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An incomplete uncertain preference relation (UPR) is typical in group decision making (GDM) for decision makers (DMs) to express preference over alternatives because of the information interaction barrier between people and decision making environment. Completing missing values can guarantee individual consistency and consensus level effectively. The operation of traditional interval preference relations (IPRs) is based only on the end point transformation, which may cause interval discretisation and information distortion easily. To overcome these limitations, pairwise comparison of alternatives in an IPR is treated as an uncertain distribution function of the subjective preference of the DM which avoids discretisation operation and handles interval numbers collectively. A belief degree is used to maintain the original information as much as possible. It guarantees the extent how people believe the estimated value is close to the incomplete original value. An uncertain chance constrained programming model is proposed herein to estimate incomplete values based on a belief degree when the preference relation obeys a linear uncertain distribution. A distance measure is defined to compute the consistency index and consensus degree. Subsequently, an iterative algorithm is presented for GDM with linear UPRs, which adjusts inconsistent preference relations and uses an operator to aggregate all individual preference relations. Furthermore, it is proven that the operation of UPRs is an extension of that of traditional IPRs under a certain belief degree.

Optimisation of Group Consistency for Incomplete Uncertain Preference Relation