Research On Some Special Fuzzy Sets Theories And Their Applications

By introducing membership function μA(x), non-membership function vA(x) and hesitancy function πA(x), the intuitionistic fuzzy sets (IFS) theory is established by Atanassov, which generalizes Zadeh’s fuzzy sets (FS). According to the IFS definition, μA(x) denotes the proportion of the support party, vA(x) denotes the proportion of the opposition party, and πA(x) denotes the proportion of the absent party. Though many scholars studied IFS and applied it widely to decision making analysis, pattern recognition, medical diagnosis, automatic control, fuzzy reasoning, etc. However, traditional researches based on IFS do not consider the detachment of the absent party, which means that the absent party is not specifically analyzed in conventional model of IFS. Therefore, these methods are suitable for static model and unsuitable for dynamic model. Then Xu and Yager presented a dynamic decision making model in2008, which was also studied by Wei, Su, et al. But they also do not study the detachment of the absent party. Moreover, the absent party may change over time in practice, while conventional IFS method cannot deal with this kind of dynamic model. Thus, the research on the variation of absent party will play an important role in dynamic decision making, dynamic pattern recognition, dynamic automatic control and dynamic fuzzy reasoning. Taking into account this, we present a series of fuzzy sets models with parameters by analyzing the hesitancy function.First, we assume that μA(x) is the firm support party of event A, vA(x) is the firm opposition party of event A, πA(x) is the maximum absent party of event A, πA*(x)=(1-λA0(x)) πA(x) is the firm absent party of event A, and πA(x)-πA*(x)=λA0(x)πA(x) denotes the convertible absent part, where λA0(x) is the proportion of the convertible absent individuals in all the absent individuals. Obviously we have μA(x)+vA(x)+πA(x)=1and0≤λA0(x)≤1. We divide the convertible absent part into two parts:λA0(x)λA1(x)πA(x) being the absent party which can be converted into the support party, and λA0(x)(1-λA1(x))πA(x) being the absent party which can be converted into the opposition party, where λA1(x) is the proportion of the convertible absent individuals being converted to the support party, and1-XλA1(x) is the proportion of the convertible absent individuals being converted to the opposition party. And we also have0≤λA1(x)≤l. According to the detachment of the absent party, we introduce a series of definitions:such as fuzzy sets with parameters (FSP), intuitionistic fuzzy sets with parameters (IFSP), interval-valued intuitionistic fuzzy sets with parameters (IVIFSP), etc. And then we also propose the construction methods of them. Secondly, we present a novel generalized interval-valued intuitionistic fuzzy sets model (GIVIFS), which is proved to be the generalization of fuzzy sets (FS), interval-valued fuzzy sets (IVFS), intuitionistic fuzzy sets (IFS), interval-valued intuitionistic fuzzy sets (IVIFS), vague sets (VS), interval-valued vague sets (IVVS). Moreover, the GIVIFS model is proved to be a closed soft algebra system for the intersection operator, the join operator and the complement operator as fuzzy sets model.Thirdly, we present some ratio distance measures and some novel score functions on IFS. And we study the properties of these measures and functions. Accordingly, we introduce the corresponding distance measures and score functions on FSP, IFSP, and IVIFSP. And then we apply them to pattern recognition and medical diagnosis, the simulation results show that the corresponding method with parameters is more effective than the conventional fuzzy sets method.Fourthly, we present some novel ranking function on IFS and IVIFS, and generalized them to FSP, IFSP, IVIFSP and GIVIFSP. And we prove that any ranking function of IFS is the special case of the ranking function of IFSDP. Finally, taking advantage of the ranking functions above, FSP model, IFSP model and IVIFSP model are applied to multiple attribute decision making. The experimental results show that we can adjust the parameters to appropriate values to obtain all feasible results. Therefore, the FS method with parameters can be applied to the dynamic decision making field, and we can predict all the possible decision making results in the future according to the variation of membership function, non-membership function, and hesitancy function.All in all, the new method proposed in this paper can expand the scope of FS and IVFS applied to pattern recognition and decision making. The simulation results show that the methods introduced in this paper are more comprehensive and flexible than the conventional FS method and IFS method.