Linear And Sysmetrical Orthopair Fuzzy Sets And Their Application In Multiple Attribute Group Decision Making

Fuzzy sets form the bedrock of uncertainty analysis and multiple attribute group decision making(MAGDM).The q-rung orthopair fuzzy sets(q-ROFSs)are extended models of fuzzy sets.There exist membership and non-membership functions in q-ROFSs,and they can deal with fuzzy and uncertain information more flexibly.q-ROFSs include intuitionistic fuzzy sets,Pythagorean fuzzy sets and Fermatean fuzzy sets.Operations are the research priority of q-ROFSs.The intersection,union,complement,distance,addition and scalar multiplication are the fundamental operations of them.T-norms and t-conorms are generalizations of intersection,union and addition.Information aggregation and MAGDM can be achieved based on addition,scalar multiplication and distance.There are complete related operations and MAGDM methods in q-ROFSs.q-ROFSs are special orthopair fuzzy sets(OFSs),but the latter are only preliminarily formulated at present.q-ROFSs carry power function conditions,and they have linear and symmetrical simplification space as OFSs.Thus,this thesis reconstructs the precise definition of OFSs and propose specific models,i.e.,linear orthopair fuzzy sets(LOFSs),symmetrical linear orthopair fuzzy sets(SLOFSs),as well as dual linear symmetrical orthopair fuzzy sets(DLSOFSs),so that we can achieve both fuzziness extensions and q-ROFS simplification.Related operations and MAGDM methods are constructed.The concrete research contents include the following three aspects.(i)LOFSs are discussed.Their operations such as intersection,union and distance are given.We emphatically discuss order-reversing involutionary negations.First,for OFSs,their basic definitions and usual operations are formally determined to extend q-ROFSs,and their order-reversing involutionary negations acquire basic properties in terms of the bijective mapping,equivalent rule,dual monotonicity,and rectangular description.Then,for LOFSs,their natural definitions and usual operations are realized by linear constraints to extend the intuitionistic fuzzy sets.They exhibit four common types,i.e.,LOFSs(1),(2),(3)and(4),and three symmetrical types,i.e.,SLOFSs(1),(2)and(3).LOFSs(4)and degenerate SLOFSs(2)are the main models.Order-reversing involutionary negations of LOFSs achieve further properties in terms of the single direction,simplified equivalent rule,and continuous bijection.Finally,the order-reversing involutionary negations are realized in LOFSs(1),(4)and all SLOFSs,but do not exist in LOFSs(2)and(3).(ii)For SLOFSs,the axiomatic definitions,general properties and construction methods of t-norms and t-conorms are got.A special type of SLOF t-norms and t-conorms motivates the addition and scalar multiplication,and related operation properties are obtained.The addition and scalar multiplication constitute relevant TOPSIS(Technique for Order Preference by Similarity to Ideal Solution),and the corresponding method of MAGDM is established to achieve good reliability.(iii)DLSOFSs are discussed.Through a given restricted condition,we define DLSOFSs.Distances are also defined.The axiomatic definitions,general properties,and construction methods of t-norms and t-conorms on DLSOFSs are provided.Specific DLSOF t-norms and t-conorms are gained,which can be used to define addition and scalar multiplication.The related TOPSIS method is designed.In summary,OFSs are explicitly defined to generalize q-ROFSs.Order-reversing involutionary negations of LOFSs are defined and researched.T-norm,t-conorms of SLOFSs and DLSOFSs are constructed,and applied in MAGDM.Results of this work benefits theoretical research and practical applications of fuzzy models.