When using linguistic approaches to solve decision problems, we need the techniques for computing with words (CW). Together with the 2-tuple fuzzy linguistic representation models (i.e., the Herrera and Martinez model and the Wang and Hao model), some computational techniques for CW are also developed. In this paper, we define the concept of numerical scale and extend the 2-tuple fuzzy linguistic representation models under the numerical scale. We find that the key of computational techniques based on linguistic 2-tuples is to set suitable numerical scale with the purpose of making transformations between linguistic 2-tuples and numerical values. By defining the concept of the transitive calibration matrix and its consistent index, this paper develops an optimization model to compute the numerical scale of the linguistic term set. The desired properties of the optimization model are also presented. Furthermore, we discuss how to construct the transitive calibration matrix for decision problems using linguistic preference relations and analyze the linkage between the consistent index of the transitive calibration matrix and one of the linguistic preference relations. The results in this paper are pretty helpful to complete the fuzzy 2-tuple representation models for CW.

In group decision making, consensus models are decision aid tools and help experts modify their individual opinions to reach a closer agreement. Based on the concept of minimum-cost consensus, this paper proposes a novel framework to achieve minimum-cost consensus under aggregation operators. Analytical results indicate that the proposed framework reduces to the consensus model of Ben-Arieh when the selected aggregation operator is the ordered weighted averaging (OWA) operator with weight vector (1/2, ..., 0, ..., 1/2)^T. Furthermore, this paper closely examines the minimum-cost consensus models with a linear cost function under the common aggregation operators (e.g., the weighted averaging operator and the OWA operator). Linear-programming-based approaches are also developed to solve these models. The results of this paper significantly contribute to efforts to develop the consensus model of Ben-Arieh et al.

Nowadays, societal and technological trends demand the management of large scale of decision makers in group decision-making (GDM) contexts. In a large-scale GDM, decision makers often have individual concerns and satisfactions, and also they will use heterogeneous preference representation structures to express their preferences. Meanwhile, it is difficult to set the numerical consensus threshold to judge whether a consensus degree can be acceptable or not in the consensus reaching process in a large-scale GDM. This study proposes a novel consensus reaching model for the heterogeneous large-scale GDM with the individual concerns and satisfactions. In this consensus reaching model, a selection process is proposed to obtain the individual preference vectors, to divide decision makers into different clusters, and to yield the preference vector of the large group. Following this, a consensus measure method that considers the individual concerns on alternatives is defined for measuring the consensus degree, and a linguistic approach is developed to measure the individual and collective satisfactions regarding the consensus degree. Finally, a feedback adjustment process is proposed and utilized to help decision makers adjust their preferences. A practical example and a simulation analysis are presented to demonstrate the validity of the proposed consensus reaching model.