Fuzzy Soft Set And Its Application In Decision-making Under Uncertainty

Artificial intelligence is rapidly coming of age,and we live in the midst of a data deluge.In this period,we may face many kinds of challenges.That is,to deal with this big data,methods in classical mathematics are not always successful because of various types of uncertainties presented in these problems,such as fuzziness,randomness,roughness,incompleteness and so on.In the process of dealing with uncertainty,many mathematical theories are proposed,such as fuzzy mathematics,probability theory,rough set theory,etc.In 1999,Molodtsov initiated the concept of soft sets which can be considered as a new mathematical tool for dealing with uncertainties.A soft set is a parameterized family of the subsets of a universal set,which describes the uncertainty from both the universal set and the parameter set.The rationale behind soft sets is founded on the idea of parameterization,which suggests that complicated objects should be perceived from various points of view.Each aspect provides an approximate description of the whole entity with high complexity.The appropriate aggregation of these approximate descriptions will lead to an accurate description of the object.The absence of any restrictions on the approximate description in soft set theory makes this theory very convenient and easily applicable in practice.In other words,the soft set theory emphasizes the parameterized description of objects and aims to provide a unified theoretical framework for various uncertainties analysis.The related research shows that there are both differences and connections between the theory of soft sets and the theory of fuzzy sets,the theory of rough sets,and they have substantial complementary as well.In this dissertation,we study the similarity measure of the fuzzy soft set,the induced soft set,and the transformation relationship between soft sets and fuzzy sets.Moreover,a learning model—the soft sets and fuzzy sets based neural networks(SF-ANN)and its applications are proposed.Further,the uncertainty decision-making problem based on fuzzy soft sets is discussed.Main research results are as follows:1.The study of uncertainty measures of fuzzy soft sets.Based on the related research results of fuzzy sets,the axiomatic definition of the similarity measure for fuzzy soft sets is proposed.A general construction method for similarity measures of fuzzy soft sets is presented by using fuzzy equivalence and fuzzy aggregation operators.The basic properties of these similarity measures are analyzed.The relationships among these similarity measures and the other proposals in the literature are surveyed.The axiomatic definition of entropy for fuzzysoft sets is proposed and the related construction approaches are presented.2.The study of induced soft sets.The soft set is based on a set-valued mapping and presents the description of the parameter by using a set of objects.Based on the inverse mapping of set-valued mapping in the soft set we proposed the notion of induced soft set.Some operations,such as extended union,extended intersection,restricted union,restricted intersection,of induced soft sets are presented.The basic properties of these operations and the lattice structures of induced soft sets are analyzed.The notion of soft equality between induced soft sets is proposed and the soft quotient algebra of induced soft sets is established.Based on a soft set and its induced soft set,we construct a reflexive and transitive relation on the parameter set and a topology space on parameter set is established with its topological properties being discussed.3.The study of the transformation between fuzzy sets and soft sets.A comparative study of fuzzy sets and soft sets is presented by considering the transformation between fuzzy sets and soft sets.It is proved that every fuzzy set on a universe can be considered as a soft set,and any soft set can be transformed to a fuzzy set as well.By using the coding approach of the binary-coded genetic algorithm(BCGA)and the induced ordered weighted averaging operators(IOWA)two transformation methods between fuzzy sets and soft sets are proposed.The related applications to decision making and similarity description are surveyed.4.The study of the fuzzy sets and soft sets based neural networks.Inspired by the idea of transformation between fuzzy sets and soft sets,we construct a neuron-like structure and extend it to multilayer neural network structures.Furthermore,a neural network learning model—SF-ANN is constructed.An illustrative example shows that the model can be used to establish the membership function of fuzzy sets.5.The study of the fuzzy soft set and ideal solution based decision-making models.Some defects in the choice value based approach and the comparison score based approach to soft set based decision-making methods are analyzed.It is shown that the rank reversal phenomenon may occur in the comparison score algorithm,and an example is provided to illustrate the rank reversal phenomenon.By analyzing the attributes and decision object,a decision-making model based on fuzzy soft set and ideal solution is proposed,and the decision-making method is presented.