Research On Multi-criteria Group Decision Making Theories And Methods With Pythagorean Fuzzy Sets

With the rapid development of the economy,more and more practical decision-making issues such as venture capital selection,supplier selection and the choice of enterprise project research and development rely on the decision of decision maker(DM).Multi-criteria decision making is an important part of modern decision-making science.It has a wide range of practical backgrounds in engineering design,economics,management and military.Nowadays,the decision-making problems faced by enterprises and organizations are becoming more and more complicated.It is difficult for a sole DM to grasp all relevant aspects of the problem.In actual decision making,multiple DMs are often involved.In complex management decision making problems,decision making conditions,decision making data and information,decision making processes,etc.involve a large number of uncertain factors,and the fuzziness in management decision making is difficult or even impossible to solve by classical mathematics and physics.Pythagorean fuzzy(PF)sets(PFSs)can describe the uncertainty of DMs on things delicately.Based on the existing research,this paper studies multi-criteria group decision problems with PF information and interval-valued Pythagorean fuzzy(IVPF)information.The key issues such as the derivation of DMs' weights,the determination of the weight of the criteria and the presentation of decision making methods have been studied in detail.Then,a series of research results have been formed.Specifically,the contents are outlined as following.(1)In order to overcome the shortcomings in the existing ranking methods of Pythagorean fuzzy numbers(PFNs),knowledge measurement and credibility are defined to characterize the quantity and quality of PF information.Then,two PFNs are compared by three indices of knowledge measurement,credibility and relative closeness degree.A ranking method of PFNs is proposed and applied to solve multi-criteria decision making problems.(2)This paper proposes an admissible order for PFNs and applies to multi-criteria group decision making(MCGDM).First,the concepts of relative distance and information reliability of PFN are proposed.Then,a new order relation is presented to compare PFNs.Moreover,the new order relation of PFNs is demonstrated to be an admissible order.Knowledge measure of PFN is defined to describe the amount of information.The desirable properties of knowledge measure of PFN are studied concretely.For MCGDM with PFNs,the comprehensive distance between individual Pythagorean fuzzy matrices and a mean one are defined.Then,the DMs' weights are obtained by the comprehensive distances.Thus,a collective Pythagorean fuzzy matrix is derived by using the Pythagorean fuzzy weighted average operator.To determine criteria weights,a multi-objective programming model is constructed by maximizing the overall knowledge measure of each alternative.This model is further transformed into a single-objective mathematical program to resolve.According to the admissible order of PFNs proposed in this paper,the ranking order of alternatives is generated by the comprehensive values of alternatives.Therefore,a new method is proposed to handle MCGDM with PFNs.Finally,an example of venture capital investment selection is provided to illustrate the effectiveness of the proposed method.(3)The effective integration of information is an aspect of the research in multi-criteria decision making problem.This paper proposes definitions of generalized PF Heronian average operator and generalized PF weighted Heronian average(GPFWHA)operator to aggregate PF information and then studies their properties.Thereby,a multi-criteria group decision making method with PF information is developed.First,DMs' weights are determined by the closeness degree of the individual PF decision matrix.Second,the PF decision matrix is obtained by using the GPFWHA operator.Then,the objective weights of the criteria are determined by the deviation of different alternatives under the criterion.Therefore,the comprehensive weights of the criteria are calculated by combining the subjective weights and the objective ones.According to the admissible relation of PFNs,the ranking order of alternatives is generated by the comprehensive values of alternatives after utilizing the GPFWHA operator again.Finally,an example of subject evaluation in university is provided to illustrate the rationality and the effectiveness of the proposed MCGDM method.(4)This paper develops a PF mathematical programming method to solve MCGDM problems under PF environments.Considering the fuzziness and hesitancy in pairwise comparisons of alternatives,we firstly introduce PF sets to depict the fuzzy truth degrees of alternative comparisons.According to the information entropy,individual subjective criteria weight vectors of DMs are calculated and integrated into a collective one by a cross entropy optimization model.Then DMs' weights are objectively derived from the collective subjective criteria weight vector.PF group consistency and inconsistency indices are defined based on PF positive ideal solution(PFPIS)and PF negative ideal solution(PFNIS),respectively.To determine comprehensive criteria weights,a bi-objective PF mathematical programming model is constructed through minimizing two inconsistency indices based on PFPIS and PFNIS simultaneously.A linear programming method is technically developed to solve this model.Using the cross entropy again,collective relative closeness degrees of alternatives are explicitly derived to rank the alternatives.Finally,an example of green supplier selection is analyzed to verify the effectiveness of the proposed method.(5)This paper investigates an interval-valued Pythagorean fuzzy(IVPF)MCGDM problem with IVPF truth degree and incomplete criteria weight information deeply.First,considering that DMs have different weights under different criteria,DMs' weights under the criteria are obtained based on the relative closeness degrees according to IVPF positive ideal solution(PIS)and IVPF negative ideal solution(NIS)of each alternative under each criterion.Then,the IVPF group consistency index and IVPF group inconsistency index are defined respectively based on IVPF PIS and IVPF NIS.By minimizing the group inconsistency indices based on IVPF PIS and IVPF NIS simultaneously,a bi-objective IVPF mathematical programming model is established to derive the criteria weights.Subsequently,the relative closeness degrees of alternatives for each DM are obtained and applied to derive the individual ranking order of alternatives.To generate the collective ranking order of alternatives,a multi-objective assignment model is established and converted into a single objective programming model to resolve.Thus,a new IVPF mathematical programming method is proposed to solve MCGDM.Finally,a software investment example is provided to demonstrate the applicability and validity of the proposed method.