Interval Fuzzy Preference Relations Research Articles

The consistency discretization adjustment method of interval fuzzy preference relationship based on individual agreement and group consensus

Constrained by the limited knowledge and fuzzy cogitation, it is often difficult to guarantee the consistency of decision makers’ preference judgments. To effectively avoid the distortion of information about the original preferences, a discrete scale consistency adjustment model based on integer variables under the individual interval fuzzy preference relationship is established. The research shows that the proposed method can not only validly enhance the differentiation between alternatives, but also drop the occurrence of zero weights to a larger extent. Furthermore, aiming at the potential multiplicity of optimal adjustment strategies for interval fuzzy preference relations, an efficient algorithm for screening multiple optimal solutions under single-person decision making is proposed by setting multiple screening dimensions. The algorithm makes up for the deficiency of ignoring potential optimal solutions in traditional methods, and greatly raises the decision flexibility of the ranking optimization process. The validity and applicability of the algorithm are verified through filtering and analyzing multiple optimal solutions under different scenarios. Considering the differences in the decision making tendencies of exclusive and altruistic groups in the interval fuzzy environment, a discrete compatible adjustment model of group consensus under different decision making behaviors is established. Finally, the efficient screening system is matched for different groups, which provides a valid path for the optimal solution selection of group consensus.

Multi criteria decision-making for distributed energy system based on multi-source heterogeneous data

The distributed energy system (DES) has been regarded as a socio-technical innovation that can improve energy efficiency and reduce carbon emission. Based on the construction and planning of a DES project in Henan Province, this paper aims to develop a comprehensive decision-making framework for selecting the most appropriate DES composition schemes with multi-source heterogeneous data. Thus, a new multi-criteria decision-making method based on fuzzy preference relations and improved Choquet integral is generated. Firstly, an evaluation criteria system is established from economic, technical, environmental and social dimensions, and 10 secondary criteria are presented. Secondly, based on the interval fuzzy preference relations of criteria given by experts, the iteration of weights of criteria is continued until the group consistency degree reaches the threshold. Thirdly, the deviation degree is used to transform the initial data, which include crisp values, interval numbers and linguistic terms to isomorphism data. Thereafter, the fuzzy measures are calculated using Mobius transformation coefficients and the final scores of the alternatives are obtained by means of Choquet integral, which can eliminate the overlap information between criteria. Finally, a real case is given to illustrate the effectiveness of the proposed evaluation criteria and technique. The result shows the multi energy complementary system as the most appropriate alternative for the project, followed by the solar-air source heat pump system, fuel cell system, natural gas system, internal combustion engine system and traditional system. Conclusively, the discussion, comparative analysis and some policy implications to promote the development of distributed energy industry are given. This study not only can assist stakeholders to obtain greater benefits of the DES, but also can serve as a reference for the government towards the development of more robust regulations and policies.

Priority weights acquisition of linear uncertain preference relations and its application in the ranking of online shopping platforms

The linear uncertain preference relations (LUPRs) uses an uncertain variable to represent the decision-maker’s pairwise comparison judgment of the scheme. The variable is subject to the linear uncertain distribution, which is the extension of the traditional fuzzy preference relations (FPRs) and interval fuzzy preference relations (IFPRs). This paper proposes a group decision modeling problem, constructs the priority weights acquisition models of the additively and multiplicatively consistent LUPRs, and especially solves the problem that the weight solution is negative value or no solution by using traditional methods to acquire weights in consistent FPRs and IFPRs. Based on these two types of consistent structure of LUPRs, this study constructs the crisp number, interval number weight solving models of LUPRs and the group decision ranking models with LUPRs. The results show that these new models are suitable for solving the weight vector of traditional FPRs and IFPRs. The case of online shopping platform selection compares the results obtained by various methods of calculating weights, further illustrating the effectiveness and rationality of the new methods.

Analysis of acceptable additive consistency and consensus of group decision making with interval-valued hesitant fuzzy preference relations

Interval-valued hesitant fuzzy sets are efficient to denote the hesitant and uncertain judgments of decision makers. To further research the utilization of this type of fuzzy sets, this paper limits to study group decision making (GDM) with interval-valued hesitant fuzzy preference relations (IVHFPRs). First, an additive consistency index is defined for interval fuzzy preference relations, by which we derive a new acceptable additive consistency concept for IVHFPRs. Then, new models based on acceptable additive consistency are built for ascertaining unknown values, and new models for judging the acceptable additive consistency of IVHFPRs are constructed. When the consistency of IVHFPRs is unacceptable, new models based on consistency and uncertain degree are constructed for obtaining acceptable additive consistency IVHFPRs that can ensure the smallest total adjustment and the minimum number of adjusted intervals. For GDM case, a new consensus index is defined to measure the agreement degree of individual judgments. When the consensus level does not satisfy requirement, new models for reaching the additive consistency and consensus requirements are built. Furthermore, a new acceptable additive consistency- and consensus-based algorithm for GDM with incomplete and unacceptable additive consistency IVHFPRs is provided. Finally, an example is given to indicate the utilization of the new algorithm and comparison analysis is made.

Consensus reaching for social network group decision making by considering leadership and bounded confidence

With the rapid development of information, communication and techniques, social network group decision making problems which allow information exchange and communication among experts are more and more common in recent years. How to use social relationships generated by social networks to promote consensus among experts has been becoming a hot topic in the field of group decision making. In this paper, we consider a new type of group decision making problems in which experts will provide his/her interval fuzzy preference relations over alternatives under social network environment and propose a new model to help experts reach consensus. In the proposed model, we first define the individual consensus measure and the group consensus measure, and then use a network partition algorithm to detect sub-networks of experts, based on which the leadership of experts can be identified. Afterwards, by considering the leadership and the bounded confidence levels of experts, a new feedback mechanism which can provide acceptable advice to experts who need to modify their opinions is devised and a consensus reaching algorithm is further developed. To demonstrate the performance of the proposed consensus model and algorithm, a hypothetical application and some simulation analysis are provided eventually.

New method for interval-valued hesitant fuzzy decision making based on preference relations

Interval-valued hesitant fuzzy preference relations (IVHFPRs) are powerful to express the judgments of decision makers (DMs) such as hesitancy and uncertainty. This paper continues to research decision making with IVHFPRs. To do this, it first defines a multiplicative consistency index for interval fuzzy preference relations (IFPRs). Then, it gives a new (acceptably) multiplicative consistency concept for IVHFPRs. After that, optimization models for judging the acceptably multiplicative consistency of IVHFPRs are built. When the multiplicative consistency is unacceptable, optimization models for improving the multiplicative consistency level and for minimizing the number of adjusted judgments are constructed, respectively. Based on the acceptably multiplicative consistency analysis, an algorithm for decision making with IVHFPRs is presented. Furthermore, formulae for determining the weights of the DMs and for measuring the consensus are offered. Based on the acceptably multiplicative consistency and consensus analysis, a new method for group decision making with IVHFPRs is proposed. Finally, an example about selecting the reservoir operation schemes is offered to show the efficiency of the new method and to compare with previous research.

Additive and Multiplicative Consistency Modeling for Incomplete Linear Uncertain Preference Relations and Its Weight Acquisition

The use of consistency methods to supplement or generate missing information so as to obtain a rational and logical ranking list has been a key point in studies of the incomplete preference relation. Existing works on incomplete interval fuzzy preference relations (IFPRs) utilize only endpoints of the original intervals and generate missing values using interval operations, resulting in the loss and distortion of information. This article considers IFPRs as linear uncertain preference relations (LUPRs), and introduces belief degree and inverse uncertainty distributions to explore the consistency and ranking problems with LUPRs. Definitions related to uncertain preference relations (UPRs) as well as their additive and multiplicative consistency are first given. Then, models to solve weight vectors are proposed under both consistency scenarios. Finally, algorithms that generate missing information in incomplete LUPRs are provided. By building mathematical relationships between the minimum deviation and the belief degree, analytic formulas for missing values are obtained when the minimum deviation is achieved. It is verified that consistent UPRs and the proposed weight-vector solving models under incomplete preference relations are extensions of the traditional IFPRs.

Consistency and consensus modeling of linear uncertain preference relations

Interval operations, as currently defined, suffer from the problem of not satisfying the conditions of global complementarity and consistency of interval fuzzy preference relations (IFPRs). In this paper, we resolve this difficulty by constructing linear uncertain preference relations (LUPRs). By considering all the information and the uncertain distribution of an interval, we propose the concept of uncertain preference relations (UPRs) for the first time. Then we apply uncertainty distributions to characterize interval judgments that are considered as a whole to participate in the uncertain operation to achieve the desired conditions of global complementarity and consistency. Based on this, we prove that IFPRs and the definitions of their additive consistency are special cases of those of LUPRs. Moreover, we investigate two types of consensus models developed based on LUPRs between the minimum deviation and belief degree. We prove that the minimum deviation is a linear, increasing function of the belief degree, and then establish sufficient and necessary conditions for the consensus model to satisfy additive consistency. Finally, the LUPRs models presented in this paper is applied, incorporating with expert assistance in decision-making, to the sensitivity assessment of the meteorological industry in a region of China, and the LUPRs models can be utilized to obtain results with smaller deviations.

A new method for interval fuzzy preference relations in group decision making based on plant growth simulation algorithm and COWA

A new method for interval fuzzy preference relations in group decision making based on plant growth simulation algorithm and COWA

Consistency-Based Algorithms for Decision-Making With Interval Fuzzy Preference Relations

This paper reviews and analyzes several consistency concepts for interval fuzzy preference relations (IFPRs). On the basis of the comparisons, one can find that Krejci's additive and multiplicative consistency concepts are more reasonable and flexible. Considering the issues that previous methods cannot fully address regarding incomplete and inconsistent IFPRs, this paper studies incomplete and inconsistent IFPRs using custom programming models based on Krejci's concepts. Meanwhile, programming models for judging the additive and multiplicative consistency of IFPRs are constructed, respectively. Considering the consensus of IFPRs in group decision-making, a consensus index is defined, and programming models for improving the consensus levels of individual IFPRs are built. On the basis of the consistency and consensus analysis, two consistency-based algorithms for group decision-making with inconsistent and incomplete IFPRs are offered. One method is based on Krejci's additive consistency concept, and the other uses Krejci's multiplicative consistency concept. Meanwhile, associated examples are provided to show the application of the new methods.

Possibility degree and divergence degree based method for interval-valued intuitionistic fuzzy multi-attribute group decision making

This paper investigates an order relation of interval-valued intuitionistic fuzzy values (IVIFVs) based on possibility degree and divergence degree and applies it to multi-attribute group decision making (MAGDM) problems. Firstly, based on the possibility degree of intuitionistic fuzzy values, the possibility degree and divergence degree of IVIFVs are defined to provide a new order relation of IVIFVs. Subsequently in interval-valued intuitionistic fuzzy MAGDM, an intuitionistic fuzzy linear programming model is constructed to derive decision makers' weights. Depending on the construction of membership and non-membership functions, three approaches are developed to solve the constructed intuitionistic fuzzy linear programming model, including the optimistic, pessimistic and mixed approaches. Utilizing decision makers’ weights, the collective decision matrix is obtained. Combining the possibility and divergence degrees of alternatives on attributes, the adjusted possibility distribution matrix for each attribute is constructed and proved to be an interval fuzzy preference relation. Then the attribute weights are determined by the negative distances and the positive distances of adjusted possibility distribution matrices. Using the interval-valued intuitionistic weighted arithmetic aggregation operator, the collective comprehensive values of alternatives are obtained. The ranking order of alternatives is generated according to the proposed order relation of IVIFVs. Therefore, an interval-valued intuitionistic fuzzy MAGDM method is proposed. Finally, the proposed method is implemented on some examples. Validation discussion and comparative analyses of this method is also done to show the effectiveness of the proposed method.

Modeling the Social Influence in Consensus Reaching Process with Interval Fuzzy Preference Relations

Consensus process plays a prominent role in decision-making problems. In traditional way, experts with low consensus degree are asked to modify their opinions according to the given advices. However, in fact, experts neither simply adopt nor completely ignore the opinions of others. In general, experts will refer to other experts’ opinions to a certain extent and their opinions will evolve due to their interactions. The aim of this paper, therefore, is to propose a consensus model based on uncertain opinion evolution to model the influence between experts during the negotiation and communication process. Firstly, we construct an influence network among experts by taking both objective similarity degree and some subjective psychological traits into consideration. Secondly, a consensus model based on opinion evolution with interval fuzzy preference relations (IFPRs) is proposed. After identifying the incompatible preference with lower consensus degree, experts will modify their preferences according to the preference evolution under the influence network. Then, we prove that the reciprocity property maintains through the dynamic consensus reaching process, and a sufficient condition guaranteeing that the consensus can be reached in the influence network is derived. Furthermore, two nonlinear programming models are proposed to justify whether the IFPR satisfies the acceptable consistency and to calculate the priority vector of the IFPR, respectively. Finally, a case study and comparative analysis are made to illustrate the effectiveness of the proposed model.

On Optimal Priority Modelling of Group Intuitionistic Fuzzy Preference Relations with Normal Uncertainty Distribution

The uncertainty distribution can more effectively express the uncertainty of decision makers’ judgments during a pairwise comparison of any alternatives. This paper investigates the priority models of group intuitionistic fuzzy preference relations with normal uncertainty distribution. The mathematical equivalence between the membership, non-membership degree interval fuzzy preference relation and the intuitionistic fuzzy preference relation is constructed, showing that there exists an inverse relationship between the priority of alternatives using these two types of interval preference relations. The new optimal models regarded the event that the deviation between the ideal judgement meeting the multiplicative consistency and the actual judgement obeying normal uncertainty distribution shall not exceed a threshold value under the given belief degree as a constraint, and regarded the minimum sum of all the threshold values as the objective function. The chance constraint was introduced to measure the degree to which multiplicative consistency can be realized under different belief degrees. The priority model provides a new method for simulating uncertainty and fuzziness in the real-world decision making environment.

Revisiting multiplicative consistency of interval fuzzy preference relation

The main aim of this paper is to investigate the multiplicative consistency of interval fuzzy preference relations. This paper proposes a new multiplicative consistency definition of interval fuzzy preference relations and discusses some of its desired properties. In addition, some goal programming models are constructed to check and improve the multiplicative consistency of interval fuzzy preference relations. Furthermore, numerical examples are provided for discussion and comparison with other existing multiplicative consistency definitions of interval fuzzy preference relations. The results demonstrate that the multiplicative consistency definition proposed is relatively compromised.

A New Approach to Derive Priority Weights from Additive Interval Fuzzy Preference Relations Based on Logarithms

This paper investigates the consistency definition and the weight-deriving method for additive interval fuzzy preference relations (IFPRs) using a particular characterization based on logarithms. In a recently published paper, a new approach with a parameter is developed to obtain priority weights from fuzzy preference relations (FPRs), then a new consistency definition for the additive IFPRs is defined, and finally linear programming models for deriving interval weights from consistent and inconsistent IFPRs are proposed. However, the discussion of the parameter value is not adequate and the weights obtained by the linear models for inconsistent IFPRs are dependent on alternative labels and not robust to permutations of the decision makers’ judgments. In this paper, we first investigate the value of the parameter more thoroughly and give the closed form solution for the parameter. Then, we design a numerical example to illustrate the drawback of the linear models. Finally, we construct a linear model to derive interval weights from IFPRs based on the additive transitivity based consistency definition. To demonstrate the effectiveness of our proposed method, we compare our method to the existing method on three numerical examples. The results show that our method performs better on both consistent and inconsistent IFPRs.

A group decision-making method considering both the group consensus and multiplicative consistency of interval-valued intuitionistic fuzzy preference relations

This paper investigates a group decision-making (GDM) method that considers group consensus and multiplicative consistency of interval-valued intuitionistic fuzzy (IVIF) preference relations (IVIFPRs). First, the mean and variance of IVIF values (IVIFVs) are defined and a ranking method for IVIFVs is proposed considering the risk attitude of the expert. Then, the group consensus is presented by the individual similarity between experts. An iteration algorithm is designed to improve the group consensus. A statistical comparative analysis validates this algorithm. Subsequently, a new multiplicative consistency of IVIFPR is defined based on the multiplicative consistency of interval fuzzy preference relation. Two single-objective programming models are established to extract the most optimistic and pessimistic interval priority weight vectors from an IVIFPR, respectively. In particular, if the feasible domains of these two models are empty, two adjusted programs are constructed to replace the originals. Combining the most optimistic and pessimistic interval priority weights, the IVIF priority weights are generated. Further, expert weights are derived from Markov model and used to derive the collective IVIFPR for generating the IVIF priority weights. Therefore, a new method is proposed to solve the GDM with IVIFPRs. Finally, two cases are analyzed to verify the effectiveness of the proposed method.

A goal programming approach to deriving interval weights in analytic form from interval Fuzzy preference relations based on multiplicative consistency

This paper focuses on how to find an analytic solution of optimal interval weights from consistent interval fuzzy preference relations (IFPRs) and obtain approximate-solution-based interval weights in analytic form from inconsistent IFPRs. The paper first analyzes the popularly used interval weight additive normalization model and illustrates its drawbacks on the existence and uniqueness for characterizing ]0, 1[-valued interval weights obtained from IFPRs. By examining equivalency of ]0, 1[-valued interval weight vectors, a novel framework of multiplicatively normalized interval fuzzy weights (MNIFWs) is then proposed and used to define multiplicatively consistent IFPRs. The paper presents significant properties for multiplicatively consistent IFPRs and their associated MNIFWs. These properties are subsequently used to establish two goal programming (GP) models for obtaining optimal MNIFWs from consistent IFPRs. By the Lagrangian multiplier method, analytic solutions of the two GP models are found for consistent IFPRs. The paper further devises a two-step procedure for deriving approximate-solution-based MNIFWs in analytic form from inconsistent IFPRs. Two visualized computation formulas are developed to determine the left and right bounds of approximate-solution-based MNIFWs of any IFPR. The paper shows that this approximate solution is an optimal solution if an IFPR is multiplicatively consistent. Three numerical examples including three IFPRs and comparative analyses are offered to demonstrate rationality and validity of the developed model.

Group decision making with interval fuzzy preference relations based on DEA and stochastic simulation

This paper proposes an integrated approach to group decision making with interval fuzzy preference relations using data envelopment analysis (DEA) and stochastic simulation. A novel output-oriented CCR DEA model is proposed to obtain the priority vector for the consistency fuzzy preference relation, in which each of the alternatives is viewed as a decision-making unit. Meanwhile, we design a consistency adjustment algorithm for the inconsistent fuzzy preference relation. Furthermore, we build an optimization model to get the weights of each fuzzy preference relation based on maximizing group consensus. Then, an input-oriented DEA model is introduced to obtain the final priority vector of the alternatives. Finally, a stochastic group preference analysis method is developed by analyzing the judgments space, which is carried out by Monte Carlo simulation. A numerical example demonstrates that the proposed method is effective.

An Approach for Group Decision Making With Interval Fuzzy Preference Relations Based on Additive Consistency and Consensus Analysis

Interval fuzzy preference relations (IFPRs) that can simply denote the lower and upper bounds of decision makers’ uncertain judgments are keenly studied by researchers and are widely used in practical decision-making problems. Consistency analysis is vitally important to avoid illogical ranking orders. First, this paper analyzes issues in previous additive consistency concepts for IFPRs. Then, it presents a new additive consistency definition that overcomes issues in previous ones. A linear programming model to judge the additive consistency of IFPRs is constructed, and a method to derive additive consistent IFPRs is proposed. Furthermore, a goal programming model to determine missing values in an incomplete IFPR is built, which have the highest consistent level with respect to known values. Regarding group decision making, a group consensus index is defined to measure the consensus of individual IFPRs, and an approach to improve the consensus level is introduced. Finally, a method for group decision making with IFPRs is developed, which can address incomplete and inconsistent cases. Associated examples are given to show the efficiency and feasibility of the developed theoretical results and comparative analysis is made.

Additive consistent interval-valued Atanassov intuitionistic fuzzy preference relation and likelihood comparison algorithm based group decision making

This paper investigates a group decision making (GDM) method based on additive consistent interval-valued Atanassov intuitionistic fuzzy (IVAIF) preference relations (IVAIFPRs) and likelihood comparison algorithm. Firstly, the likelihood of IVAIF values (IVAIFVs) is defined by the likelihood of intervals. Then a likelihood comparison algorithm is designed to rank IVAIFVs. According to the additive consistent interval fuzzy preference relation, we define the additive consistency of an IVAIFPR. Two special interval fuzzy preference relations are extracted from an IVAIFPR. They can be regarded as the lowest and highest preferred matrices of the IVAIFPR, respectively. Using a parametric linear program, the IVAIF priority weights of an IVAIFPR are generated from these two extracted special interval fuzzy preference relations. For the GDM with IVAIFPRs, the group consensus is defined by the distances between the individual IVAIFPRs and the collective one. To derive decision makers' weights, an optimization model is constructed by maximizing the group consensus and transformed into a linear program to resolve. Subsequently, utilizing the IVAIF weighted averaging operator, the collective IVAIFPR is obtained and applied to obtain the IVAIF priority weights. The order of alternatives is generated by ranking the IVAIF priority weights. At length, an enterprise resource planning system selection example is analyzed to verify the effectiveness of the proposed method.