Distances Between Partial Preference Orderings

TL;DR

The paper tackles measuring distances between partial preference orderings by contrasting a brute-force combinatorial approach with a belief-function framework. It shows that Frobenius distance on pairwise preference matrices can be extended to PPOs and that a direct belief-function method using 8N×8N BBA matrices avoids combinatorial growth while modeling uncertainty explicitly. Through two worked examples, it demonstrates that direct BBA-based distances align with average-based intuitions and often outperform indirect BB A distance measures which may yield divergent conclusions. The work provides a practical, scalable tool for distance computation in MCDM under uncertainty and outlines future extension to hybrid partial preferences.

Abstract

This paper proposes to establish the distance between partial preference orderings based on two very different approaches. The first approach corresponds to the brute force method based on combinatorics. It generates all possible complete preference orderings compatible with the partial preference orderings and calculates the Frobenius distance between all fully compatible preference orderings. Unfortunately, this first method is not very efficient in solving high-dimensional problems because of its big combinatorial complexity. That is why we propose to circumvent this problem by using a second approach based on belief functions, which can adequately model the missing information of partial preference orderings. This second approach to the calculation of distance does not suffer from combinatorial complexity limitation. We show through simple examples how these two theoretical methods work.