Interval Fuzzy Preference Relations Research Articles

Consistency test and weight generation for additive interval fuzzy preference relations

Some simple yet pragmatic methods of consistency test are developed to check whether an interval fuzzy preference relation is consistent. Based on the definition of additive consistent fuzzy preference relations proposed by Tanino (Fuzzy Sets Syst 12:117---131, 1984), a study is carried out to examine the correspondence between the element and weight vector of a fuzzy preference relation. Then, a revised approach is proposed to obtain priority weights from a fuzzy preference relation. A revised definition is put forward for additive consistent interval fuzzy preference relations. Subsequently, linear programming models are established to generate interval priority weights for additive interval fuzzy preference relations. A practical procedure is proposed to solve group decision problems with additive interval fuzzy preference relations. Theoretic analysis and numerical examples demonstrate that the proposed methods are more accurate than those in Xu and Chen (Eur J Oper Res 184:266---280, 2008b).

TOPSIS-Based Consensus Model for Group Decision-Making With Incomplete Interval Fuzzy Preference Relations

Due to the vagueness of real-world environments and the subjective nature of human judgments, it is natural for experts to estimate their judgements by using incomplete interval fuzzy preference relations. In this paper, based on the technique for order preference by similarity to ideal solution method, we present a consensus model for group decision-making (GDM) with incomplete interval fuzzy preference relations. To do this, we first define a new consistency measure for incomplete interval fuzzy preference relations. Second, a goal programming model is proposed to estimate the missing interval preference values and it is guided by the consistency property. Third, an ideal interval fuzzy preference relation is constructed by using the induced ordered weighted averaging operator, where the associated weights of characterizing the operator are based on the defined consistency measure. Fourth, a similarity degree between complete interval fuzzy preference relations and the ideal one is defined. The similarity degree is related to the associated weights, and used to aggregate the experts' preference relations in such a way that more importance is given to ones with the higher similarity degree. Finally, a new algorithm is given to solve the GDM problem with incomplete interval fuzzy preference relations, which is further applied to partnership selection in formation of virtual enterprises.

SOME ICOWA OPERATORS AND THEIR APPLICATIONS TO GROUP DECISION MAKING WITH INTERVAL FUZZY PREFERENCE RELATIONS

We develop some new cases of the induced continuous ordered weighted averaging (ICOWA) operator and study their desirable properties, which are very suitable to deal with group decision making (GDM) with interval fuzzy preference relations. First, we present the consensus indicator ICOWA (CI-ICOWA) operator which uses the consensus indicator of the interval fuzzy preference as the order inducing variable in the ICOWA operator. Then the concept of compatibility degree (CD) for two interval fuzzy preference relations is defined based on the continuous ordered weighted averaging (COWA) operator and the compatibility degree ICOWA (CD-ICOWA) operator is proposed which uses the CD as the order inducing variable in the ICOWA operator. Next, we investigate some desirable properties of the CD-ICOWA operator. Additionally, we construct an optimization model to obtain the weights of experts by minimizing the compatibility degree in the GDM. Finally, an illustrative numerical example is used to verify the developed approaches.

A CONSENSUS MODEL FOR GROUP DECISION-MAKING PROBLEMS WITH INTERVAL FUZZY PREFERENCE RELATIONS

Interval fuzzy preference relations can be useful to express decision makers' preferences in group decision-making problems. Usually, we apply a selection process and a consensus process to solve a group decision situation. In this paper, we present a consensus model for group decision-making problems with interval fuzzy preference relations. This model is based on two consensus criteria, a consensus measure and a proximity measure, and also on the concept of coincidence among preferences. We compute both consensus criteria in the three representation levels of a preference relation and design an automatic feedback mechanism to guide experts in the consensus reaching process. We show an application example in social work.

A consensus model for group decision making problems with linguistic interval fuzzy preference relations

Sometimes, we find decision situations in which it is difficult to express some preferences by means of concrete preference degrees. In this paper, we present a consensus model for group decision making problems in which the experts use linguistic interval fuzzy preference relations to represent their preferences. This model is based on two consensus criteria, a consensus measure and a proximity measure, and on the concept of coincidence among preferences. We compute both consensus criteria in the three representation levels of a preference relation and design an automatic feedback mechanism to guide experts in the consensus reaching process.

Goal programming approaches to deriving interval weights based on interval fuzzy preference relations

This article investigates the consistency of interval fuzzy preference relations based on interval arithmetic, and new definitions are introduced for additive consistent, multiplicative consistent and weakly transitive interval fuzzy preference relations. Transformation functions are put forward to convert normalized interval weights into consistent interval fuzzy preference relations. By analyzing the relationship between interval weights and consistent interval fuzzy preference relations, goal-programming-based models are developed for deriving interval weights from interval fuzzy preference relations for both individual and group decision-making situations. The proposed models are illustrated by a numerical example and an international exchange doctoral student selection problem.

Some programming models to derive priority weights from additive interval fuzzy preference relation

Deriving priority weights from fuzzy preference relations is a significant issue in decision making problems. In this paper, based on the definition of additive consistent fuzzy preference relations proposed by Tanino, a new approach with a parameter is developed to obtain priority weights, and properties of the new approach are explored. Then, a method for correcting inconsistent fuzzy preference relations is derived, and a new definition for the additive consistent interval fuzzy preference relations is obtained for the interval complementary pairwise comparison matrix. From these, linear programming models for generating interval priority weights from additive consistent or inconsistent interval fuzzy preference relations are established. Finally, three numerical examples are examined to show the feasibility of the developed method, and comparisons are also made between this new approach and the methods proposed by Xu and Chen [15]. Through the numerical examples, the ranking of interval priority weights using the different methods was found to be the same but with a slightly different degree of possibility. However, for the same interval complementary pairwise comparison matrix, the new definition for additive consistent interval fuzzy preference relations proposed in this paper was found to have more consistent information.

SOME ISSUES ON MULTIPLICATIVE CONSISTENCY OF INTERVAL RECIPROCAL RELATIONS

Consistency plays an important part in decision making with interval reciprocal relation also called interval fuzzy preference relation. In this paper, we give the definition of perfect multiplicative consistent interval reciprocal relation and discuss its properties. A method is proposed to construct a set of multiplicative consistent reciprocal relations from a perfect consistent interval reciprocal relation. Then we develop methods to get the perfect multiplicative consistent interval reciprocal relation from the inconsistent one and estimate the missing values from an incomplete interval reciprocal relation. Finally, a group decision-making method is proposed based on perfect multiplicative consistent interval reciprocal relations. Some examples are also given to compare the proposed methods with the existing ones.

A new method of obtaining the priority weights from an interval fuzzy preference relation

In analyzing a multiple criteria decision-making problem, the decision maker may express her/his opinions as an interval fuzzy or multiplicative preference relation. Then it is an interesting and important issue to investigate the consistency of the preference relations and obtain the reliable priority weights. In this paper, a new consistent interval fuzzy preference relation is defined, and the corresponding properties are derived. The transformation formulae between interval fuzzy and multiplicative preference relations are further given, which show that two preference relations, consistent interval fuzzy and multiplicative preference relations, can be transformed into each other. Based on the transformation formula, the definition of acceptably consistent interval fuzzy preference relation is given. Furthermore a new algorithm for obtaining the priority weights from consistent or inconsistent interval fuzzy preference relations is presented. Finally, three numerical examples are carried out to compare the results using the proposed method with those using other existing procedures. The numerical results show that the given procedure is feasible, effective and not requisite to solve any mathematical programing.

Deriving interval weights from an interval multiplicative consistent fuzzy preference relation

In this paper, relations between a multiplicative consistent interval fuzzy preference relation and an additive consistent interval fuzzy preference relation are established. Based on the new relations, a new method ( Algorithm 3) is proposed to derive interval weights by transforming a multiplicative consistent interval fuzzy preference relation into an additive consistent interval fuzzy preference relation, collecting additive consistent information ( Algorithm 2), transforming back into multiplicative consistent information and calculating the interval weights by Eq. (20). Finally, two numerical examples are given to illustrate the new method.

An approach to group decision making with interval fuzzy preference relations based on induced generalized continuous ordered weighted averaging operator

The aim of this paper is to develop a new class of operator called the induced generalized continuous ordered weighted averaging (IGCOWA) operator, which extends the GOWA operator, the IGOWA operator and the ICOWG operator. We study the desirable properties of the IGCOWA operator and describe families of IGCOWA operator according to the special weighting vector and generalized mean parameter. Particularly, we present the consistency induced generalized continuous ordered weighted averaging (C-IGCOWA) operator, in which the consistency indicator is the induced ordering variable to reflect the importance of interval fuzzy preference relation. Finally, the C-IGCOWA operator is applied to group decision making with interval fuzzy preference relations in a numerical example.

New approach to determine the priorities from interval fuzzy preference relations

An approach is proposed to solve the problem how to obtain the priorities from interval fuzzy preference relations. Firstly, another expression of interval numbers is given. Then, some basic definitions on consistency and weak transitivity of real and interval fuzzy preference relations are described. Based on these definitions, a two-phase process for determining the priorities from interval fuzzy preference relations is presented. Finally, two examples are used to illustrate the use of the proposed approach.

Consistency of interval fuzzy preference relations in group decision making

Interval fuzzy preference relations are an extension of fuzzy preference relations, which are usually used by experts to express their uncertain preference information over objects in group decision making. In this paper, we focus our attention on the investigation of consistency of interval fuzzy preference relations. We first establish a quadratic programming model by minimizing all the deviations of individual interval fuzzy preference relations and collective interval fuzzy preference relation, from which an exact solution can be found to derive the importance weights of experts. Then, we give two approaches to constructing additive and multiplicative consistent interval fuzzy preference relations, respectively, and show the relationship between the consistency of individual interval fuzzy preference relations and the consistency of collective interval fuzzy preference relation. At last, a practical application is given to our models and approaches.

Consolidating SWOT analysis with nonhomogeneous uncertain preference information

SWOT analysis is an important support tool for decision-making, and is commonly used to systematically analyze organizations’ internal and external environments. However, one of its deficiencies is in the measurement and evaluation of prioritization of the factors and strategies. This paper is aimed to present a novel quantified SWOT analytical method based multiple criteria group decision-making (MCGDM) concept, in which the priorities of SWOT factors and groups are derived by multiple decision makers (DMs) with nonhomogeneous uncertain preference information (NUPI), such as interval multiplicative preference relations, interval fuzzy preference relations, and uncertain linguistic preference relations. In this method, the SWOT analysis provides a basic frame within which to perform analyses of decision situations, in turn, MCGDM methods assist in carrying out SWOT more analytically and in elaborating the results of the analyses so that SWOT factors and groups can be prioritized with respect to the entire SWOT. The uniform and aggregation of the NUPI and the derivation of priorities for SWOT groups and factors are investigated in detail. Finally, an example is to validate the procedure of the proposed method.

Uncertain Power Average Operators for Aggregating Interval Fuzzy Preference Relations

In this paper, we investigate group decision making problems based on interval fuzzy preference relations. We define an uncertain power weighted average (UPWA) operator and an uncertain power ordered weighted average (UPOWA) operator, on the basis of the power average operator of Yager (IEEE Trans Syst Man Cybern A 31:724–731, 1988) and the uncertain geometric mean. In the situations where the weights of experts are known, we develop a method based on the UPWA operator for group decision making with interval fuzzy preference relations; and in the situations where the weights of experts are unknown, we develop a method based on the UPOWA operator for group decision making with interval fuzzy preference relations.

Interval multiplicative transitivity for consistency, missing values and priority weights of interval fuzzy preference relations

In this paper, the concept of multiplicative transitivity of a fuzzy preference relation, as defined by Tanino [T. Tanino, Fuzzy preference orderings in group decision-making, Fuzzy Sets and Systems 12 (1984) 117–131], is extended to discover whether an interval fuzzy preference relation is consistent or not, and to derive the priority vector of a consistent interval fuzzy preference relation. We achieve this by introducing the concept of interval multiplicative transitivity of an interval fuzzy preference relation and show that, by solving numerical examples, the test of consistency and the weights derived by the simple formulas based on the interval multiplicative transitivity produce the same results as those of linear programming models proposed by Xu and Chen [Z.S. Xu, J. Chen, Some models for deriving the priority weights from interval fuzzy preference relations, European Journal of Operational Research 184 (2008) 266–280]. In addition, by taking advantage of interval multiplicative transitivity of an interval fuzzy preference relation, we put forward two approaches to estimate missing value(s) of an incomplete interval fuzzy preference relation, and present numerical examples to illustrate these two approaches.

A Method for Estimating Criteria Weights from Intuitionistic Preference Relations

An intuitionistic preference relation is a powerful means to express decision makers' information of intuitionistic preference over criteria in the process of multi-criteria decision making. In this paper, we first define the concept of its consistence and give the equivalent interval fuzzy preference relation of it. Then we develop a method for estimating criteria weights from it, and then extend the method to accommodate group decision making based on them And finally, we use some numerical examples to illustrate the feasibility and validity of the developed method.

MAGDM Linear-Programming Models With Distinct Uncertain Preference Structures

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.

Some models for deriving the priority weights from interval fuzzy preference relations

Interval fuzzy preference relation is a useful tool to express decision maker’s uncertain preference information. How to derive the priority weights from an interval fuzzy preference relation is an interesting and important issue in decision making with interval fuzzy preference relation(s). In this paper, some new concepts such as additive consistent interval fuzzy preference relation, multiplicative consistent interval fuzzy preference relation, etc., are defined. Some simple and practical linear programming models for deriving the priority weights from various interval fuzzy preference relations are established, and two numerical examples are provided to illustrate the developed models.

Multiple-Attribute Group Decision Making With Different Formats of Preference Information on Attributes

Interval utility values, interval fuzzy preference relations, and interval multiplicative preference relations are three common uncertain-preference formats used by decision-makers to provide their preference information in the process of decision making under fuzziness. This paper is devoted in investigating multiple-attribute group-decision-making problems where the attribute values are not precisely known but the value ranges can be obtained, and the decision-makers provide their preference information over attributes by three different uncertain-preference formats i.e., 1) interval utility values; 2) interval fuzzy preference relations; and 3) interval multiplicative preference relations. We first utilize some functions to normalize the uncertain decision matrix and then transform it into an expected decision matrix. We establish a goal-programming model to integrate the expected decision matrix and all three different uncertain-preference formats from which the attribute weights and the overall attribute values of alternatives can be obtained. Then, we use the derived overall attribute values to get the ranking of the given alternatives and to select the best one(s). The model not only can reflect both the subjective considerations of all decision-makers and the objective information but also can avoid losing and distorting the given objective and subjective decision information in the process of information integration. Furthermore, we establish some models to solve the multiple-attribute group-decision-making problems with three different preference formats: 1) utility values; 2) fuzzy preference relations; and 3) multiplicative preference relations. Finally, we illustrate the applicability and effectiveness of the developed models with two practical examples.