Consistent Interval Fuzzy Preference Relations Research Articles

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.

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.

On extension of multiplicative consistency to interval fuzzy preference relations

Various definitions of consistency for interval fuzzy preference relations have been proposed in the literature. The aim of this paper is to review the definitions of multiplicative consistency based on the extension of Tanino’s multiplicative-transitivity property and to point out their drawbacks. In particular, some of the definitions proposed in the literature are not invariant under permutation of objects in interval fuzzy preference relations and some of them violate reciprocity of the pairwise comparisons of objects, which is not acceptable. The weak form of multiplicative-consistency defined by Xu and Chen (Eur J Oper Res 184:266–280, 2008) is approved as the only one appropriate among all definitions examined in the paper. Further, a new definition of multiplicative consistency that is much stronger than the definition proposed by Xu and Chen (Eur J Oper Res 184:266–280, 2008) is introduced. Tools for verifying both the multiplicative consistency defined in this paper and the multiplicative weak consistency defined by Xu and Chen (Eur J Oper Res 184:266–280, 2008) are proposed, and some interesting properties of both multiplicatively consistent and multiplicatively weakly consistent interval fuzzy preference relations are demonstrated. Finally, numerical examples are provided in order to illustrate and compare both types of consistency.

On additive consistency of interval fuzzy preference relations

Various definitions of consistent interval fuzzy preference relations have been proposed in the literature. This paper aims to review the definitions of additive consistency based on the interval extension of Tanino’s additive-transitivity property. It is demonstrated that some of the definitions are not invariant under permutation of objects in interval fuzzy preference relations and some of them violate the reciprocity of pairwise comparisons of objects, which is not acceptable. Afterwards, two new definitions of consistency, additive consistency and additive weak consistency, based on a proper interval extension of Tanino’s additive-transitivity property are proposed. Both definitions are invariant under permutation of objects and do not violate the additive reciprocity of pairwise comparisons in interval fuzzy preference relations. Further, useful tools for verifying the consistency are given, and some interesting properties of both additively consistent and additively weakly consistent interval fuzzy preference relations are demonstrated. The new definitions of consistency are compared with the reviewed definitions on several illustrative examples and on a real-life decision-making problem.

Multiplicative consistency analysis for interval fuzzy preference relations: A comparative study

To examine the multiplicative consistency of interval fuzzy preference relations (IFPRs), this paper first analyzes the limitations associated with the previous consistency concepts. Accordingly, a new consistency concept is defined that is an extension of the crisp case and overcomes limitations in the previous concepts. Next, a linear programming model to judge the consistency of IFPRs is constructed, and an approach to derive multiplicative consistent IFPRs is introduced. Furthermore, goal-programming models to determine missing values in an incomplete IFPR are constructed that have the highest consistent level with respect to known values. Moreover, a multiplicative consistency and consensus based method for group decision making with IFPRs is developed that can address incomplete and inconsistent cases. Finally, two practical decision-making problems are offered to demonstrate the feasibility and efficiency of the new method, and an analysis of a numerical and theoretical comparison with several related methods is performed.

Approaches to improving consistency of interval fuzzy preference relations

This article introduces a consistency index for measuring the consistency level of an interval fuzzy preference relation (IFPR). An approach is then proposed to construct an additive consistent IFPR from a given inconsistent IFPR. By using a weighted averaging method combining the original IFPR and the constructed consistent IFPR, a formula is put forward to repair an inconsistent IFPR to generate an IFPR with acceptable consistency. An iterative algorithm is subsequently developed to rectify an inconsistent IFPR and derive one with acceptable consistency and weak transitivity. The proposed approaches can not only improve consistency of IFPRs but also preserve the initial interval uncertainty information as much as possible. Numerical examples are presented to illustrate how to apply the proposed approaches.

A note on “Incomplete interval fuzzy preference relations and their applications”

In a very recent paper by Xu et al. (2014), an interval-arithmetic-based equation is introduced to define additively consistent interval fuzzy preference relations (IFPRs), and some properties of consistent IFPRs are put forward and used to define additive consistency for incomplete IFPRs. This note shows that such additive consistency definitions are technically wrong, and corrects errors in the definitions and properties of additively consistent IFPRs.

Logarithmic least squares prioritization and completion methods for interval fuzzy preference relations based on geometric transitivity

This article introduces the notion of geometric transitivity, which may be used to define consistent interval fuzzy preference relations. Some useful properties are presented for consistent interval fuzzy preference relations. A close investigation reveals that existing methods for determining the analogous property of consistency are not robust to permutations of the pairwise judgments and not able to reflect the hesitancy in the decision-maker’s preference. A parametric transformation formula is put forward to convert a normalized interval weight vector into a consistent interval fuzzy preference relation. By minimizing the squared difference between the logarithm of the ratio of the original judgment and the logarithm of the ratio of the converted consistent one, a logarithmic least squares model is developed to derive interval weights from any interval fuzzy preference relation. Based on geometric transitivity, a logarithmic least squares model is established to rectify inconsistency for complete and inconsistent interval fuzzy preference relations, and a logarithmic least squares completion approach is further developed to estimate missing values for incomplete interval fuzzy preference relations. Numerical examples are furnished to show the validity of the proposed models and compare with other existing methods.

Note on “Some models for deriving the priority weights from interval fuzzy preference relations”

Deriving accurate interval weights from interval fuzzy preference relations is key to successfully solving decision making problems. Xu and Chen (2008) proposed a number of linear programming models to derive interval weights, but the definitions for the additive consistent interval fuzzy preference relation and the linear programming model still need to be improved. In this paper, a numerical example is given to show how these definitions and models can be improved to increase accuracy. A new additive consistency definition for interval fuzzy preference relations is proposed and novel linear programming models are established to demonstrate the generation of interval weights from an interval fuzzy preference relation.

Incomplete interval fuzzy preference relations and their applications

This paper investigates incomplete interval fuzzy preference relations. A characterization, which is proposed by Herrera-Viedma et al. (2004), of the additive consistency property of the fuzzy preference relations is extended to a more general case. This property is further generalized to interval fuzzy preference relations (IFPRs) based on additive transitivity. Subsequently, we examine how to characterize IFPR. Using these new characterizations, we propose a method to construct an additive consistent IFPR from a set of n−1 preference data and an estimation algorithm for acceptable incomplete IFPRs with more known elements. Numerical examples are provided to illustrate the effectiveness and practicality of the solution process.

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).

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.

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.

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.

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.

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.