On Possibly Optimal Tradeoffs in Multicriteria Spanning Tree Problems

International audienceIn this paper, we propose an interactive approach to determine a compromise solution in the multicriteria spanning tree problem. We assume that the Decision Maker’s preferences over spanning trees can be represented by a weighted sum of criteria but that weights are imprecisely known. In the first part of the paper, we propose a generalization of Prim’s algorithm to determine the set of possibly optimal tradeoffs. In the second part, we propose an incremental weight elicitation method to reduce the set of feasible weights so as to identify a necessary optimal tradeoff. Numerical tests are given to demonstrate the practical feasibility of the approach

Scaling-invariant maximum margin preference learning

One natural way to express preferences over items is to represent them in the form of pairwise comparisons, from which a model is learned in order to predict further preferences. In this setting, if an item a is preferred to the item b, then it is natural to consider that the preference still holds after multiplying both vectors by a positive scalar (e.g., ). Such invariance to scaling is satisfied in maximum margin learning approaches for pairs of test vectors, but not for the preference input pairs, i.e., scaling the inputs in a different way could result in a different preference relation being learned. In addition to the scaling of preference inputs, maximum margin methods are also sensitive to the way used for normalizing (scaling) the features, which is an essential pre-processing phase for these methods. In this paper, we define and analyse more cautious preference relations that are invariant to the scaling of features, or preference inputs, or both simultaneously; this leads to computational methods for testing dominance with respect to the induced relations, and for generating optimal solutions (i.e., best items) among a set of alternatives. In our experiments, we compare the relations and their associated optimality sets based on their decisiveness, computation time and cardinality of the optimal set

Minimality and comparison of sets of multi-attribute vectors

In a decision-making problem, there is often some uncertainty regarding the user preferences. We assume a parameterised utility model, where in each scenario we have a utility function over alternatives, and where each scenario represents a possible user preference model consistent with the input preference information. With a set A of alternatives available to the decision-maker, we can consider the associated utility function, expressing, for each scenario, the maximum utility among the alternatives. We consider two main problems: firstly, finding a minimal subset of A that is equivalent to it, i.e., that has the same utility function. We show that for important classes of preference models, the set of possibly strictly optimal alternatives is the unique minimal equivalent subset. Secondly, we consider how to compare A to another set of alternatives B , where A and B correspond to different initial decision choices. This is closely related to the problem of computing setwise max regret. We derive mathematical results that allow different computational techniques for these problems, using linear programming, and especially, with a novel approach using the extreme points of the epigraph of the utility function