Interactive algorithms for improving incomplete linguistic preference relations based on consistency measures

Incomplete linguistic preference relations (InLPRs) are generally inevitable in group decision making problems due to several reasons. Two vital issues of InLPRs are the consistency and the estimation of missing entries. The initial InLPR may be not consistent, which means that some of its entries do not reflect the real opinions of the experts accurately. Thus, there are deviations between some initial provided values and real opinions. Therefore, it is valuable to elicit the providers to realize and repair the deviations. In this paper, we discuss the consistency and the completing algorithms of InLPRs by interacting with the experts. Servicing as the minimum condition of consistency, the weak consistency of InLPRs is defined and a weak consistency reaching algorithm is designed to guarantee the logical correctness of InLPRs. Then two distinct completing algorithms are presented to estimate the missing entries. The former not only estimates all possible linguistic terms and represents them by the extended hesitant fuzzy linguistic terms sets but also keeps weak consistency during the computing procedures. The later can automatically revise the existing entries using the new opinions supplemented by the experts during interactions. All the proposed algorithms interact with the experts to elicit and mine their actual opinions more accurately. A real case study is also presented to clarify the advantages of our proposal. Moreover, these algorithms can serve as assistant tools for the experts to present their preferences.