Probabilistic linguistic term sets (PLTSs) provide a flexible manner to model complex linguistic expressions in qualitative decision making. Some fundamental aspects of PLTSs, such as order relations and basic operations, are not built in a solid way. The lack of required properties make their applications be not sufficiently reliable. In this study, a PLTS is mapped to a stochastic variable. A partial order of PLTSs is defined based on the idea of stochastic order. The concept of a total order is then provided by refining the partial order. Moreover, aggregation functions of PLTSs are proposed by defining an aggregation function in the set of linguistic terms. Operations and aggregation operators of PLTSs are further carried out based on the aggregation functions. The above theoretical developments own some merits: (1) the partial order is reasonable because the ranking results coincide with rational decision-makers who consider increasing utility functions; (2) the operations preserve desirable properties which are satisfied by the linguistic aggregation functions; and (3) the operations and aggregation operators of PLTSs could be specified according to practical applications based on the proposed theoretical framework.

This paper proposes an approach to linguistic three-way decision making problem with double hierarchy linguistic term evaluation information. Double hierarchy linguistic term set consists of the first hierarchy and second hierarchy linguistic term set, which can describe uncertainty and fuzziness more flexibly. First, the Hamacher operational rules, score function and distance measure of double hierarchy linguistic elements are defined. Next, we construct the double hierarchy linguistic decision-theoretic rough set model. And the conditional probability is calculated based on the double hierarchy linguistic term environment with grey relational analysis, which makes the process of decisions more rational. Then the loss functions are aggregated by the double hierarchy linguistic Hamacher weighted averaging operator, which takes into account different decision attitudes of decision makers. And the results of decision are deduced by the minimum-loss principle. Finally, a case study about the selection of cooperation companies during the COVID-19 is used to demonstrate the practicability of our proposed method.

The hesitant Pythagorean fuzzy set is frequently considered as a solution for decision making under uncertainty. Whereas the representation of uncertain information might be not sufficient in the hesitant Pythagorean fuzzy environment, thus the concept of interval-valued hesitant Pythagorean fuzzy sets (IVHPFSs) is proposed. Specifically, we first propose the concept of IVHPFSs and then study the operational rules and distance measures of IVHPFSs in detail. To ease the possible application, we explore two decision-making processes in the setting of IVHPFSs by drawing support from the technique for order preference by similarity to ideal solution and Choquet integral-based method. Finally, the selecting processes of project private partner are also presented to demonstrate the decision-making processes based on IVHPFSs and compared with some similar techniques.

Modeling linguistic information is vital for qualitative decision making (QDM). Compared with single linguistic terms, the complex linguistic expressions (CLEs) are more powerful and flexible to express linguistic opinions under uncertainties. Among the existing types of CLEs, the linguistic terms with weakened hedges (LTWHs), which focus on the uncertainty of using single terms, can be used to model the linguistic expressions in natural languages. This paper concentrates on the application of LTWHs in the framework of QDM with preference relations. The concept of linguistic preference relations with hedges is presented after a new computational model of LTWHs is formed. Some consistency concepts, such as weak consistency and additive consistency, are then defined and their properties are studied. Theories and algorithms for consistency checking and improving are proposed. Finally, the availability of the proposed technique is demonstrated by a real application. Different from many studies related to consistency measures, we make use of fuzzy weighted digraphs to develop the theories and algorithms in a visible manner. Moreover, for consistency improving, the degree of consistency is measured by linguistic terms rather than numerical values so that the threshold of satisfactory consistency is interpretable.