Such order relations can be, e.g., comparing alternatives by the values of the evaluation functions In this paper, we construct and compare algorithmic approaches lexicographically [15], by Pareto order, weighted sums [6], to solve the Preference Consistency Problem for based on hierarchical models [16] or by conditional preferences preference statements based on hierarchical models. Instances structures as CP-nets [2] and partial lexicographic preference of this problem contain a set of preference statements that are trees [11]. Here, the choice of the order relation can direct comparisons (strict and non-strict) between some alternatives, lead to stronger or weaker inferences and can make solving and a set of evaluation functions by which all alternatives PDP computationally more or less challenging. In a recommender can be rated. An instance is consistent based on hierarchical system or in a multi-objective decision making scenario, preference models, if there exists an hierarchical model the user should only be presented with a relatively small on the evaluation functions that induces an order relation on number of solutions, hence, a strong order relation is required.

Preference Inference involves inferring additional user preferences from elicited or observed preferences, based on assumptions regarding the form of the user's preference relation. In this paper we consider a situation in which alternatives have an associated vector of costs, each component corresponding to a different criterion, and are compared using a kind of lexicographic order, similar to the way alternatives are compared in a Hierarchical Constraint Logic Programming model. It is assumed that the user has some (unknown) importance ordering on criteria, and that to compare two alternatives, firstly, the combined cost of each alternative with respect to the most important criteria are compared; only if these combined costs are equal, are the next most important criteria considered. The preference inference problem then consists of determining whether a preference statement can be inferred from a set of input preferences. We show that this problem is co-NP-complete, even if one restricts the cardinality of the equal-importance sets to have at most two elements, and one only considers non-strict preferences. However, it is polynomial if it is assumed that the user's ordering of criteria is a total ordering; it is also polynomial if the sets of equally important criteria are all equivalence classes of a given fixed equivalence relation. We give an efficient polynomial algorithm for these cases, which also throws light on the structure of the inference.