Qualitative hesitant fuzzy group decision making: An additively consistent probability and consensus-based perspective

Hesitant fuzzy linguistic preference relations (HFLPRs) can efficiently denote the hesitant qualitative judgments of decision makers. Consistency and consensus are two critical topics in group decision making (GDM) with preference relations. This paper uses the additively consistent concept for linguistic fuzzy preference relations (LFPRs) to give an additive consistency definition for HFLPRs. To judge the additive consistency of HFLPRs, 0-1 mixed programming models (0-1-MPMs) are constructed. Meanwhile, additive-consistency-based 0-1-MPMs to ascertain missing values in incomplete HFLPRs are established. Following the consistent probability of LFPRs, an algorithm to calculate the linguistic priority weighting vector is presented. In consideration of the consensus of GDM, a consistency-probability-distance-measure-based consensus index is defined, and an interactive improving consensus method is provided. Finally, a method for GDM with HFLPRs is offered that can address incomplete and inconsistent cases. Meanwhile, numerical examples are offered, and comparative analysis is made.     

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.     

Consistency and consensus models for group decision-making with uncertain 2-tuple linguistic preference relations

Due to the uncertainty of the decision environment and the lack of knowledge, decision-makers may use uncertain linguistic preference relations to express their preferences over alternatives and criteria. For group decision-making problems with preference relations, it is important to consider the individual consistency and the group consensus before aggregating the preference information. In this paper, consistency and consensus models for group decision-making with uncertain 2-tuple linguistic preference relations (U2TLPRs) are investigated. First of all, a formula which can construct a consistent U2TLPR from the original preference relation is presented. Based on the consistent preference relation, the individual consistency index for a U2TLPR is defined. An iterative algorithm is then developed to improve the individual consistency of a U2TLPR. To help decision-makers reach consensus in group decision-making under uncertain linguistic environment, the individual consensus and group consensus indices for group decision-making with U2TLPRs are defined. Based on the two indices, an algorithm for consensus reaching in group decision-making with U2TLPRs is also developed. Finally, two examples are provided to illustrate the effectiveness of the proposed algorithms.     

Consistency and consensus reaching process for group decision making based on complete interval distributed preference relations under social network analysis

Group decision-making (GDM) problems often consist of many indeterminacy factors in realistic situation. How to cope with consistency and consensus under uncertain circumstance are two critical issues in pairwise comparison based GDM problems. In this paper, we firstly propose the model of complete interval distributed preference relation (CIDPR) based on the concept of linguistic distribution with interval symbolic proportions, distribution linguistic preference relation (DLPR) and IDPR. Secondly, the additive consistency index of CIDPR is defined to measure the consistency level of expert's judgment, and an adjustment algorithm is proposed for converting inconsistent CIDPR to an acceptable consistent level. Thirdly, since trust relation is a critical factor in the generation of experts’ weights and the adjustment of experts’ opinions, consensus reaching process (CRP) is designed to take into account distributed linguistic trust relations under social network analysis (SNA). In the proposed adjustment mechanism, non-consensus individual should modify opinion towards his/her trusted and highly weighted expert. The advantage of the proposed inconsistent CIDPR adjustment model can maximally retain the information in the original distribution, while the CRP has a relatively fast convergent speed and good practicality. An illustrative example of strategic new product selection is conducted to demonstrate the applicability of the proposed method and its potential in supporting realistic GDM problems.     

Consistency and consensus improvement models driven by a personalized normalization method with probabilistic linguistic preference relations

Probabilistic linguistic preference relation (PLPR) provides an effective and flexible tool with which preference degrees of decision-makers can be captured when they vacillatingly express linguistic preference values among several linguistic terms. Individual consistency and group consensus are two important research topics of PLPRs in group decision making (GDM). Considering the problems associated with these two topics, this study proposes a novel GDM framework with consistency-driven and consensus-driven optimization models based on a personalized normalization method for managing complete and incomplete PLPRs. First, existing limitations of the traditional normalization method for probabilistic linguistic term sets (PLTSs) managing ignorance information are specifically discussed. Given the potential valuable information hidden in PLTSs, a personalized normalization method is newly proposed through a two-stage decision-making process with a comprehensive fusion mechanism. Then, based on the proposed normalization method for PLTSs, consistency-driven optimization models that aim to minimize the overall adjustment amount of a PLPR are constructed to improve consistency. Moreover, the developed models are extended to improve consistency and estimate the missing values of an incomplete PLPR. Subsequently, a consensus-driven optimization model that aims to maximize group consensus by adjusting experts’ weights is constructed to support the consensus-reaching process. Finally, an illustrative example, followed by some comparative analyses is presented to demonstrate the application and advantages of the proposed approach.     

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.     

Adaptive consensus model with multiplicative linguistic preferences based on fuzzy information granulation

An adaptive consensus model based on fuzzy information granulation (fuzzy IG) is presented for group consensus decision-making problems with multiplicative linguistic preference relations (MLPRs). Firstly, a granular representation of linguistic terms is concerned with the triangular fuzzy formation of a family of information granules over given Analytical Hierarchy Process (AHP) numerical scales. On this basis, the individual consistency and group consensus measure indices using fuzzy granulation technique are constructed, respectively. Then, the optimal cut-off points of fuzzy information granules are obtained by establishing a multi-objective optimization model together with a multi-objective particle swarm optimization (MOPSO) algorithm. A novel group consensus decision-making approach where consensus reaching process (CRP) is achieved by adaptively adjusting individual preferences through the optimization of the cut-off points is proposed. After conflict elimination, the obtained group preference gives the ranking of the alternatives. Finally, a real emergency decision-making case for liquid ammonia leak is given to illustrate the application steps of the proposed method and comparative analysis with the existing GDM methods. Comparative results demonstrate that the proposed method has some advantages in aspects of avoiding information loss or distortion and improving consensus performance.     

An approach to group decision making with heterogeneous incomplete uncertain preference relations

For practical group decision making problems, decision makers tend to provide heterogeneous uncertain preference relations due to the uncertainty of the decision environment and the difference of cultures and education backgrounds. Sometimes, decision makers may not have an in-depth knowledge of the problem to be solved and provide incomplete preference relations. In this paper, we focus on group decision making (GDM) problems with heterogeneous incomplete uncertain preference relations, including uncertain multiplicative preference relations, uncertain fuzzy preference relations, uncertain linguistic preference relations and intuitionistic fuzzy preference relations. To deal with such GDM problems, a decision analysis method is proposed. Based on the multiplicative consistency of uncertain preference relations, a bi-objective optimization model which aims to maximize both the group consensus and the individual consistency of each decision maker is established. By solving the optimization model, the priority weights of alternatives can be obtained. Finally, some illustrative examples are used to show the feasibility and effectiveness of the proposed method.     

The induced linguistic continuous ordered weighted geometric operator and its application to group decision making

In this paper, based on the induced linguistic ordered weighted geometric (ILOWG) operator and the linguistic continuous ordered weighted geometric (LCOWG) operator, we develop the induced linguistic continuous ordered weighted geometric (ILCOWG) operator, which is very suitable for group decision making (GDM) problems taking the form of uncertain multiplicative linguistic preference relations. We also present the consistency of uncertain multiplicative linguistic preference relation and study some properties of the ILCOWG operator. Then we propose the relative consensus degree ILCOWG (RCD-ILCOWG) operator, which can be used as the order-inducing variable to induce the ordering of the arguments before aggregation. In order to determine the weights of experts in group decision making (GDM), we define a new distance measure based on the LCOWG operator and develop a nonlinear model on the basis of the criterion of minimizing the distance of the uncertain multiplicative linguistic preference relations. Finally, we analyze the applicability of the new approach in a financial GDM problem concerning the selection of investments.     

Deriving the personalized individual semantics of linguistic information from flexible linguistic preference relations

In group decision making, flexible linguistic preference relations (FLPRs) are very useful with the pairwise comparisons taking the form of flexible linguistic expressions (FLEs). Due to the fact that different decision makers have different understandings of words, this paper investigates the personalized individual semantics (PISs) of the linguistic information in FLPRs. Two optimization models are constructed to compute a linguistic distribution which is closest to an incomplete FLE. The FLPRs are transformed into fuzzy preference relations by using optimization models which maximize consistency and consensus of the fuzzy preference relations. The PISs of linguistic terms and subsets of the linguistic term set are obtained in this process. A group decision making model based on FLPRs is presented and a green supplier selection problem in automotive industry is solved by using the proposed model. The comparative analysis is presented to show the feasibility of the group decision making model.     

A Consensus Model for Group Decision Making With Incomplete Fuzzy Preference Relations

Two processes are necessary to solve group decision making problems: A consensus process and a selection process. The consensus reaching process is necessary to obtain a final solution with a certain level of agreement between the experts; and the selection process is necessary to obtain such a final solution. In a previous paper, we present a selection process to deal with group decision making problems with incomplete fuzzy preference relations, which uses consistency measures to estimate the incomplete fuzzy preference relations. In this paper we present a consensus model. The main novelty of this consensus model is that of being guided by both consensus and consistency measures. Also, the consensus reaching process is guided automatically, without moderator, through both consensus and consistency criteria. To do that, a feedback mechanism is developed to generate advice on how experts should change or complete their preferences in order to reach a solution with high consensus and consistency degrees. In each consensus round, experts are given information on how to change their preferences, and to estimate missing values if their corresponding preference relation is incomplete. Additionally, a consensus and consistency based induced ordered weighted averaging operator to aggregate the experts' preferences is introduced, which can be used in consensus models as well as in selection processes. The main improvement of this consensus model is that it supports the management of incomplete information and it allows to achieve consistent solutions with a great level of agreement.     

A decision support model for group decision making with intuitionistic fuzzy linguistic preferences relations

As a new preference structure, the intuitionistic fuzzy linguistic preference relation (IFLPR) was introduced to efficiently cope with situations in which the membership degree and non-membership degree are represented as linguistic terms. For group decision making (GDM) problems with IFLPRs, two significant and challenging issues are individual consistency and group consensus before deriving the reliable priority weights of alternatives. In this paper, a novel decision support model is investigated to simultaneously deal with the individual consistency and group consensus for GDM with IFLPRs. First, the concepts of multiplicative consistency and weak transitivity for IFLPRs are introduced and followed by a discussion of their desirable properties. Then, a transformation approach is developed to convert the normalized intuitionistic fuzzy priority weights into multiplicative consistent IFLPR. Based on the distance of IFLPRs, the consistency index, individual consensus degree and group consensus degree for IFLPRs are further defined. In addition, two convergent automatic iterative algorithms are proposed in the investigated decision support model. The first algorithm is utilized to convert an unacceptable multiplicative consistent IFLPR to an acceptable one. The second algorithm can assist the group decision makers to achieve a predefined consensus level. The main characteristic of the investigated decision support model is that it guarantees each IFLPR is still acceptable multiplicative consistent when the predefined consensus level is achieved. Finally, several numerical examples are provided, and comparative analyses with existing approaches are performed to demonstrate the effectiveness and practicality of the investigated model.

     

A Consensus Support System Model for Group Decision-Making Problems With Multigranular Linguistic Preference Relations

The group decision-making framework with linguistic preference relations is studied. In this context, we assume that there exist several experts who may have different background and knowledge to solve a particular problem and, therefore, different linguistic term sets (multigranular linguistic information) could be used to express their opinions. The aim of this paper is to present a model of consensus support system to assist the experts in all phases of the consensus reaching process of group decision-making problems with multigranular linguistic preference relations. This consensus support system model is based on i) a multigranular linguistic methodology, ii) two consensus criteria, consensus degrees and proximity measures, and iii) a guidance advice system. The multigranular linguistic methodology permits the unification of the different linguistic domains to facilitate the calculus of consensus degrees and proximity measures on the basis of experts' opinions. The consensus degrees assess the agreement amongst all the experts' opinions, while the proximity measures are used to find out how far the individual opinions are from the group opinion. The guidance advice system integrated in the consensus support system model acts as a feedback mechanism, and it is based on a set of advice rules to help the experts change their opinions and to find out which direction that change should follow in order to obtain the highest degree of consensus possible. There are two main advantages provided by this model of consensus support system. Firstly, its ability to cope with group decision-making problems with multigranular linguistic preference relations, and, secondly, the figure of the moderator, traditionally presents in the consensus reaching process, is replaced by the guidance advice system, and in such a way, the whole group decision-making process is automated     

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.     

An interindividual iterative consensus model for fuzzy preference relations

Consensus reaching models are widely used to derive a representative solution in group decision-making problems. Current models present limitations regarding the achievement of the agreement and keeping enough consistency for achieving valid solutions. Therefore, this paper proposed a new consensus model based on the deviation degree of two fuzzy preference relations (FPRs), in which a novel consistency index (CI) is defined to measure whether an FPR is of acceptable consistency. Additionally, an interindividual similarity index (ISI) is devised to measure the consensus degree of two FPRs. In the proposed consensus reaching process, ISI is also used to guide the two most incompatible decision-makers (DMs) to modify their judgments. The proposed iterative consensus reaching algorithm is convergent, CI preservation. After that, a stationary vector method is adopted to determine DMs’ weights for the aggregation process based on DMs’ opinion transition probabilities. Finally, an illustrative example and comparative analysis is given to demonstrate the effectiveness of the proposed model.     

On compatibility of uncertain multiplicative linguistic preference relations based on the linguistic COWGA

The aim of this work is to develop a new compatibility for the uncertain multiplicative linguistic preference relations and utilize it to determine the optimal weights of experts in the group decision making (GDM). First, the compatibility degree and compatibility index for the two multiplicative linguistic preference relations are proposed. Then, based on the linguistic continuous ordered weighted geometric averaging (LCOWGA) operator, some concepts of the compatibility degree and compatibility index for the two uncertain multiplicative linguistic preference relations are presented. We prove the property that the synthetic uncertain linguistic preference relation is of acceptable compatibility under the condition that the uncertain multiplicative linguistic preference relations given by experts are all of acceptable compatibility with the ideal uncertain multiplicative linguistic preference relation, which provides a theoretic basis for the application of the uncertain multiplicative linguistic preference relations in GDM. Next, an optimal model is constructed to determine the weights of experts based on the criterion of minimizing the compatibility index in GDM. Moreover, an approach to GDM with uncertain multiplicative linguistic preference relations is developed, and finally, an application of the approach to supplier selection problem with uncertain multiplicative linguistic preference relations is pointed out.     

On compatibility of uncertain additive linguistic preference relations based on the linguistic COWA operator

We develop a new compatibility for the uncertain additive linguistic preference relations and study its properties which are very suitable to deal with group decision making (GDM) problems involving uncertain additive linguistic preference relations. Based on the linguistic continuous ordered weighted averaging (LCOWA) operator, we present some concepts of the compatibility degree and compatibility index for the two uncertain additive linguistic preference relations. Then, we study some desirable properties including the property that the synthetic uncertain linguistic preference relation is of acceptable compatibility under the condition that uncertain additive linguistic preference relations given by experts are all of acceptable compatibility with the ideal uncertain linguistic preference relation, which provides a theoretic basis for the application of the uncertain additive linguistic preference relations in GDM. In order to determine the weights of experts, we construct an optimal model based on the criterion of minimizing the compatibility index in GDM. Finally, we propose a new approach based on the compatibility index and the expected additive linguistic preference relation to GDM and develop an application of the optimal weights approach compared with the equal weights approach where we analyze a GDM regarding the evaluation of schools in a university.     

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.     

An Adaptive Consensus Support Model for Group Decision-Making Problems in a Multigranular Fuzzy Linguistic Context

Different consensus models for group decision-making (GDM) problems have been proposed in the literature. However, all of them consider the consensus reaching process a rigid or inflexible one because its behavior remains fixed in all rounds of the consensus process. The aim of this paper is to improve the consensus reaching process in GDM problems defined in multigranular linguistic contexts, i.e., by using linguistic term sets with different cardinality to represent experts' preferences. To do that, we propose an adaptive consensus support system model for this type of decision-making problem, i.e., a process that adapts its behavior to the agreement achieved in each round. This adaptive model increases the convergence toward the consensus and, therefore, reduces the number of rounds to reach it.     

A dynamically weight adjustment in the consensus reaching process for group decision-making with hesitant fuzzy preference relations

The consensus reaching process is a dynamic and iterative process for improving group's consensus level before making a final decision in group decision-making (GDM). As the experts will express their opinions under their own intellectual level from different aspects, it is natural that the experts’ weights should reflect their judgment information. This paper proposes a dynamic way to adjust weights of decision-makers (DMs) automatically when they are asked to give original judgment information for GDM problems, in which the DMs express their judgment information by hesitant fuzzy preference relations (HFPRs). Two indices, an individual consensus index of hesitant fuzzy preference relation (ICIHFPR) and a group consensus index of hesitant fuzzy preference relation (GCIHFPR), are introduced. Normalisation of HFPRs with different numbers of possible values is taken into consideration for better computation and comparison. An iterative consensus reaching algorithm is presented with DMs’ weighting vector changing in each consensus reaching process and the process terminates until both the ICIHFPR and GCIHFPR are controlled within predefined thresholds. Finally, an example is illustrated and comparative analyses demonstrate the effectiveness of the proposed methods.