Common probability-based interactive algorithms for group decision making with normalized probability linguistic preference relations

Probabilistic linguistic variable is a kind of powerful qualitative fuzzy sets, which permits the decision makers (DMs) to apply several linguistic variables with probabilities to denote a judgment. This paper studies group decision making (GDM) with normalized probability linguistic preference relations (NPLPRs). To achieve this goal, an acceptably multiplicative consistency based interactive algorithm is provided to derive common probability linguistic preference relations (CPLPRs) from PLPRs, by which a new acceptably multiplicative consistency concept for NPLPRs is defined. When the multiplicative consistency of NPLPRs is unacceptable, models for deriving acceptably multiplicatively consistent NPLPRs are constructed. Then, it studies incomplete NPLPRs (InNPLPRs) and offers a common probability and acceptably multiplicative consistency based interactive algorithm to determine missing judgments. Furthermore, a correlation coefficient between CPLPRs is provided, by which the weights of the DMs are ascertained. Meanwhile, a consensus index based on CPLPRs is defined. When the consensus does not reach the requirement, a model to increase the level of consensus is built that can ensure the adjusted LPRs to meet the multiplicative consistency and consensus requirement. Moreover, an interactive algorithm for GDM with NPLPRs is provided, which can address unacceptably multiplicatively consistent InNPLPRs. Finally, an example about the evaluation of green design schemes for new energy vehicles is provided to indicate the application of the new algorithm and comparative analysis is conducted.