Research Of Fuzzy Multiple Attributes Decision Making Method Based On S-shaped Utility Function

Multiple attributes decision making(MADM)is the process to find the optimal alternative among a set of feasible and limited alternatives by using the existing decision making information,which is based on some incommensurable and conflicting attributes.The theory and method of MADM is established in the operation research and economics disciplines,which have extensive significance and application value in the field of economy,management,engineering and military.As the decision making problems are more and more complicated,the decision makers sometimes are difficult to describe the evaluation of the attributes by using the exactly number since the complexity of the objective things and the capacity limitations of the decision makers.In this case,it is more reasonable to describe the attributes with the fuzzy numbers or linguistic values.Therefore,the study of the fuzzy multiple attributes decision making(FMADM)problem has the great significance.At present,the existing studies of the FMADM are usually based on the classical expected utility theory.However,the expected utility theory supposed that the decision makers are completely rational.In fact,the decision makers are always limited rational while facing the risk,and decision makers have different subjective risk preferences under different risk environment.In the field of behavioral finance,the S-shaped utility function can depict the subjective risk attitudes of decision makers,such as the prospect value function,which is one of the typical S-shaped utility function.The S-shaped utility function is the convex function in the loss area which is under the reference point,and it is the concave function in the gain area which is above the reference point.The S-shaped utility function can reflect the risk attitude of the decision makers and describe the actual behavior of the decision makers.Therefore,the study of the FMADM problem with the S-shaped utility function has important theoretical significance and practical value.The main research contents and contributions are shown as follows.(1)The fuzzy multiple attributes decision making problem is studied where the attributes values are the intuitionistic fuzzy numbers.A new intuitionistic fuzzy score function is proposed,which takes account into the hesitation degree of the intuitionistic fuzzy numbers.First,in order to transform the intuitionistic fuzzy numbers into the corresponding real numbers,after analyzing the existing scoring function,a precise score function(Pscore function)is developed based on the information of the hesitate degree,and corresponding properties of the P-score function and the analysis of the relationship between the existing score function and P-score function are studied.Second,regarding that the information of the attributes weights is incompletely known or completely unknown,two new maximizing integrated utility optimization weighting models are developed to determine the attributes weights by considering the subjective and objective factors.Third,in order to depict the subjective risk attitudes of the decision makers,the hyperbolic absolute risk aversion(HARA)function is introduced into the framework of the S-shaped utility function,and the S-shaped hyperbolic absolute risk aversion(S-HARA)utility function is correspondingly defined.Based on the intuitionistic fuzzy P-score function and the S-shaped utility function,the intuitionistic fuzzy utility calculation formulas are respectively developed under the prospect value function and the S-HARA utility function.Finally,based on the P-score function,the integrated weighting model and the intuitionistic fuzzy utility,the intuitionistic fuzzy numbers multiple attributes decision making methods under the prospect value function and the S-HARA utility function are respectively proposed.In addition,the new methods are applied in the practical multiple attributes decision making problems and corresponding parameters sensitivity analysis and comparative analysis are respectively presented.(2)The fuzzy multiple attributes decision making problem is studied where the attributes values are the linguistic values.A new two tuple linguistic S-shaped utility function is defined and the variable form of the utility function is changed from the real number to the two tuple linguistic,which expands the application range of utility theory.First,in order to determine the relation degree between the alternative and the ideal alternative,the two tuple linguistic utility and two tuple linguistic utility relational degree are defined based on the basic principle of the grey system theory,and corresponding calculation formula of the two tuple linguistic S-shaped utility function is proposed.Second,regarding that the information of the attributes weights is incompletely known or completely unknown,two new maximizing two tuple linguistic utility optimization weighting models are developed to determine the attributes weights by considering the subjective and objective factors.Finally,according to the two tuple linguistic utility relational degree and weighting models,the linguistic value multiple attributes decision making methods under the prospect value function and the S-HARA utility function are respectively proposed.(3)The fuzzy multiple attributes decision making problem is studied where the attributes values are the intuitionistic linguistic numbers.A new intuitionistic linguistic distance is developed by considering the relative membership information of the intuitionistic linguistic numbers.First,regarding the problem of the existing intuitionistic linguistic Hamming distance,a new relative distance(R-distance)is defined by considering the relative membership degree of the intuitionistic linguistic numbers.Additionally,the properties of the R-distance are studied and the establishment conditions of R-distance are verified.Second,based on the intuitionistic linguistic R-distance and the utility function,the intuitionistic linguistic utility function is developed and corresponding calculation formulas are respectively proposed under the prospect value function and the S-HARA utility function.Third,a minimizing R-distance optimization weighting model is established by considering the subjective and objective factors.Finally,based on the intuitionistic linguistic R-distance,the intuitionistic linguistic utility and the minimizing R-distance optimization weighting models,the intuitionistic linguistic numbers multiple attributes decision making methods under the prospect value function and the S-HARA utility function are respectively developed.(4)For the hybrid fuzzy multiple attributes decision making problem where the attributes values are the intuitionistic fuzzy numbers,linguistic values and intuitionistic linguistic numbers,a new fuzzy multiple attributes decision making method based on the utility projection are developed.It can find the optimal alternative based on the original information,which avoids the transformation of the hybrid decision making information.First,the gain and loss values of the intuitionistic fuzzy numbers,linguistic values and intuitionistic linguistic numbers are calculated based on the P-score function and R-distance.Second,regarding that the information of the attributes weights is incompletely known or completely unknown,two new optimization weighting models are developed to determine the attributes weights by considering the subjective and objective factors.Furthermore,based on the principle of the projection,the utility projection values of each alternative are calculated according to the gain and loss values and the weighting model.Finally,an example analysis is provided to illustrate the application of the developed approach,which shows the effectiveness and feasibility of the proposed method.