Completing 2-tuple linguistic preference relations based on upper bound condition

The purpose of this paper is to propose a method to estimate missing preference values when handling incomplete linguistic preference relations assessed by using 2-tuple linguistic preference values. We assert that additive transitivity alone may result in estimated values that do not qualify as 2-tuple linguistic preference values. Therefore, an upper bound condition based on additive transitivity is proposed to deal with incomplete 2-tuple linguistic preference relations (2TLPRs). We prove that if experts with incomplete information abide by this property, then the missing preference values can be estimated such that they are expressible. Additionally, because of this property, the resultant complete 2TLPR is proved to be consistent. Based on additive transitivity and the upper bound condition, an interactive completing algorithm is developed to estimate the missing entries in incomplete 2TLPR. When the upper bound condition is not satisfied or there exist entries that do not coincide with the real opinions of the experts, the proposed algorithm can automatically adjust the known entries by interacting with the experts. By using the proposed algorithm, the completed 2TLPR is not only additively consistent, but also can accurately reflect the actual preference of the experts.