Connecting the numerical scale model to the unbalanced linguistic term sets

Herrera and Martinez initiated a 2-tuple fuzzy linguistic representation model for computing with words (CWW). In addition to the Herrera and Martinez model, two different models based on linguistic 2-tuples (i.e., the model of Herrera et al. and the numerical scale model) have been developed to deal with term sets that are not uniformly and symmetrically distributed, i.e., unbalanced linguistic term sets (ULTSs). Both the model of Herrera et al. and the numerical scale model can deal with ULTSs, so a challenge is naturally proposed to analysts: how to compare these two different models. In this study, we provide a connection between the model of Herrera et al. and the numerical scale model. The results show that the model of Herrera et al. provides a new approach to set a numerical scale. Furthermore, we prove the equivalence of the linguistic computational models between the model of Herrera et al. and the numerical scale model, if the numerical scale is set based on the model of Herrera et al.