On group decision making with four formats of incomplete preference relations

In this paper, a new approach is proposed to solve group decision making (GDM) problems where the preference information on alternatives provided by decision makers (DMs) is represented in four formats of incomplete preference relations, i.e., incomplete multiplicative preference relations, incomplete fuzzy preference relations, incomplete additive linguistic preference relations, incomplete multiplicative linguistic preference relations. In order to make the collective opinion close each decision maker’s opinion as near as possible, an optimization model is constructed to integrate the four different formats of incomplete preference relations and to compute the collective ranking values of the alternatives. The ranking of alternatives or selection of the most desirable alternative(s) is directly obtained from the derived collective ranking values. A numerical example is also used to illustrate the applicability of the proposed approach.     

Group decision making based on incomplete multiplicative and fuzzy preference relations

The main aim of this paper is to investigate the group decision making on incomplete multiplicative and fuzzy preference relations without the requirement of satisfying reciprocity property. This paper introduces a new characterization of the multiplicative consistency condition, based on which a method to estimate unknown preference values in an incomplete multiplicative preference relation is proposed. Apart from the multiplicative consistency property among three known preference values, the method proposed also takes the multiplicative consistency property among more than three values into account. In addition, two models for group decision making with incomplete multiplicative preference relations and incomplete fuzzy preference relations are presented, respectively. Some properties of the collective preference relation are further discussed. Numerical examples are provided to make a discussion and comparison with other similar methods.     

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.     

Intuitionistic preference relations and their application in group decision making

Intuitionistic fuzzy set, characterized by a membership function and a non-membership function, was introduced by Atanassov [Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87-96]. In this paper, we define the concepts of intuitionistic preference relation, consistent intuitionistic preference relation, incomplete intuitionistic preference relation and acceptable intuitionistic preference relation, and study their various properties. We develop an approach to group decision making based on intuitionistic preference relations and an approach to group decision making based on incomplete intuitionistic preference relations respectively, in which the intuitionistic fuzzy arithmetic averaging operator and intuitionistic fuzzy weighted arithmetic averaging operator are used to aggregate intuitionistic preference information, and the score function and accuracy function are applied to the ranking and selection of alternatives. Finally, a practical example is provided to illustrate the developed approaches.     

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.     

Linear optimization modeling of consistency issues in group decision making based on fuzzy preference relations

The consistency measure is a vital basis for group decision making (GDM) based on fuzzy preference relations, and includes two subproblems: individual consistency and consensus consistency. This paper proposes linear optimization models for solving some issues on consistency of fuzzy preference relations, such as individual consistency construction, consensus model and management of incomplete fuzzy preference relations. Our proposal optimally preserves original preference information in constructing individual consistency and reaching consensus (in Manhattan distance sense), and maximizes the consistency level of fuzzy preference relations in calculating the missing values of incomplete fuzzy preference relations. Linear optimization models can be solved in very little computational time using readily available softwares. Therefore, the results in this paper are also of simplicity and convenience for the application of consistent fuzzy preference relations in GDM problems.     

Estimating incomplete information in group decision making: A framework of granular computing

A general assumption in group decision making scenarios is that of all individuals possess accurate knowledge of the entire problem under study, including the abilities to make a distinction of the degree up to which an alternative is better than other one. However, in many real world scenarios, this may be unrealistic, particularly those involving numerous individuals and options to choose from conflicting and dynamics information sources. To manage such a situation, estimation methods of incomplete information, which use own assessments provided by the individuals and consistency criteria to avoid discrepancy, have been widely employed under fuzzy preference relations. In this study, we introduce the information granularity concept to estimate missing values supporting the objective of obtaining complete fuzzy preference relations with higher consistency levels. We use the concept of granular preference relations to form each missing value as a granule of information in place of a crisp number. This offers the flexibility that is required to estimate the missing information so that the consistency levels related to the complete fuzzy preference relations are as higher as possible.     

Two flexibility degrees-driven consensus model in group decision making with intuitionistic fuzzy preference relations

A group of experts are commonly invited to find an optimal solution to a complex decision making problem. When the bipolarity of decision information should be considered in group decision making (GDM), intuitionistic fuzzy values (IFVs) have the capability to model such opinions of decision makers (DMs). This paper develops a consensus model in GDM under intuitionistic fuzzy environments with flexibility. First, it is assumed that the initial opinions of DMs are expressed as intuitionistic fuzzy preference relations (IFPRs). A novel additive consistency index is constructed to measure the deviation degree of IFPRs from fuzzy preference relations (FPRs) with additive consistency, where the non-determinacy degree of IFPRs is incorporated. The thresholds of the proposed index corresponding to IFPRs with acceptable additive consistency are discussed and computed. Second, the consensus level of DMs is defined using the similarity degree between two IFVs. An optimization problem is established by maximizing the fitness function, which is constructed by linearly combining the proposed additive consistency index and consensus level. Two flexibility degrees are offered to each DM such that the initial opinions with the bipolarity can be adjusted correspondingly. Third, individual IFPRs in GDM are optimized using the particle swarm optimization (PSO) algorithm. Numerical examples are carried out to illustrate the proposed consensus model by comparing with the existing ones. The obtained results reveal that the proposed additive consistency index can reflect the inherent property of IFPRs. Different with the previous studies, two original flexibility degrees are proposed to characterize the multi-granularity of decision information in GDM.     

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.     

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.     

Deriving the priority weights from hesitant multiplicative preference relations in group decision making

In this paper, we investigate the deviation of the priority weights from hesitant multiplicative preference relations (HMPRs) in group decision-making environments. As basic elements of HMPRs, hesitant multiplicative elements (HMEs) usually have different numbers of possible values. To correctly compute or compare HMEs, there are two principles to normalize them, i.e., the α-normalization and the β-normalization. Based on the α-normalization, we develop a new goal programming model to derive the priority weights from HMPRs in group decision-making environments. Based on the β-normalization, a consistent HMPR and an acceptably consistent HMPR are defined, and their desired properties are studied. A convex combination method is then developed to obtain interval weights from an acceptably consistent HMPR. This approach is further extended to group decision-making situations in which the experts evaluate their preferences as several HMPRs. Finally, some numerical examples are provided to illustrate the validity and applicability of the proposed models.     

Interval programming method for hesitant fuzzy multi-attribute group decision making with incomplete preference over alternatives

The aim of this study is to employ the main structure of LINMAP (LINear programming technique for Multidimensional Analysis of Preference) to propose an interval programming method for solving multi-attribute group decision making (MAGDM) problems in which the ratings of alternatives are taken as hesitant fuzzy elements (HFEs) and all pair-wise comparison judgments over alternatives are represented by interval numbers. The contribution of this study is fivefold: (1) we define the new consistency and inconsistency indices; (2) we construct an interval programming model to determine the hesitant fuzzy positive ideal solution and the optimal weights of attributes, and at the same time present a decision algorithm; (3) we discuss several special cases of the proposed model in detail; (4) we show that compared with intuitionistic fuzzy LINMAP method (Li et al., 2010), the proposed approach reveals more useful information including the interval preference information, and does not need to transform HFEs into intuitionistic fuzzy numbers but directly deals with MAGDM problems and thus obtains better final decision results; and (5) we demonstrate the applicability and implementation process of the proposed approach by using an energy project selection example.     

The fusion process with heterogeneous preference structures in group decision making: A survey

In group decision making (GDM), decision makers who have different experiential, cultural and educational backgrounds will naturally provide their preference information by heterogeneous preference structures (e.g., utility values, preference orderings, numerical preference relations and multigranular linguistic preference relations). To date, many studies have discussed GDM problems with heterogeneous preference structures. To provide a clear perspective on the fusion process with heterogeneous preference structures in GDM, this paper presents a review of three types of fusion approaches: the indirect approach, the optimization-based approach and the direct approach. Moreover, with respect to insights gained from prior researches, several open problems are proposed for the future research.     

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.     

Group decision making based on multiple types of linguistic preference relations

In this paper, we investigate group decision making problems with multiple types of linguistic preference relations. The paper has two parts with similar structures. In the first part, we transform the uncertain additive linguistic preference relations into the expected additive linguistic preference relations, and present a procedure for group decision making based on multiple types of additive linguistic preference relations. By using the deviation measures between additive linguistic preference relations, we give some straightforward formulas to determine the weights of decision makers, and propose a method to reach consensus among the individual preferences and the group’s opinion. In the second part, we extend the above results to group decision making based on multiple types of multiplicative linguistic preference relations, and finally, a practical example is given to illustrate the application of the results.     

Group decision making with fuzzy linguistic preference relations via cooperative games method

When we consider the weighting approach for group decision making with fuzzy linguistic preference relations, the groupment of experts has merely been studied. In this paper, a novel weighting approach on the basis of cooperative games method is developed. The group decision error matrix is built to reflect the deviations of all experts with given initial weighting vector. An iterative algorithm is designed to lower the sum of the decision error so that a final convergence result can be obtained. The advantage of the weighting algorithm is that it can consider the contribution of each expert and reduce the sum of decision error with increasing iteration numbers. Then an optimization model using triangular fuzzy numbers as alternatives’ weights is constructed, whose results are used to rank the alternatives. Finally, a numerical example of subjective evaluation of vehicle sound quality is considered to illustrate the feasibility and validity of the proposed weighting approach in the group decision making problem.     

A web based consensus support system for group decision making problems and incomplete preferences

Reaching a high level of consensus among experts is critical in group decision making problems. Usually, it is the moderator task to assure that the consensus process is carried out properly and, if possible, to offer recommendations to the expert in order to change their opinions and narrow their differences.In this paper we present an implemented web based consensus support system that is able to help, or even replace, the moderator in a consensus process where experts are allowed to provide their preferences using one of many types (fuzzy, linguistic and multi-granular linguistic) of incomplete preference relations.This system is based on both consistency and consensus measures and it has been designed to provide advice to the experts to increase group consensus level while maintaining the individual consistency of each expert. The consistency measures are characterized by and computed using uninorm operators. When appropriate, the system also helps experts to reduce the incompleteness of their preference relations. The web interface allows to carry out distributed consensus processes and thus, experts do not necessarily need to physically meet together.     

Prioritized multi-criteria decision making based on preference relations

There may exist priority relationships among criteria in multi-criteria decision making (MCDM) problems. This kind of problems, which we focus on in this paper, are called prioritized MCDM ones. In order to aggregate the evaluation values of criteria for an alternative, we first develop some weighted prioritized aggregation operators based on triangular norms (t-norms) together with the weights of criteria by extending the prioritized aggregation operators proposed by Yager (Yager, R. R. (2004). Modeling prioritized multi-criteria decision making. IEEE Transactions on Systems, Man, and Cybernetics, 34, 2396–2404). After discussing the influence of the concentration degrees of the evaluation values with respect to each criterion to the priority relationships, we further develop a method for handling the prioritized MCDM problems. Through a simple example, we validate that this method can be used in more wide situations than the existing prioritized MCDM methods. At length, the relationships between the weights associated with criteria and the preference relations among alternatives are explored, and then two quadratic programming models for determining weights based on multiplicative and fuzzy preference relations are developed.     

A novel group decision making method with intuitionistic fuzzy preference relations for RFID technology selection

A more scientific decision making process for radio frequency identification (RFID) technology selection is important to increase success rate of RFID technology application. RFID technology selection can be formulated as a kind of group decision making (GDM) problem with intuitionistic fuzzy preference relations (IFPRs). This paper develops a novel method for solving such problems. First, A technique for order preference by similarity to ideal solution (TOPSIS) based method is presented to rank intuitionistic fuzzy values (IFVs). To achieve higher group consensus as well as possible, we construct an intuitionistic fuzzy linear programming model to derive experts’ weights. Depending on the construction of membership and non-membership functions, the constructed intuitionistic fuzzy linear programming model is solved by three kinds of approaches: optimistic approach, pessimistic approach and mixed approach. Then to derive the ranking order of alternatives from the collective IFPR, we extend quantifier guided non-dominance degree (QGNDD) and quantifier guided dominance degree (QGDD) to intuitionistic fuzzy environment. A new two-phase ranking approach is designed to generate the ordering of alternatives based on QGNDD and QGDD. Thereby, the corresponding method is proposed for the GDM problems with IFPRs. Some generalizations on the constructed intuitionistic fuzzy linear programming model are further discussed. At length, the validity of the proposed method is illustrated with a real-world RFID technology selection example.     

Optimal consensus models for group decision making under linguistic preference relations

This paper proposes an optimal consensus model to derive weights for linguistic preference relations (LPRs). Two indexes, an individual‐to‐group consensus index (ICI) and a collective consensus index (CCI), are introduced. An iterative algorithm is presented to describe the consensus reaching process. By changing the weights and modifying a pair of individuals' comparison judgments—which have largest deviation value to the group judgments—the consensus reaching process can terminate, while both ICI and CCI are controlled with predefined thresholds. The algorithm aims to preserve the decision makers’ original information as much as possible. The model and algorithm are then extended to handle the uncertain additive LPRs. Finally, two examples are given to show the effectiveness of the proposed methods.