A Better Group Consensus Ranking via a Min-transitive Fuzzy Linear Ordering

Abstract

The current fuzzy methods for determining a group consensus ranking (GCR) or, equivalently, a linear order of items of a finite set S from a given set of weighted linear orders $\mathcal {L}= \{L_1, L_2, \ldots , L_M\}$ on S are ad hoc in nature. This is because the underlying key fuzzy relations used in determining the GCR are not min-transitive. We present here a better method for GCR based on a min-transitive fuzzy linear order on S obtained from $\mathcal {L}$. We define a collection of probability distributions $P_x(j), x \in S$, on the rank set $\{1, 2, \ldots , |S|\}$ based on the linear orders $\mathcal {L}$. The distributions $P_x(j)$ give a min-transitive fuzzy partial order $\mu _{\mathcal {L}}(\cdot , \cdot )$ on S, where $\mu _{\mathcal {L}}(x, y)$ says how “left” the distribution $P_x(\cdot )$ is to $P_y(\cdot )$. We then extend $\mu _{\mathcal {L}}(\cdot , \cdot )$ to a best possible min-transitive fuzzy linear order $\mu _{\mathcal {L}}^\star (\cdot , \cdot )$ on S, which gives the desired ranking of items in S.