Multicriteria decision making based on bi-direction Choquet integrals

A bi-direction performance evaluation model for water pollution treatment engineering under the intuitionistic multiplicative linguistic environment

As the increasingly serious water pollution problem affects the sustainable development of the ecological environment, the research of water pollution treatment engineering cannot be delayed. Among them, the performance evaluation of water pollution treatment engineering is a major focus. After reading the existing studies, it is found that most of the existing performance evaluation indicators of water pollution treatment engineering have qualitative indicators and there is an unbalanced preference representation. Intuitionistic multiplicative linguistic sets can be a good representation of the qualitative preference and non-preference of decision-makers in the context of decision-making containing unbalanced phenomena. Therefore, to better solve the problem of water pollution treatment engineering, this paper introduces intuitionistic multiplicative linguistic sets to the problem of water pollution treatment engineering and proposes an effective theory for it. First, considering the multiplicative nature of the intuitionistic multiplicative linguistic set, a new score function and accuracy function are defined, and on this basis, the priority rules of intuitionistic multiplicative linguistic set are given to prepare for the subsequent water pollution treatment engineering performance ranking. And the distance measure of intuitionistic multiplicative linguistic set is introduced and a CRITIC attribute weight determination model under intuitionistic multiplicative linguistic set is obtained on this basis. Secondly, the Choquet integral operator is applied to better represent the correlation between elements. However, the nature of membership degree and non-membership degree shows that it is not reasonable to aggregate the information of intuitionistic multiplicative linguistic sets with a single increasing and decreasing transformation. Therefore, in this paper, we propose the IMLS bi-direction exponent Choquet integral operator, which is inspired by the bi-direction Choquet integral. Lastly, we improve the original preference function of the classical PROMETHEE II method to obtain the bi-directional PROMETHEE II method in intuitionistic multiplicative linguistic information. Finally, a numerical case is also provided to illustrate the scientific and rational application of the bi-directional PROMETHEE II method in intuitionistic multiplicative linguistic information for the performance evaluation of water pollution treatment engineering.

Picture Fuzzy Einstein Hybrid-Weighted Aggregation Operator and Its Application to Multicriteria Group Decision Making

As an extension of intuitionistic fuzzy sets (IFSs), picture fuzzy sets (PFSs) can better model and represent the hesitancy and uncertainty of decision makers’ preference information. In this study, we propose a multicriteria group decision making (MCGMD) method based on picture fuzzy sets. We first define some basic picture Einstein operations with closed properties among PFSs on the basis of the Einstein t-norms and t-conorms. Then, utilizing the hybrid-weighted operator and the developed picture Einstein operations laws, we put forward a picture fuzzy Einstein hybrid-weighted aggregation operator for aggregating PFSs and discuss its several important properties. Furthermore, we present a new MCGMD method based on the proposed picture fuzzy Einstein hybrid-weighted aggregation operator. Finally, an example is conducted to validate the effectiveness of the proposed MCGMD method.