A Hybrid Stochastic-Full Enumeration Approach to a Ranking Problem with Insufficient Information

When comparing n objects pairwise, at least (n−1) comparisons have to be performed (assuming that a corresponding directed graph is connected) for a derivation of a ranking (a total or partial order) of all objects. The aim of the paper is to introduce a novel algorithm for a case with insufficient information, that is the case when the number of available pairwise comparisons ranges from 1 to (n − 2). It is assumed that the comparisons are performed via the following two non-numerical binary relations: preference relation (≻) and indifference relation(∼). The algorithm provides a probability of each possible ranking (permutation) of all compared objects based on the revealed pairwise comparisons, while missing comparisons are modeled via full enumeration of all feasible cases (for a small number of objects), or via Monte Carlo simulations (for a large number of objects).

I. Introduction

Pairwise comparisons (PCs), that is comparisons of (only) two objects at a time, constitute a fundamental part of many multiple criteria decision/aiding methods such as the AHP/ANP, BWM, ELECTRE, MACBETH, PAPRIKA, or PROMETHEE, see e.g. [2], [4], [9], [10], [11], [19], [23], [24], [25], or [26].