The Evaluation of the Entropy of Decision Makers' Preferences in Ordinal Consensus Ranking Problem

The aim of the article is to introduce Shannon entropy and the relative entropy into ordinal consensus ranking problem (OCRP) context as a measure of uncertainty of preferences within a group of decision makers (DMs). An existence of a consensus in the OCRP crucially depends on initial decision makers' preferences: rankings of n objects from the 1st to the nth place. When those rankings are markedly different, there is high entropy (uncertainty) associated with DMs' preferences and a unique consensus within a group is unlikely to be achieved. However, when DMs' rankings are similar, the entropy is small and a consensus is much more likely to be obtained. In the article a relationship between the relative entropy of DMs preferences and an agreement among four classic methods' for OCRP solutions is discussed on several examples. Moreover, experimental evaluation of the relationship is performed and our results indicate that there is a negative relation between the relative entropy and the agreement among methods' solutions, e.g. the higher is the relative entropy, the lower is the methods' agreement. The question remains whether a theoretical relation connecting the entropy and a solution existence (or agreement among methods' solutions) to OCRP can be derived in general.