Consistency measures for hesitant fuzzy linguistic preference relations

Hesitant fuzzy linguistic term sets (HFLTSs) are used to deal with situations in which the decision makers (DMs) think of several possible linguistic values or richer expressions than a single term for an indicator, alternative, variable, etc. Compared with fuzzy linguistic approaches, they are more convenient and flexible to reflect the DMs' preferences in decision making. For further applications of HFLTSs to decision making, we develop a concept of hesitant fuzzy linguistic preference relations (HFLPRs) as a tool to collect and present the DMs' preferences. Due to the importance of the consistency measures using preference relations in decision making, we develop some consistency measures for HFLPRs to ensure that the DMs are being neither random nor illogical. A consistency index is defined to establish the consistency thresholds of HFLPRs to measure whether an HFLPR is of acceptable consistency. For HFLPRs with unacceptable consistency, we develop two optimization methods to improve the consistency until they are acceptable. Several illustrative examples are given to validate the consistency measures and the optimization methods.