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

Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1154)

## 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.

## Keywords

Group ranking Weighted linear orders Fuzzy partial order Fuzzy linear order Min-transitivity

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