A consistency model for group decision making problems with interval multiplicative preference relations

The main aim of this paper is to present a consistency model for interval multiplicative preference relation (IMPR). To measure the consistency level for IMPR, a referenced consistent IMPR of a given IMPR is defined, which has the minimum logarithmic distance from the given IMPR. Based on the referenced consistent IMPR, the consistency level of an IMPR can be measured and an IMPR with unacceptable consistency can be adjusted by a proposed algorithm such that the revised IMPR is of acceptable consistency. A consistency model for group decision making (GDM) problems with IMPRs is proposed to obtain the collective IMPR with highest consistency level. Numerical examples are provided to illustrate the validity of the proposed approaches in decision making.     

A consistency and consensus based decision support model for group decision making with multiplicative preference relations

In group decision making (GDM) with multiplicative preference relations (also known as pairwise comparison matrices in the Analytical Hierarchy Process), to come to a meaningful and reliable solution, it is preferable to consider individual consistency and group consensus in the decision process. This paper provides a decision support model to aid the group consensus process while keeping an acceptable individual consistency for each decision maker. The concept of an individual consistency index and a group consensus index is introduced based on the Hadamard product of two matrices. Two algorithms are presented in the designed support model. The first algorithm is utilized to convert an unacceptable preference relation to an acceptable one. The second algorithm is designed to assist the group in achieving a predefined consensus level. The main characteristics of our model are that: (1) it is independent of the prioritization method used in the consensus process; (2) it ensures that each individual multiplicative preference relation is of acceptable consistency when the predefined consensus level is achieved. Finally, some numerical examples are given to verify the effectiveness of our model.     

A consensus model for group decision making under interval type-2 fuzzy environment

We propose a new consensus model for group decision making (GDM) problems, using an interval type-2 fuzzy environment. In our model, experts are asked to express their preferences using linguistic terms characterized by interval type-2 fuzzy sets (IT2 FSs), because these can provide decision makers with greater freedom to express the vagueness in real-life situations. Consensus and proximity measures based on the arithmetic operations of IT2 FSs are used simultaneously to guide the decision-making process. The majority of previous studies have taken into account only the importance of the experts in the aggregation process, which may give unreasonable results. Thus, we propose a new feedback mechanism that generates different advice strategies for experts according to their levels of importance. In general, experts with a lower level of importance require a larger number of suggestions to change their initial preferences. Finally, we investigate a numerical example and execute comparable models and ours, to demonstrate the performance of our proposed model. The results indicate that the proposed model provides greater insight into the GDM process.     

Ratio-based similarity analysis and consensus building for group decision making with interval reciprocal preference relations

Similarity analysis and preference information aggregation are two important issues for consensus building in group decision making with preference relations. Pairwise ratings in an interval reciprocal preference relation (IRPR) are usually regarded as interval-valued And-like representable cross ratios (i.e., interval-valued cross ratios for short) from the multiplicative perspective. In this paper, a ratio-based formula is introduced to measure similarity between a pair of interval-valued cross ratios, and its desirable properties are provided. We put forward ratio-based similarity measurements for IRPRs. An induced interval-valued cross ratio ordered weighted geometric (IIVCROWG) operator with interval additive reciprocity is developed to aggregate interval-valued cross ratio information, and some properties of the IIVCROWG operator are presented. The paper devises an importance degree induced IRPR ordered weighted geometric operator to fuse individual IRPRs into a group IRPR, and discusses the derivation of its associated weights. By employing ratio-based similarity measurements and IIVCROWG-based aggregation operators, a soft consensus model including a generation mechanism of feedback recommendation rules is further proposed to solve group decision making problems with IRPRs. Three numerical examples are examined to illustrate the applicability and effectiveness of the developed models.     

Group decision making with 2-tuple intuitionistic fuzzy linguistic preference relations

The aim of this paper is to propose a new type of preference relation, the intuitionistic fuzzy linguistic preference relation (IFLPR). Taking as base the 2-tuple fuzzy linguistic representation model, we introduce the definition of the IFLPR, and its transitivity properties. We present an approach to group decision making based on IFLPRs and incomplete-IFLPRs, respectively. The score function and accuracy function are applied to the ranking and selection of alternatives. Finally, we give an example of IFLPRs in group decision making, and a comparative of the exploitation of the IFLPR with the exploitation of the traditional fuzzy linguistic preference relations.     

Multi-criteria group decision making with incomplete hesitant fuzzy preference relations

In order to simulate the hesitancy and uncertainty associated with impression or vagueness, a decision maker may give her/his judgments by means of hesitant fuzzy preference relations in the process of decision making. The study of their consistency becomes a very important aspect to avoid a misleading solution. This paper defines the concept of additive consistent hesitant fuzzy preference relations. The characterizations of additive consistent hesitant fuzzy preference relations are studied in detail. Owing to the limitations of the experts’ professional knowledge and experience, the provided preferences in a hesitant fuzzy preference relation are usually incomplete. Consequently, this paper introduces the concepts of incomplete hesitant fuzzy preference relation, acceptable incomplete hesitant fuzzy preference relation, and additive consistent incomplete hesitant fuzzy preference relation. Then, two estimation procedures are developed to estimate the missing information in an expert's incomplete hesitant fuzzy preference relation. The first procedure is used to construct an additive consistent hesitant fuzzy preference relation from the lowest possible number, (n − 1), of pairwise comparisons. The second one is designed for the estimation of missing elements of the acceptable incomplete hesitant fuzzy preference relations with more known judgments. Moreover, an algorithm is given to solve the multi-criteria group decision making problem with incomplete hesitant fuzzy preference relations. Finally, a numerical example is provided to illustrate the solution processes of the developed algorithm and to verify its effectiveness and practicality.     

A new consensus model for group decision making using fuzzy ontology

Involving many people in decision making does not guarantee success. In practice, there are always individuals who try to exert pressure in order to persuade others who could easily be influenced. In these situations, classical group decision making models fail. Thus, there is still the necessity of developing tools to help users reach collective decisions as if they participated in a real face to face meeting. In such a way, a proper negotiation process can lead to successful solutions. Therefore, we propose a new consensus model to deal with the psychology of negotiation by using the power of a fuzzy ontology as weapon of influence in order to improve group decision scenarios making them more precise and realistic. In addition, the use of a fuzzy ontology gives us the possibility to take into account large sets of alternatives.     

Linguistic labels for expressing fuzzy preference relations infuzzy group decision making

This paper proposes improvements to pairwise group decision making based on fuzzy preference relations in three ways. First, it extends the fuzzy preference relation representation using linguistic labels. Decision makers may express their preference relations in linguistic labels which are more practically implementable for solving group decision making problems. Second, it modifies the computational procedures by using fuzzy sets representation and computation, and by avoiding the use of strict threshold values. This allows natural representation, preserves the preference accuracy, and produces more intuitively meaningful solutions. Finally, it considers fuzzy criteria of the alternatives explicitly. Solutions are first derived based on each criterion, and then by using neat ordered weighted average (OWA) operator the final solutions which accommodate all criteria are determined. The proposed method is verified for solving fuzzy group decision making problems, i.e., advertising media selection cases.     

A fuzzy group decision making model with trapezoidal fuzzy preference relations based on compatibility measure and COWGA operator

This paper proposes a fuzzy group decision-making model based on a logarithm compatibility measure with multiplicative trapezoidal fuzzy preference relations (MTFPRs) based on a continuous ordered weighted geometric averaging (COWGA) operator. New concepts are presented to measure deviation between MTFPR and its expected fuzzy preference relation. Then, an iterative algorithm is developed to help individual MTFPR reach acceptable compatibility. To determine the weights of decision makers, an optimal model is constructed using group logarithm compatibility index COWGA operator. Finally, we illustrate an example to show how it works and compare it with the existing methods. The main advantages of the proposed approach are the following: (1) The COWGA operator makes decision making more flexible; (2) an iterative and convergent algorithm is proposed to improve the compatibility of MTFPR; (3) decision makers’ weights in group decision making are determined by an optimal model based on a logarithm compatibility measure.     

Solving multi-criteria decision making with incomplete linguistic preference relations

When there are n criteria or alternatives in a decision matrix, a pairwise comparison methodology of analytic hierarchy process (AHP) with the time of n(n ? 1)/2 is frequently used to select, evaluate or rank the neighboring alternatives. But while the number of criteria or comparison level increase, the efficiency and consistency of a decision matrix decrease. To solve such problems, this study therefore uses horizontal, vertical and oblique pairwise comparisons algorithm to construct multi-criteria decision making with incomplete linguistic preference relations model (InLinPreRa). The use of pairwise comparisons will not produce the inconsistency, even allows every decision maker to choose an explicit criterion or alternative for index unrestrictedly. When there are n criteria, only n ? 1 pairwise comparisons need to be carried out, then one can rest on incomplete linguistic preference relations to obtain the priority value of alternative for the decision maker’s reference. The decision making assessment model that constructed by this study can be extensively applied to every field of decision science and serves as the reference basis for the future research.     

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

Compatibility is a very efficient tool for measuring the consensus level in group decision making (GDM) problems. The lack of acceptable compatibility can lead to unsatisfied or even incorrect results in GDM problems. Preference relations can be given in various forms, one of which called intuitionistic multiplicative preference relation is a new developed preference structure that uses an unsymmetrical scale (Saaty's 1–9 scale) to express the decision maker's preferences instead of the symmetrical scale in an intuitionistic fuzzy preference relation. This new preference relation can reflect our intuition more objectively. In this paper, we first develop some compatibility measures for intuitionistic multiplicative values and intuitionistic multiplicative preference relations in GDM. Their desirable properties are also studied in detail. Furthermore, based on compatibility measures, we further develop two different consensus models with respect to intuitionistic multiplicative preference relations for checking, reaching and improving the group consensus level. Finally, a numerical example is given to illustrate the effectiveness of our measures and models.     

Compatibility of interval fuzzy preference relations with the COWA operator and its application to group decision making

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.     

A consensus model for group decision making under trapezoidal fuzzy numbers environment

Natural linguistic terms can preferably express the opinions of decision makers in complicated decision environment. Group decision making with multiplicative trapezoidal fuzzy preference relations transforming from natural linguistic terms attracts the attention of researchers for its important research significant. The developed approach is based on consensus improving process by using a new similarity measure and a trapezoidal fuzzy power ordered weighted geometric averaging (TFPOWGA) operator. In order to introduce this approach, firstly, trapezoidal fuzzy power geometric averaging operator and TFPOWGA operator are presented to aggregate trapezoidal fuzzy numbers (TFNs). Secondly, a new similarity measure for TFNs is introduced by combining centroids and areas of TFNs. We further propose a new consensus improving algorithm that consists of consensus measure and a dynamic feedback mechanism containing a multi-objective optimization model and some indirect rules. And then a selection stage is described to rank the alternatives. At last, an example is implemented to demonstrate effectiveness of the approach.

     

A consensus model for multiperson decision making with different preference structures

In this paper, we present a consensus model for multiperson decision making (MPDM) problems with different preference structures based on two consensus criteria: 1) a consensus measure which indicates the agreement between experts' opinions and 2) a measure of proximity to find out how far the individual opinions are from the group opinion. These measures are calculated by comparing the positions of the alternatives between the individual solutions and collective solution. In such a way, the consensus situation is evaluated in each moment in a more realistic way. With these measures, we design a consensus support system that is able to substitute the actions of the moderator. In this system, the consensus measure is used to guide the consensus process until the final solution is achieved while the proximity measure is used to guide the discussion phases of the consensus process. The consensus support system has a feedback mechanism to guide the discussion phases based on the proximity measure. This feedback mechanism is based on simple and easy rules to help experts change their opinions in order to obtain a degree of consensus as high as possible. The main improvement of this consensus model is that it supports consensus process automatically, without moderator, and, in such a way, the possible subjectivity that the moderator can introduce in the consensus process is avoided.     

A new approach to fuzzy group decision making with trapezoidal fuzzy preference relations by using compatibility measure

This paper aims to develop a new approach to deal with fuzzy group decision making (GDM) with additive trapezoidal fuzzy preference relations (ATFPRs) by using compatibility measure. We firstly present some concepts of compatibility index and expected preference relation (PR) for ATFPR and then propose a compatibility improving algorithm to help each individual PR achieve acceptable compatibility . Moreover, a least deviation model is provided to obtain the priority vector. Besides, based on the criterion of minimizing the compatibility index, we put forward an optimal model to determine the weights of experts in GDM. Finally, the GDM process with compatibility of ATFPRs is presented, and an illustrative example is utilized to verify the developed approach . The main features of our approach are that: (1) It guarantees that each individual ATFPR is acceptably compatible by using compatibility improving algorithm. (2) It ensures that experts’ weights in group aggregation are determined objectively by optimal model.

     

A distance-based aggregation approach for group decision making with interval preference orderings

Xu (2013) proposed a nonlinear programming model to derive an exact formula to determine the experts’ relative importance weights for the group decision making (GDM) with interval preference orderings. However, in this study, we show that the exact formula to determine the weight vector which always equals to w = (1/m, 1/m, … , 1/m)^T (m is the number of experts). In this paper, we propose a distance-based aggregation approach to assess the relative importance weights for GDM with interval preference orderings. Relevant theorems are offered to support the proposed approach. After that, by using the weighted arithmetic averaging operator, we obtain the aggregated virtual interval preference orderings. We propose a possibility degree formula to compare two virtual interval preference orderings, then rank and select the alternatives. The proposed method is tested by two numerical examples. Comparative analysis are provided to show the advantages and effectiveness of the proposed method.     

A flexible consensus scheme for multicriteria group decision making under linguistic assessments

The present paper proposes a flexible consensus scheme for group decision making, which allows one to obtain a consistent collective opinion, from information provided by each expert in terms of multigranular fuzzy estimates. It is based on a linguistic hierarchical model with multigranular sets of linguistic terms, and the choice of the most suitable set is a prerogative of each expert. From the human viewpoint, using such model is advantageous, since it permits each expert to utilize linguistic terms that reflect more adequately the level of uncertainty intrinsic to his evaluation. From the operational viewpoint, the advantage of using such model lies in the fact that it allows one to express the linguistic information in a unique domain, without losses of information, during the discussion process.The proposed consensus scheme supposes that the moderator can interfere in the discussion process in different ways. The intervention can be a request to any expert to update his opinion or can be the adjustment of the weight of each expert’s opinion. An optimal adjustment can be achieved through the execution of an optimization procedure that searches for the weights that maximize a corresponding soft consensus index.In order to demonstrate the usefulness of the presented consensus scheme, a technique for multicriteria analysis, based on fuzzy preference relation modeling, is utilized for solving a hypothetical enterprise strategy planning problem, generated with the use of the Balanced Scorecard methodology.     

A consensus reaching model for 2-tuple linguistic multiple attribute group decision making with incomplete weight information

The aim of this paper is to put forward a consensus reaching method for multi-attribute group decision-making (MAGDM) problems with linguistic information, in which the weight information of experts and attributes is unknown. First, some basic concepts and operational laws of 2-tuple linguistic label are introduced. Then, a grey relational analysis method and a maximising deviation method are proposed to calculate the incomplete weight information of experts and attributes respectively. To eliminate the conflict in the group, a weight-updating model is employed to derive the weights of experts based on their contribution to the consensus reaching process. After conflict elimination, the final group preference can be obtained which will give the ranking of the alternatives. The model can effectively avoid information distortion which is occurred regularly in the linguistic information processing. Finally, an illustrative example is given to illustrate the application of the proposed method and comparative analysis with the existing methods are offered to show the advantages of the proposed method.     

Group decision making with linguistic preference relations with application to supplier selection

Linguistic preference relation is a useful tool for expressing preferences of decision makers in group decision making according to linguistic scales. But in the real decision problems, there usually exist interactive phenomena among the preference of decision makers, which makes it difficult to aggregate preference information by conventional additive aggregation operators. Thus, to approximate the human subjective preference evaluation process, it would be more suitable to apply non-additive measures tool without assuming additivity and independence. In this paper, based on λ-fuzzy measure, we consider dependence among subjective preference of decision makers to develop some new linguistic aggregation operators such as linguistic ordered geometric averaging operator and extended linguistic Choquet integral operator to aggregate the multiplicative linguistic preference relations and additive linguistic preference relations, respectively. Further, the procedure and algorithm of group decision making based on these new linguistic aggregation operators and linguistic preference relations are given. Finally, a supplier selection example is provided to illustrate the developed approaches.     

Managing the consensus in group decision making in an unbalanced fuzzy linguistic context with incomplete information

To solve group decision-making problems we have to take in account different aspects. On the one hand, depending on the problem, we can deal with different types of information. In this way, most group decision-making problems based on linguistic approaches use symmetrically and uniformly distributed linguistic term sets to express experts’ opinions. However, there exist problems whose assessments need to be represented by means of unbalanced linguistic term sets, i.e., using term sets which are not uniformly and symmetrically distributed. On the other hand, there may be cases in which experts do not have an in-depth knowledge of the problem to be solved. In such cases, experts may not put their opinion forward about certain aspects of the problem and, as a result, they may present incomplete information. The aim of this paper is to present a consensus model to help experts in all phases of the consensus reaching process in group decision-making problems in an unbalanced fuzzy linguistic context with incomplete information. As part of this consensus model, we propose an iterative procedure using consistency measures to estimate the incomplete information. In addition, the consistency measures are used together with consensus measures to guided the consensus model. The main novelty of this consensus model is that it supports the management of incomplete unbalanced fuzzy linguistic information and it allows to achieve consistent solutions with a great level of agreement.