We report approaches for measuring the transitivity properties of fuzzy preference relations. The concept of transitivity measurement is proposed to quantify the transitivity degree of fuzzy preference relations. We reveal the equivalence conditions of three typical transitivity properties: weak stochastic transitivity, moderate stochastic transitivity, and strong stochastic transitivity. The likelihoods of the Condorcet paradox and inverted-order voting paradox occurring in a voting profile are measured by our defining a possibility degree index. When a fuzzy preference relation is not transitive, an iteration algorithm is proposed to obtain a new matrix with transitivity. It is observed that different transitivity properties of fuzzy preference relations correspond to different measurement methods. The occurrence of the voting paradoxes could be characterized by the development of a transitivity measurement.

我们报告了测量模糊偏好关系的传递特性的方法。提出传递性测度的概念来量化模糊偏好关系的传递性程度。我们揭示了三种典型传递性属性的等价条件：弱随机传递性、中等随机传递性和强随机传递性。Condorcet悖论和倒序投票悖论发生在投票概况中的可能性通过我们定义的可能性程度指数来衡量。当模糊偏好关系不具有传递性时，提出一种迭代算法来获得具有传递性的新矩阵。可以看出，模糊偏好关系的不同传递性质对应于不同的测量方法。