Additive Reciprocal Matrices Research Articles

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

The existing works usually assume that fuzzy preference relations (FPRs) exhibit additively reciprocal property. In this study, we investigate the situation that FPRs have no additively reciprocal property and address its implication to decision making under uncertainty. First, the considered situation is captured by proposing the novel concept of additively reciprocal property breaking (ARPB) for FPRs. It is proved that FPRs with ARPB, interval additive reciprocal preference relations (IARPRs) and intuitionistic fuzzy preference relations (IFPRs) are transformed each other. Second, the equivalence between FPRs with ARPR and additive reciprocal matrices (ARMs) is defined by keeping the inconsistency level unchanged. An optimization model is proposed to adjust an FPR without weak transitivity to an ARM with weak transitivity. Third, a novel decision making model is developed by considering weak transitivity of FPRs as the minimum requirement of rational choices. A new algorithm is elaborated on where decision information could be expressed as FPRs with ARPR, IARPRs and IFPRs, respectively. Finally, numerical results are reported to show the advantages of the developed model by comparing with the existing ones. The observations reveal that the concept of ARPB offers a novel understanding for uncertainty in decision information.

Measuring consistency of interval-valued preference relations: comments and comparison

The concepts of consistency definition and consistency index are usually used to measure the consistency of a preference relation. When interval numbers are used to express the preference information, the consistency of the derived interval-valued preference relations (IVPRs) is worth being investigated. In this study, a comment is provided for the ideas behind consistency definitions and consistency indexes of interval multiplicative reciprocal matrices (IMRMs) and interval additive reciprocal matrices (IARMs), respectively. A comparison is made by considering the two kinds of consistency definitions of IVPRs. It is found that the method of defining the consistency of IVPRs in terms of the imaginary intervals is equivalent to that of defining the approximate consistency. Numerical examples are reported to illustrate the differences of the two consistency definitions of IVPRs. The observations illustrate that the fundamental inconsistency of IVPRs is compatible with the underlying idea of fuzzy sets. It is revealed that a consistent preference relation is only a particular case with a fixed value of the defined consistency index. In general, the consistency index could be used to quantify the deviation degree from a consistent real-valued preference relation.

On weak consistency of interval additive reciprocal matrices

When one estimates the importance of alternatives under rational choice, it is natural to avoid self-contradiction from the viewpoint of psychology. Due to the vagueness encountered in a manner analogous to human thought, decision makers always exhibit limited rationality. The judgements could be expressed as interval-valued comparison matrices within the framework of analytic hierarchy process. In this study, for additive reciprocal matrices (ARMs), three axiomatic properties are proposed to characterize the additive consistency and the multiplicative consistency under fully rational behavior. For interval additive reciprocal matrices (IARMs), the concept of weak consistency is used to capture the limited rationality. By weakening some axiomatic properties of consistent ARMs, the reasonable properties of IARMs with weak consistency are presented. Two kinds of IARMs satisfying the properties of weak consistency are analyzed and some comparisons are offered. It is observed that the consistency of ARMs can be defined exactly and characterized by using the axiomatic properties. The properties of characterizing the consistency degree of IARMs should be captured by weakening the axiomatic ones of consistent ARMs. The proposed approach visualizes the development process starting from cardinal consistency of numeric-valued preference relations to weak consistency of interval-valued comparison matrices.

An inconsistency index of interval additive reciprocal matrices with application to group decision making

Inconsistency measure and aggregation of decision information are two important issues in group decision making. In this study, interval additive reciprocal matrices are used to express the opinions of decision makers within the framework of fuzzy analytic hierarchy process (FAHP). Some existing consistency indexes of interval additive reciprocal matrices are reviewed and the existing shortcomings are shown. By considering the randomness experienced by decision makers in comparing alternatives, a novel inconsistency index named as the weak-consistency index is introduced. Some properties of the proposed weak-consistency index are studied in detail. Consistent additive reciprocal matrices are most useful for the ideal cases, while interval-valued matrices are their softened versions. In order to aggregate individual interval additive reciprocal matrices, a new aggregation operator based on the novel consistency index is proposed. The properties of the collective interval additive reciprocal matrix are further addressed. Numerical results are reported to illustrate the proposed weak-consistency index and group decision making model. Comparisons with some existing consistency indexes and group decision making models reveal that the known shortcomings are overcome.

A consensus model for group decision making under additive reciprocal matrices with flexibility

A consensus model is proposed for group decision making (GDM) with additive reciprocal preference relations based on the technique for order preference by similarity to an ideal solution (TOPSIS). The initial positions of decision makers are expressed as additive reciprocal matrices. A method of obtaining a collective ideal additive reciprocal matrix is proposed in terms of individual ones. A novel fitness function is constructed through a linear combination of two objective functions involving the consensus degree within the group of experts and the consistency measure of individual preference relations. A new algorithm based on particle swarm optimization (PSO) is proposed to model the dynamic and iterative consensus process of GDM with flexibility. Numerical results are reported to illustrate the proposed model through some comparisons with the existing ones. The weight variance is proposed to characterize the dispersion degree of the weights of alternatives, and analyzed by considering the effects of the flexibility degree and the constraint conditions. The observations reveal that with an increasing of the flexibility degree, the dominant alterative may be chosen, or any solution may not be reached by a group of experts.

A group decision making model based on triangular fuzzy additive reciprocal matrices with additive approximation-consistency

A group decision making (GDM) model is proposed when the experts evaluate their opinions through triangular fuzzy numbers. First, it is pointed out that the preference relations with triangular fuzzy numbers are inconsistent in nature. In order to distinguish the typical consistency, the concept of additive approximation-consistency is proposed for triangular fuzzy additive reciprocal matrices. The properties of triangular fuzzy additive reciprocal matrices with additive approximation-consistency are studied in detail. Second, using (n − 1) restricted preference values, a triangular fuzzy additive reciprocal preference relation with additive approximation-consistency is constructed. Third, a novel compatibility degree among triangular fuzzy additive reciprocal preference relations is defined. It is further applied to introduce the compatibility-degree induced ordered weighted averaging (CD-IOWA) operator for generating a collective triangular fuzzy additive reciprocal matrix with additive approximation-consistency. Finally, a new algorithm for the group decision-making problem with triangular fuzzy additive reciprocal preference relations is presented. A numerical example is carried out to illustrate the proposed definitions and algorithm.

A decision-making model based on interval additive reciprocal matrices with additive approximation-consistency

In order to capture the rationality experienced by decision makers in choosing the best alternative(s), of much interest is to study the consistency of preference relations in the analytic hierarchy process (AHP). When the typical AHP is extended by using fuzzy numbers to evaluate the opinions of decision makers, the consistency of fuzzy judgments is worth to be considered. In this paper, we analyze some definitions of interval additive reciprocal matrices with additive consistency. It is concluded that interval additive reciprocal matrices are inconsistent in essence. The concept of additive approximation-consistency of interval additive reciprocal matrices is proposed. Moreover, a novel exchange method is designed to enumerate all permutations of alternatives for checking the approximation-consistency. By considering the randomness exhibited in pairwisely comparing alternatives, a method of obtaining the interval weight vector is given. A new algorithm of solving the decision making problem with interval additive reciprocal matrices is proposed. Finally, two numerical examples are carried out to illustrate the new definition and some comparisons are offered.

A novel group decision-making model based on triangular neutrosophic numbers

In group decision-making (GDM) model, experts often evaluate their opinion by using triangular fuzzy numbers. The preference relations with triangular fuzzy numbers are in consistent in nature, so we turned to neutrosophic in this paper. It is very important to take into account consistency of expert opinion and consensus degree in GDM. In order to distinguish the typical consistency, the concept of additive approximation consistency is proposed for triangular neutrosophic additive reciprocal matrices. The properties of triangular neutrosophic additive reciprocal matrices with additive approximation consistency are studied in detail. Second, by using $$(n-1)$$ restricted preference values, a triangular neutrosophic additive reciprocal preference relation with additive approximation consistency is constructed. The differences among expert’s opinions are measured using consensus degree. For generating a collective triangular neutrosophic additive reciprocal matrix with additive approximation consistency, the neutrosophic triangular weighted aggregation operator is used. Finally, a novel algorithm for the group decision-making problem with triangular neutrosophic additive reciprocal preference relations is presented. A numerical example is carried out to illustrate the proposed definitions and algorithm.

On Consistency in AHP and Fuzzy AHP

Abstract The analytic hierarchy process (AHP) is used widely for analyzing decisions made in various real-world applications. Its basic idea is to construct a hierarchy of concepts encountered in a given decision problem and to choose the best alternative according to pairwise comparison matrices given by the decision maker. Under the assumption of fully rational economics, a reasonable decision should be consistent. It becomes an important issue on how to analyze and ensure the consistency of comparison matrices together with the judgments of the decision maker. The main objectives of the present paper are threefold. First, we review the basic idea and methods used to define the consistency and the transitivity of multiplicative reciprocal matrices, additive reciprocal matrices and comparison matrices with fuzzy interval and triangular fuzzy numbers. The existing controversy behind the applications of fuzzy set theory to the AHP in the literature is presented. Second, the consistency of the collective comparison matrices in group decision making based on AHP and fuzzy AHP is further analyzed. We point out that the weak consistency of preference relations with fuzzy numbers in fuzzy AHP and group decision making should be investigated comprehensively. Third, under the consideration of the vagueness in the process of evaluating the judgements, a new concept of fuzzy consistency of comparison matrices in the AHP is given.

A Modified TOPSIS Method for Obtaining the Associated Weights of the OWA-Type Operators

The ordered weighted averaging OWA operator proposed by Yager and its some extensions have been extensively utilized to perform the mean-type aggregation of individual preference relations in group decision making. An important issue in the theory of the OWA-type operators is the determination of the associated weights. In the present paper, the technique for order preference by similarity to ideal solution is modified to propose a new method for generating the associated weights under the environment of a group of additive reciprocal matrices. First, a consistent additive reciprocal matrix is obtained from each additive reciprocal matrix. It is further used to generate an ideal additive reciprocal matrix by using the arithmetic averaging operator. Then, a similarity degree between every additive reciprocal matrix and the ideal one is defined. Based on the similarity degree, the associated weights are determined by making use of an exponential function and they show that more importance is given to that with more similarity degree. Finally, a numerical example is carried out to illustrate the given method.

A group decision making model based on a generalized ordered weighted geometric average operator with interval preference matrices

This paper presents a model for a group decision-making problem with interval preference matrices. First, a new definition of interval additive reciprocal matrices with multiplicative consistency is given. Transformation methods between interval additive and multiplicative reciprocal matrices are proposed and applied to homogenize interval preference matrices. Second, the consistency of interval multiplicative reciprocal matrices is utilized to propose a generalized ordered weighted geometric averaging operator, which permits the aggregation of interval multiplicative reciprocal matrices in such a way that more important weight is given to that with more consistency. In order to avoid a misleading solution, the consistency and acceptable consistency of the collective interval multiplicative reciprocal matrix are studied in detail. Finally, a new algorithm is presented to solve the group decision-making problem with interval preference matrices. Numerical results are carried out to illustrate the given definitions, methods and algorithm, respectively.