Order Relations and Operations on the Set of Probabilistic Linguistic Term Sets

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 makes 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 total order is then provided by refining the partial order. Moreover, aggregation functions of PLTSs are proposed by defining an aggregation function in a 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 such as the partial order is reasonable because the ranking results coincide with rational decision-makers who consider increasing utility functions, the operations preserve desirable properties that are satisfied by the linguistic aggregation functions, and the operations and aggregation operators of PLTSs could be specified according to practical applications based on the proposed theoretical frame work.