Research On Picture Fuzzy Information Theory And Its Application In Multi-Criteria Decision-making

With the advent of information age and the rapid development of computer technology,large and complex fuzzy data appear in various fields of people's lives.Compared with fuzzy sets and intuitionistic fuzzy sets,picture fuzzy sets,which contain four dimensions of opinion or attitude such as positive,neutral,negative and rejection,are more advantages in describing and analyzing fuzzy information,which makes them widely used for information evaluation of multi-criteria decision-making problems in disciplines such as environmental science,information science,management engineering,applied science and medical systems.However,the existing picture fuzzy information theory and its applications have some limitations.This paper takes picture fuzzy information system as the research object,and integrates the knowledge of picture fuzzy set theory,rough set theory,decision theory,tensor theory,measurement theory and statistical analysis methods,and proposes some improved methods and techniques.The main research content and innovation points are summarized as follows:1.As an important tool for quantitative analysis of picture fuzzy informa-tion,measure theory of picture fuzzy information is an important foundation for decision theory and methods.By classifying the existing score functions,distance measures and similarity measures,it is found that the existing score functions,distance measures and similarity measures are not developed on the basis of bounded lattice,and the lack of theoretical foundation makes them invalid or unreasonable to some extent.To overcome these shortcomings,this paper integrates the ideas of lattice theory to build a measure system of picture fuzzy information on the complete lattice from a probabilistic perspec-tive.Firstly,by introducing the parametersand the balance coefficient2),the probability-based^-1-accuracy score function is constructed on the complete lattice,and the comparative analysis with the existing score functions shows that the proposed score function is more flexible,effective and reasonable for ranking picture fuzzy numbers.Secondly,the concepts of generalized picture fuzzy distance measure and generalized picture fuzzy similarity are introduced for the first time based on the equivalence relation,and the proposed score func-tion is used to construct two types of generalized picture fuzzy distance measure and generalized picture fuzzy similarity on the complete lattice.Finally,some examples are presented to demonstrate validity and feasibility of the proposed distance and similarity.2.Information aggregation operators are an important part of information fusion theory and one of the classical methods to solve multi-criteria decision-making problems.However,most of the existing picture fuzzy information aggregation operators cannot effectively solve multi-criteria decision-making problems in covered picture fuzzy information systems.To solve multi-criteria decision-making problems in covered picture fuzzy information system,this pa-per proposes covering-based picture fuzzy rough set(CPFRS)model by fusing picture fuzzy sets with rough sets,and combine the ideas of TOPSIS(Tech-nique for order preference by similarity to ideal solution)method to construct CPFRS-TOPSIS method.The validity and feasibility of the proposed method are verified through case studies,sensitivity analysis and comparative analysis.3.Real and fuzzy information systems usually present uncertainty and in-consistency due to the complexity of the decision environment,measure meth-ods,decision levels and personal preferences.To address the multi-criteria decision-making problem in real or fuzzy information systems,this paper inte-grates the proposed picture fuzzycovering rough set(PFCRS)model with the VIKOR(Vlsekriterijumska optimizacija i kompromisno resenje)method to construct the PFCRS-VIKOR method.Firstly,an improved PFCRS model is proposed to address the limitations of the three PFCRS models and to explore some of its basic properties.Secondly,by combining the improved PFCRS model,the probability-based^-1-accuracy picture fuzzy distance measure and the VIKOR method,the PFCRS-VIKOR method is given for dealing with uncertain multi-criteria decision-making problems.Finally,the validity,flex-ibility and superiority of the proposed method are verified by case analysis,sensitivity analysis and comparison analysis.4.In a practical multi-criteria decision-making environment,the crite-ria evaluation value of an object is often influenced by multiple factors,but the existing decision methods based on matrix theory or(intuitionistic)fuzzy tensor are difficult to handle multi-criteria decision-making problems with high-dimensional picture fuzzy information characteristics.In this paper,the concept of picture fuzzy tensor is proposed for the first time by fusing picture fuzzy set with tensor,and the information aggregation method based on picture fuzzy tensor is constructed by exploring its basic operations and fundamental proper-ties.The feasibility and effectiveness of the proposed method are verified by two case analyses.