Computing Version Spaces in the Qualitative Approach to Multicriteria Decision Aid

Abstract

We consider a lattice-based model in multiattribute decision making, where preferences are represented by global utility functions that evaluate alternatives in a lattice structure (which can account for situations of indifference as well as of incomparability). Essentially, this evaluation is obtained by first encoding each of the attributes (nominal, qualitative, numeric, etc.) of each alternative into a distributive lattice, and then aggregating such values by lattice functions. We formulate version spaces within this model (global preferences consistent with empirical data) as solutions of an interpolation problem and present their complete descriptions accordingly. Moreover, we consider the computational complexity of this interpolation problem, and show that up to 3 attributes it is solvable in polynomial time, whereas it is NP complete over more than 3 attributes. Our results are then illustrated with a concrete example.

Learning rule sets and Sugeno integrals for monotonic classification problems

“…Improvements towards more compact representations seem feasible, even if the minimal SUF-interpolation problem is NP-hard [18]:…”

“…Several studies [23,24,31,43], relying on empirical or noisy artificial data, show that in certain cases, relabeling the data improves model accuracy. In our case, relabeling D allows to subsequently perform interpolation, in the sense of [18].…”

“…They can only represent selection and rejection rules involving a single threshold for all attributes. This problem can be partially circumvented using qualitative decomposable models based on Sugeno integrals, that we call Sugeno Utility Functionals (SUFs), first introduced in [38] and extensively studied in [17,19,20,21,18], where local mappings from attribute scales to the set of classes are used to make the former commensurate with latter.…”

Interpolation by lattice polynomial functions: A polynomial time algorithm

“…However, in many cases, the ranges of the criteria differ from the set of utility values. To overcome these limitations, the authors of [8,10] introduced generalizations of Sugeno integrals, such as Sugeno Utility Functionals (SUF) or maxima of SUFs. These models allow to merge local evaluations when the ranges of criteria differ from each other.…”

“…Furthermore we intend to extend our interpolation framework to SUFs (see Section 3). In fact a solution to the interpolation problem generalized to a certain class of restricted SUFs was already proposed in [10]. However it was shown that the interpolation problem in that case is NP-complete.…”