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

The breaking of additively reciprocal property of fuzzy preference relations and its implication to decision making under uncertainty

In this paper, we propose a method that improves the multiplicative consistency and minimizes indeterminacy (the sum of widths of all interval membership degrees) by using only off-diagonal elements of an interval fuzzy preference relation (IFPR). In addition, this method keeps the initial information as much as possible. To do so, we formulate a concept of the multiplicative consistency that satisfies the additive reciprocity between the related preferences of the IFPR and is invariant under any permutation of objects. Next, the equations which are equivalent to the multiplicative consistency for the IFPR and uses only off-diagonal elements are derived. Based on these equations, the linear models to judge the multiplicative consistency of the IFPR and calculate multiplicatively consistent IFPR minimizing indeterminacy by using only off-diagonal elements are constructed. Based on linear models, we construct an algorithm that calculates the acceptable consistent IFPR keeping the initial information as much as possible and prove that a consistency index of algorithm converges to zero. The proposed method can reduce a large amount of calculations and is correct in judging and improving the multiplicative consistency for the IFPR in comparison with previous results because it uses only off-diagonal elements of the initial IFPR. In addition, a numerical example is provided to show the feasibility and efficiency of the proposed method.

In the analytical hierarchy process (AHP), it needs the decision maker to establish a pairwise comparison matrix requires n(n–1)/2 judgments for a level with n criteria (or alternatives). In some instances, the decision maker may have to deal with the problems in which only partial information and uncertain preference relation is available. Consequently, the decision maker may provide interval fuzzy preference relation with incomplete information. In this paper, we focus our attention on the investigation of incomplete interval fuzzy preference relation. We first extend a characterization to the interval fuzzy preference relation which is based on the additive transitivity property. Using the characterization, we propose a method to construct interval additive consistent fuzzy preference relations from a set of n–1 preference data. The study reveals that the proposed method can not only alleviate the comparisons, but also ensure interval preference relations with the additive consistent property. We also develop a novel procedure to deal with the analytic hierarchy problem for group decision making with incomplete interval fuzzy preference relations. Finally, a numerical example is illustrated and a supplier selection case in supply chain management is investigated using the proposed method.

Group decision making with preference information on alternatives is an interesting and important research topic which has been receiving more and more attention in recent years. The purpose of this paper is to investigate multiple-attribute group decision-making (MAGDM) problems with distinct uncertain preference structures. We develop some linear-programming models for dealing with the MAGDM problems, where the information about attribute weights is incomplete, and the decision makers have their preferences on alternatives. The provided preference information can be represented in the following three distinct uncertain preference structures: 1) interval utility values; 2) interval fuzzy preference relations; and 3) interval multiplicative preference relations. We first establish some linear-programming models based on decision matrix and each of the distinct uncertain preference structures and, then, develop some linear-programming models to integrate all three structures of subjective uncertain preference information provided by the decision makers and the objective information depicted in the decision matrix. Furthermore, we propose a simple and straightforward approach in ranking and selecting the given alternatives. It is worth pointing out that the developed models can also be used to deal with the situations where the three distinct uncertain preference structures are reduced to the traditional ones, i.e., utility values, fuzzy preference relations, and multiplicative preference relations. Finally, we use a practical example to illustrate in detail the calculation process of the developed approach.

The intuitionistic fuzzy preference relation (IFPR), whose elements are intuitionistic fuzzy values (IFVs), is more powerful than the traditional multiplicative preference relation and the fuzzy preference relation in expressing comprehensive preference information of a decision maker. The aim of this paper is to investigate a new approach to derive the priority weights from an IFPR. To do so, we give a new definition of multiplicative consistent IFPR, which is based on the membership and nonmembership degrees of the intuitionistic fuzzy judgments. After that, a formula, which involves the underlying intuitionistic fuzzy weights of the IFPR, is proposed to construct such a multiplicative consistent IFPR. Based on the formula, some fractional programming models are built to generate the intuitionistic fuzzy priority weighting vector of the IFPR. Several numerical examples are given to illustrate the validity and applicability of the proposed method.

We develop a new compatibility for the interval fuzzy preference relations based on the continuous ordered weighted averaging (COWA) operator and use it to determine the weights of experts in group decision making (GDM). We define some concepts of the compatibility degree and the compatibility index for the two interval fuzzy preference relations based on the COWA operator. We study some desirable properties of the compatibility index and investigate the relationship between the each expert's interval fuzzy preference relation and the synthetic interval fuzzy preference relation. The prominent characteristic of the compatibility index based on the COWA operator is that it can deal with the compatibility of all the arguments by using a controlled parameter considering the attitude of decision maker rather than the compatibility of the simply two points in intervals. To determine the experts' weights in the GDM with the interval fuzzy preference relations, we propose an optimal model based on the criterion of minimizing the compatibility index. In the end, we give a numerical example to develop the new approach to GDM with interval fuzzy preference relations.

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.

Szmidt and Kacprzyk (Lecture Notes in Artificial Intelligence 3070:388---393, 2004a) introduced a similarity measure, which takes into account not only a pure distance between intuitionistic fuzzy sets but also examines if the compared values are more similar or more dissimilar to each other. By analyzing this similarity measure, we find it somewhat inconvenient in some cases, and thus we develop a new similarity measure between intuitionistic fuzzy sets. Then we apply the developed similarity measure for consensus analysis in group decision making based on intuitionistic fuzzy preference relations, and finally further extend it to the interval-valued intuitionistic fuzzy set theory.

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.

This paper presents a new method for group decision making using group recommendations based on interval fuzzy preference relations and consistency matrices. First, it constructs consistency matrices from interval fuzzy preference relations. Then, it constructs a collective consistency matrix, constructs a weighted collective preference relation, and constructs a group collective preference relation. Then, it constructs a consensus relation for each expert and calculates the group consensus degree for the experts based on the constructed consensus relations. If the group consensus degree is smaller than a predefined threshold value, then it modifies the interval fuzzy preference values in the interval fuzzy preference relations. The above process is performed repeatedly, until the group consensus degree is larger than or equal to the predefined threshold value. Finally, based on the group collective preference relation, it calculates the score of each alternative. The larger the score of the alternative, the better the preference order of the alternative. The proposed method can overcome the drawbacks of the existing methods for group decision making using group recommendations.

Group decision making based on the modified probability calculation method and DEA cross-efficiency with probabilistic hesitant fuzzy preference relations

Mathematical programming methods for consistency and consensus in group decision making with intuitionistic fuzzy preference relations

Compatibility measures and consensus models for group decision making with intuitionistic multiplicative preference relations

There are plenty of researches on group decision making (GDM) problem and most of them assume that all the experts are independent. However, the social network connection is an important characteristic among experts, and should been taken into account in the GDM decision process. The social network analysis (SNA) is a rapidly developing technology to deal with the problem about social network connections. In this paper, we address GDM problem with fuzzy preference relations where the experts have directed social network connections. And we use in-degree and out-degree centrality indices in SNA to build the social characteristics of the experts. Then we utilize the importance induced ordered weighted averaging(I-IOWA) operator to aggregate all the individual fuzzy preference relations with social indices. Furthermore,we investigate the reciprocity, additive consistency and acceptable consistency properties about the obtained group fuzzy preference relations. A procedure of addressing GDM problem with fuzzy preference relations where the experts are within directed social network connections is developed. Moreover, we compare the results of the case where all the experts are independent and the case where all the experts are within directed social network connections. Some useful tips that affect final outranking result in the directed graphical social network connection are obtained. In the end, we summarize the main results of this paper and point out some research directions for future research.

Considering the influence of undesirable outputs in environmental assessments of the practical production process, this paper proposes a DEA cross-efficiency method and fuzzy preference relation based on semi-disposability of undesirable outputs for environmental assessments, which combines self-evaluation with peer-evaluation and avoids potentially unrealistic weighting scheme. We first develop a new input-oriented DEA model based on semi-disposability of undesirable outputs. Then, the corresponding DEA cross-efficiency method is proposed to evaluate the homogeneous decision-making units (DMUs) and the additive fuzzy preference relation is constructed using the DEA cross-efficiency scores. An output-oriented DEA model is adopted to derive the priority vector from the constructed additive fuzzy preference relation. Furthermore, we summarize the specific procedures for environmental assessments based on DEA cross-efficiency and fuzzy preference relation. Finally, the practical example of industrial environment assessment of the Yangtze River Economic Belt is provided to show the applicability and effectiveness of the proposed method.