Hesitant fuzzy linguistic preference relation (HFLPR) is a new preference structure that the decision makers (DMs) are hesitant about several possible linguistic terms of preference information for pairwise comparison between alternatives each other. The purpose of this paper is to investigate the additive consistency of hesitant fuzzy linguistic preference relation (HFLPR) with a new expansion principle for hesitant fuzzy linguistic term sets (HFLTSs). In order to normalize HFLTSs with different numbers of linguistic terms, a least common multiple extended (LCME) principle is proposed. According to the LCME principle, the additive consistent index of a HFLPR is defined to measure the consistency level of the HFLPR. For improving the unacceptable additive consistency of a HFLPR, a pure integer programming model is constructed to derive an acceptable additive consistent HFLPR. The similarities between DMs are calculated based on their individual HFLPR with acceptable additive consistency. Subsequently, the confidence degrees of DMs are defined to derive DMs' weights, and some examples are analyzed to verify the effectiveness of the proposed method.

Large-scale group decision making (LSGDM) is a common decision-making activity in which experts elicit their preferences through preference relations with consistencies based on additive properties. LSGDM has become a popular topic in decision making because of its necessity and applicability in multiple fields. In LSGDM, the consensus reaching process (CRP) ensures that decision makers (DMs) agree on the final decision-making results. Therefore, it is significant to investigate the CRP to improve the LSGDM process. In this paper, a new method for LSGDM that includes a clustering algorithm, a weight determination method and a CRP is developed. First, based on the consensus and the consistency of the additive preference relation, the DMs are classified into different subgroups by using the k-means clustering algorithm. Then, because different subgroups have distinct decision-making interests, a weight determination method based on a cooperative game is proposed. Furthermore, to measure the consensus of each subgroup more comprehensively, the intra- and inter-consensus levels are defined. These consensus levels are divided into four scenarios (acceptable-acceptable, acceptable-unacceptable, unacceptable-acceptable, and unacceptable-unacceptable) according to predetermined thresholds. Furthermore, different feedback mechanisms based on multiobjective optimization models corresponding to different CRP scenarios are presented. Finally, an illustrative example and comparative analyses are provided to verify the feasibility and validity of the proposed method.

In most existing linguistic distribution multiple criteria group decision-making (MCGDM) models, the sample size of group linguistic distribution assessment (LDA) is often neglected, and the decision makers are viewed to be independent. The probabilistic distribution is closely related to the sample size, and there exist more or fewer interference effects among different individual opinions. Thus, this article develops a quantum framework for modeling interference effects in the linguistic distribution MCGDM process. First, considering the sample size information, we redefine the LDAs and provide a new computational model for them. Second, to integrate a series of LDAs, the linguistic distribution weighted averaging (LDWA) operator is presented. Mathematic proofs reveal that the new LDWA operator can not only ensure the information integrality but also can avoid the conjunctive and disjunctive operations. Third, to explore the interference effects among individual opinions, a quantum framework is constructed. In this process, individual opinions are viewed as various wave functions occurring synchronously. They interfere with each and influence the aggregation result. Finally, an illustrative example of Internet finance soft power evaluation is provided to verify the effectiveness. Sensitivity and comparative analyses are also implemented to assess the stability and validity of our method.

Air pollution forecasting plays a vital role in environment pollution warning and control. Air pollution forecasting studies can also recommend pollutant emission control strategies to mitigate the number of poor air quality days. Although various literature works have focused on the decomposition-ensemble forecasting model, studies concerning the endpoint effect of ensemble empirical mode decomposition (EEMD) and the forecasting model of sub-series selection are still limited. In this study, a hybrid forecasting approach (EEMD-MM-CFM) is proposed based on integrated EEMD with the endpoint condition mirror method and combined forecasting model for sub-series. The main steps of the proposed model are as follows: Firstly, EEMD, which sifts the sub-series intrinsic mode functions (IMFs) and a residue, is proposed based on the endpoint condition method. Then, based on the different individual forecasting methods, an optimal combined forecasting model is developed to forecast the IMFs and residue. Finally, the outputs are obtained by summing the forecasts. For illustration and comparison, air quality index (AQI) data from Hefei in China are used as the sample, and the empirical results indicate that the proposed approach is superior to benchmark models in terms of some forecasting assessment measures. The proposed hybrid approach can be utilized for air quality index forecasting.