Incomplete preference relations: An upper bound condition

In decision making, consistency in fuzzy preference relations is associated with the study of transitivity property. While using additive consistency property to complete incomplete preference relations, the preference values found may lie outside the interval [0, 1] or the resultant relation may itself be inconsistent. This paper proposes a method that avoids inconsistency and completes an incomplete preference relation using an upper bound condition. Additionally, the paper extends the upper bound condition for multiplicative reciprocal preference relations. The proposed methods ensure that if (n − 1) preference values are provided by an expert, such that they satisfy the upper bound condition, then the preference relation is completed such that the estimated values lie inside the unit interval [0, 1] in the case of preference relations and [1/9, 9] in the case of multiplicative preference relation. Moreover, the resultant preference relation obtained using the proposed method is transitive.