Some New Results Of Fuzzy Soft Sets And Their Applications In Disease Diagnosis

Fuzzy sets play a very significant role in multi attribute decision-making problems.An intuitionistic fuzzy set is an extension of fuzzy set by adding non-membership degree into the analysis such that sum of membership degree and non-membership degree is not greater than one.As an extension of intuitionistic fuzzy sets,picture fuzzy sets consist of three membership degree of an element named as the positive membership degree,the neutral membership degree,and the negative membership degree,respectively such that their sum is not greater than unity.In the literature,researchers are working on the picture fuzzy sets and successfully developed several algorithms to solve different problems such as clustering,fuzzy inference,and decision-making.However,fuzzy set theory,intuitionistic fuzzy set theory and picture fuzzy set theory can only solve a part of uncertain problems in real-world.To this end,Smarandache initiated neutrosophic set theory by fusing the non-standard analysis and a tri-component set.A NS consists of three membership functions(truth membership function,indeterminacy membership function and falsity membership function),where every function value is a real standard or non-standard subset of the nonstandard unit interval]0-,1+[.NSs has achieved good applications in image processing and cluster analysis.Molodtsov initiated the theory of soft sets as a mathematical tool to dealing with un-certainties and shown several applications of this theory in solving many practical problems in engineering,physics,computer sciences,economics,social sciences,medical sciences and many other diverse fields.He established the fundamental results of this new theory and successfully applied the soft set theory into several directions,such as smoothness of func-tions,game theory,operations research,Riemann integration,Perron integration,theory of probability,theory of measurement and so on.Soft sets have achieved a lot of development in both theory and application.Maji et al.presented and studied an application of soft sets in decision making problems that is based on the reduction of parameters to keep the optimal choice objects.Further,Maji et al.first introduced the concepts of soft subset,soft superset,soft equality,null soft set,and absolute soft set.They also gave some operations on soft set and verified De Morgan's laws.Afterwards,Ali et al.further studied some important properties associated with the new operations and investigated some algebraic structures of soft sets.Sezgin and Atagun extended the theoretical aspect of operations on soft sets.Kernels and closures of soft set relations,and soft set relation mappings are pre-sented by Yang and Guo.Chen et al.presented a new definition of soft set parametrization reduction.Kong et al.introduced the definition of normal parameter reduction into soft sets and then presented a heuristic algorithm to compute normal parameter reduction of soft sets.Babitha et al.introduced a sub soft set of the cartesian product of the soft set and many related concepts.Xiau et al.proposed the notion of exclusive disjunctive soft sets and studied some of its operations.Gong et al.defined the concept of bijective soft set and some of its operations and studied its several properties,Feng et al.established an interesting connection between two mathematical approaches to vagueness:rough sets and soft sets.They also defined new types of soft sets such as full soft sets,intersection complete soft sets and partition soft sets.Maji et al.extended soft set theory by joining it with ex-isting fuzzy set approach and developed the idea of fuzzy soft set.Roy and Maji presented an application of fuzzy soft set in decision making.Yang et al.defined the interval-valued fuzzy soft set which is based on a combination of the interval-valued fuzzy set and soft set.Majumdar and Samanta generalized the concept of fuzzy soft sets;that is,a degree of which is attached with the parameterization of fuzzy sets while defining a fuzzy soft set.Several extension models of soft,sets are rapidly developed such as generalized interval-valued fuzzy soft sets,interval-valued intuitionistic fuzzy soft sets,interval-valued hesitant fuzzy soft sets,and dual hesitant fuzzy soft set.Furthermore,Maji et al.defined the intuitionistic fuzzy soft sets,followed by studies on picture fuzzy soft set,and generalized intuitionistic fuzzy soft sets.Peng and Garg proposed three algorithms to solve interval-valued fuzzy soft decision making problem by weighted distance based approximation,combinative distance-based as-sessment and similarity measure.Peng and Yang presented a novel interval valued fuzzy soft set approaches.They initiated a new axiomatic definition of interval-valued fuzzy distance measure,which is expressed by interval valued fuzzy number.Then,they determined t.he objective weights of various parametrers via normal distribution.Peng and Liu presented some novel single-valued neutrosophic soft set methods.They initiated a new axiomatic def-inition of single-valued neutrosophic similarity measure,which is expressed by single-valued neutrosophic number.Then,they determined the objective weights of various parameters via grey system theory.Peng and Dai presented some novel hesitant fuzzy soft set methods.They determined the objective weights of various parameters via Shannon entropy theory.Then,they developed the combined weights,which can show both the subjective information and the objective information.Also they proposed three algorithms to solve hesitant fuzzy soft decision making problem by multi-attributive border approximation area comparison,weighted aggregated sum product assessment and complex proportional assessment.Zhu and Zhan proposed the concept of fuzzy parameterized fuzzy soft sets,along with decision making.Zhao et al.presented a novel decision making approach based on intuitionistic fuzzy soft sets.Das introduced the notion of weighted fuzzy soft multiset and decision-making.Garg and Arora presented an approach to solve the decision making problems by formulating a nonlinear programming model with interval-valued intuitionistic fuzzy softset information.Deli introduced the notion of interval valued neutrosophic soft sets and applications to the decision making process.Garg and Arora presented the t-norm operations based Maclaurin Symmetric mean aggregation operators for solving the decision making problems under the dual hesitant fuzzy soft set environment.Among the most illness leading the death in the entire world is cancer.Lung cancer has become one of the majority occurring diseases in the world,it has exponentiation trend in its incidence in future.In order to avoid such a life-threatening difficulty,one of the able solutions is to make people aware of their respective lung cancer risks previously and ought to take preventive measures suitably.In light of this,people try to develop medical expert systems or disease diagnosis expert systems of lung cancer with the help of mathematics.As uncertainty always appears in the course of diagnosis,fuzzy rule-based expert systems of lung cancer are developed.A fuzzy rule-based expert system includes a set of fuzzy rules and membership functions.So far,the fuzzy inference system has become a vigorous area of research in many sciences.This thesis constructs some new fuzzy soft sets(i.e.an interval-valued picture fuzzy soft set,possibility m-polar fuzzy soft set,time-neutrosophic soft set,and n-valued refined neutrosophic soft set),and studies related properties.We also use examples about multi-attribute decision making to illustrate effectiveness and feasibility of the proposed models.We first introduce briefly fuzzy set theory and soft set theory to be used.Then a new concept of the interval-valued picture fuzzy soft set theory and some operations(e.g.subset,equal,complement,inf product,sup product,union,and intersection)of interval-valued picture fuzzy soft set are defined.After that an algorithm using an interval-valued picture fuzzy soft set to solve the decision-making problem is constructed.The concept of possibility m-polar fuzzy soft set(because it is useful in decision-making and other similar problems)is also introduced.Two algorithms by using inf product or sup product operations of possibility m-polar fuzzy soft sets for fuzzy decision-making problem are presented.Then,some new concepts describing“subset”and“equal”of time-neutrosophic soft set and n-valued re-fined neutrosophic soft set are proposed.We also studies some of their properties.Finally,we establish a fuzzy soft expert system which is verified useful in real practical applications by an example.This paper contains the following five chapters:Chapter 1:In this chapter,we introduce some basic concepts,such as fuzzy set,m-polar fuzzy set,neutrosophic set,picture fuzzy set,interval-valued picture fuzzy set,soft set,fuzzy soft set,picture fuzzy soft set,time-neutrosophic soft set,and n-valued refined neutrosophic soft set.Chapter 2:In this chapter,we define the notion of the interval-valued picture fuzzy soft set,particularly,some new operations(such as subset,equal,complement,inf product,sup product,union,and intersection)of interval-valued picture fuzzy soft sets;we also investigate properties of these operations.Further,we construct an algorithm using an interval-valued picture fuzzy soft set to solve the decision-making problems and illustrate,through a numerical example,its effectiveness applicability to handling uncertainties during the decision-making process.Chapter 3:In this chapter,we first present the notion of possibility m-polar fuzzy soft set.For convenience of practical applications,several operations(such as subset,equal,complement,union,intersection,inf product,and sup product)over the possibility m-polar fuzzy soft sets are introduced.We give out two algorithms by using inf product and sup product operations of possibility,m-polar fuzzy soft sets,respectively,for fuzzy decision-making problems.Applicability to handling uncertainties during the decision-making process is also demonstrated through a numerical example.Chapter 4:We studies time-neutrosophic soft sets and n-valued refined neutrosophic soft sets.The main content is about dealing with 'subset' of a time-neutrosophic soft set(resp.,n-valued refined neutrosophic soft set)and 'equal' of two time-neutrosophic soft sets(resp.,n-valued refined neutrosophic soft sets).Some examples are given,and related properties are also discuss.Chapter 5:In this chapter,we develop a new fuzzy soft expert system.A prediction process is composed of four main steps:(1)Transform real-valued inputs into fuzzy numbers.(2)Transform fuzzy numbers of data into fuzzy soft sets.(3)Reduce,using normal parameter reduction method,the obtained family of fuzzy soft sets into a new family of fuzzy soft sets.(4)Use the proposed algorithm to get the output data.The research objective is forty five patients(thirty males,fifteen females,all are cigarette smokers)who endure treatment in the Respiratory Department of Nanjing Chest Hospital,China.The number of training data among taken data is 55 records,and the remaining 45 records are used for the testing process in our system.The attribute contains six parameters:weight loss,shortness of breath,chest pain,persistence a cough,blood in sputum,and age of patients.The quantized accuracies of the proposed system is found to be 100%,which shows the fuzzy soft expert system developed is useful to the expert doctor to decide if a patient has lung cancer or not.Finally,we make comparisons between our proposed system and the fuzzy inference system.