Consistency and consensus driven for hesitant fuzzy linguistic decision making with pairwise comparisons

•Define the hesitant fuzzy linguistic geometric consistency index (HFLGCI).•Build a feedback consensus algorithm for hesitant fuzzy linguistic decision making.•Analyze the critical values of HFLGCI with different orders of preference relations.•Provide an online decision-making portal based on the proposed algorithm.•Present a case concerning performance evaluation of venture capital guiding funds.