. In the first case, the higher the evaluation gj (a), the better a is with respect to criterion gj ; in the other case, the higher the evaluation gj (a), the worse a is with respect to criterion gj . For example, evaluating a car involves both quantitative and qualitative criteria having increasing or decreasing direction of preference. Price and acceleration are typical quantitative criteria while comfort and safety are qualitative criteria. Among these, acceleration, comfort and safety have an increasing direction of preference, while price has a decreasing direction of preference. In the following, without loss of generality, we shall suppose that criteria have increasing direction of preference. According to Roy [51], in MCDA the following three most important decision problems are considered: