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Supplier selection is a complex decision-making process which is influenced by uncertainty and DM hesitation. To deal with this problem, various studies have proposed the use of methods based on Fuzzy Set Theory and its extensions. There are several varieties of Hesitant Fuzzy Linguistic VIKOR (HFLVIKOR) methods in the literature which can use two or more linguistic terms in each evaluation and can generate solutions based on the group's utility or individual regret. Despite these benefits, we have not found comparative studies involving HFLVIKOR methods to supplier selection. Given this gap, this study applies and compares Extended Hesitant Fuzzy Linguistic VIKOR (EHFLVIKOR) methods and Hesitant Fuzzy Linguistic Term Set VIKOR (PDHFLVIKOR) methods with Possibility Distributions within the supplier selection context. This application was realized using a service company. We also performed 8 scenario simulations to analyze the effect of including other criteria and alternatives. Despite the identified limitations of each method, we conclude that both are appropriate for dealing with group decision making under uncertainty and require little effort in their data collection. The PDHFLVIKOR results were more consistent in maintaining the ranking and the compromise solution in the real application as well as simulated scenarios. Unlike PDHFLVIKOR, EHFLVIKOR loses information when there is a consensus. On the other hand, because it makes it possible to create subgroups of DMs, EHFLVIKOR appears to be more appropriate for companies in which there is more than one DM involved in supplier selection for each functional area. This study's results will help managers and academics in selecting the most appropriate method for supplier selection. • Compares EHFLVIKOR and PDHFLVIKOR methods for supplier selection. • Analyzes the characteristics and points out the benefits and limitations of each method. • Both methods are consistent when new alternatives or criteria are included. • EHFLVIKOR is more appropriate when there is no consensus among decision maker subgroups. • PDHFLVIKOR does not lose information when there is a consensus and deals better with uncertainty. [ABSTRACT FROM AUTHOR]