Linguistic Group Decision-Making (LiGDM) aims at solving decision situations involving human decision-makers (DMs) whose opinions are modeled by using linguistic information. To achieve agreed solutions that increase DMs' satisfaction towards the collective solution, Linguistic Consensus Reaching Processes (LiCRPs) have been developed. These LiCRPs aim at suggesting DMs to change their original opinions to increase the group consensus degree, computed by a certain consensus measure. In recent years, these LiCRPs have been a prolific research line, and consequently numerous proposals have been introduced in the specialized literature. However, it has been pointed out the non-existence of objective metrics to compare these models and decide which one presents the best performance for each LiGDM problem. Therefore, this paper aims at introducing a metric to evaluate the performance of LiCRPs that takes into account the resulting consensus degree and the cost of modifying DMs' initial opinions. Such a metric is based on a linguistic Comprehensive Minimum Cost Consensus (CMCC) model based on ELICIT (Extended Comparative Linguistic Expressions with Symbolic Translation) information that models DMs' hesitancy and provides accurate Computing with Words processes. Additionally, the linguistic CMCC optimization model is linearized to speed up the computational model and improve its accuracy.