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.     

On compatibility of uncertain additive linguistic preference relations and its application in the group decision making

We develop a new compatibility for the uncertain additive linguistic preference relations and utilize it to determine the optimal weights of experts in the group decision making (GDM). Based on some operational laws for the uncertain additive linguistic preference labels, we propose some new concepts of the compatibility degree and acceptable compatibility index for the two uncertain additive linguistic preference relations. We also prove the properties that the synthetic preference relation is also of acceptable compatibility under the condition that additive linguistic preference relations provided by experts are all of acceptable compatibility with the specific linguistic preference relation, which provides a theoretic basis for the application of the uncertain additive linguistic preference relations in the GDM. Furthermore, we establish a mathematical model to obtain the weights of experts based on the criterion of minimizing the compatibility in the GDM, and we discuss the solution to the model. Finally, we give a numerical example to make comparative analysis on compatibility index using the optimal experts’ weights approach and the equal experts’ weights approach, which indicates that the model is feasible and effective.     

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.     

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.

     

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.     

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.     

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.     

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.     

Group consensus algorithms based on preference relations

In many group decision-making situations, decision makers’ preferences for alternatives are expressed in preference relations (including fuzzy preference relations and multiplicative preference relations). An important step in the process of aggregating preference relations, is to determine the importance weight of each preference relation. In this paper, we develop a number of goal programming models and quadratic programming models based on the idea of maximizing group consensus. Our models can be used to derive the importance weights of fuzzy preference relations and multiplicative preference relations. We further develop iterative algorithms for reaching acceptable levels of consensus in group decision making based on fuzzy preference relations or multiplicative preference relations. Finally, we include an illustrative example.     

Analysis of self-confidence indices-based additive consistency for fuzzy preference relations with self-confidence and its application in group decision making

Preference relations have been widely used in group decision-making (GDM) problems. Recently, a new kind of preference relations called fuzzy preference relations with self-confidence (FPRs-SC) has been introduced, which allow experts to express multiple self-confidence levels when providing their preferences. This paper focuses on the analysis of additive consistency for FPRs-SC and its application in GDM problems. To do that, some operational laws for FPRs-SC are proposed. Subsequently, an additive consistency index that considers both the fuzzy preference values and self-confidence is presented to measure the consistency level of an FPR-SC. Moreover, an iterative algorithm that adjusts both the fuzzy preference values and self-confidence levels is proposed to repair the inconsistency of FPRs-SC. When an acceptable additive consistency level for FPRs-SC is achieved, the collective FPR-SC can be computed. We aggregate the individual FPRs-SC using a self-confidence indices-based induced ordered weighted averaging operator. The inherent rule for aggregation is to give more importance to the most self-confident experts. In addition, a self-confidence score function for FPRs-SC is designed to obtain the best alternative in GDM with FPRs-SC. Finally, the feasibility and validity of the research are demonstrated with an illustrative example and some comparative analyses.     

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.     

Incomplete interval-valued intuitionistic fuzzy preference relations

The aim of this paper is to investigate decision making problems with interval-valued intuitionistic fuzzy preference information, in which the preferences provided by the decision maker over alternatives are incomplete or uncertain. We define some new preference relations, including additive consistent incomplete interval-valued intuitionistic fuzzy preference relation, multiplicative consistent incomplete interval-valued intuitionistic fuzzy preference relation and acceptable incomplete interval-valued intuitionistic fuzzy preference relation. Based on the arithmetic average and the geometric mean, respectively, we give two procedures for extending the acceptable incomplete interval-valued intuitionistic fuzzy preference relations to the complete interval-valued intuitionistic fuzzy preference relations. Then, by using the interval-valued intuitionistic fuzzy averaging operator or the interval-valued intuitionistic fuzzy geometric operator, an approach is given to decision making based on the incomplete interval-valued intuitionistic fuzzy preference relation, and the developed approach is applied to a practical problem. It is worth pointing out that if the interval-valued intuitionistic fuzzy preference relation is reduced to the real-valued intuitionistic fuzzy preference relation, then all the above results are also reduced to the counterparts, which can be applied to solve the decision making problems with incomplete intuitionistic fuzzy preference information.     

Group decision making model and approach based on interval preference orderings

In group decision making under uncertainty, interval preference orderings as a type of simple uncertain preference structure, can be easily and conveniently used to express the experts’ evaluations over the considered alternatives. In this paper, we investigate group decision making problems with interval preference orderings on alternatives. We start by fusing all individual interval preference orderings given by the experts into the collective interval preference orderings through the uncertain additive weighted averaging operator. Then we establish a nonlinear programming model by minimizing the divergences between the individual uncertain preferences and the group’s opinions, from which we derive an exact formula to determine the experts’ relative importance weights. After that, we calculate the distances of the collective interval preference orderings to the positive and negative ideal solutions, respectively, based on which we use a TOPSIS based approach to rank and select the alternatives. All these results are also reduced to solve group decision making problems where the experts’ evaluations over the alternatives are expressed in exact preference orderings. A numerical analysis of our model and approach is finally carried out using two illustrative examples.     

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 new method of obtaining the priority weights from an interval fuzzy preference relation

In analyzing a multiple criteria decision-making problem, the decision maker may express her/his opinions as an interval fuzzy or multiplicative preference relation. Then it is an interesting and important issue to investigate the consistency of the preference relations and obtain the reliable priority weights. In this paper, a new consistent interval fuzzy preference relation is defined, and the corresponding properties are derived. The transformation formulae between interval fuzzy and multiplicative preference relations are further given, which show that two preference relations, consistent interval fuzzy and multiplicative preference relations, can be transformed into each other. Based on the transformation formula, the definition of acceptably consistent interval fuzzy preference relation is given. Furthermore a new algorithm for obtaining the priority weights from consistent or inconsistent interval fuzzy preference relations is presented. Finally, three numerical examples are carried out to compare the results using the proposed method with those using other existing procedures. The numerical results show that the given procedure is feasible, effective and not requisite to solve any mathematical programing.     

Non-dominance and attitudinal prioritisation methods for intuitionistic and interval-valued intuitionistic fuzzy preference relations

A novel intuitionistic fuzzy set (IFS) score function and an intuitionistic fuzzy preference relation (IFPR) quantifier guided non-dominance based prioritisation method are introduced. Based on Yager’s continuous OWA (COWA) operator, the interval-valued intuitionistic fuzzy COWA (IVIF-COWA) operator is defined, and a new attitudinal expected score function for interval-valued intuitionistic fuzzy numbers (IVIFNs) is introduced. The novelty of this attitudinal expected score function is that it allows the comparison of IVIFNs by taking into account of the decision makers’ attitudinal character. Moreover, we show that the new attitudinal expected score function extends: (i) the IFS score function introduced in this paper, which is mathematically equivalent to Chen and Tan’s score function (Chen and Tan, 1994); and (ii) Xu and Chen’s score function for IVIFNs (Xu and Chen, 2007). Using the proposed score functions, a method is developed to construct FPRs from a given IFPR and IVIFPR, respectively. When the hesitancy degree function is null, we prove that the score FPRs coincide with their respective IFPR and IVIFPR. Finally, a ranking sensitivity analysis of the attitudinal expected score function with respect to the attitudinal parameter is provided.     

MAGDM Linear-Programming Models With Distinct Uncertain Preference Structures

Group decision making with preference information on alternatives is an interesting and important research topic which has been receiving more and more attention in recent years. The purpose of this paper is to investigate multiple-attribute group decision-making (MAGDM) problems with distinct uncertain preference structures. We develop some linear-programming models for dealing with the MAGDM problems, where the information about attribute weights is incomplete, and the decision makers have their preferences on alternatives. The provided preference information can be represented in the following three distinct uncertain preference structures: 1) interval utility values; 2) interval fuzzy preference relations; and 3) interval multiplicative preference relations. We first establish some linear-programming models based on decision matrix and each of the distinct uncertain preference structures and, then, develop some linear-programming models to integrate all three structures of subjective uncertain preference information provided by the decision makers and the objective information depicted in the decision matrix. Furthermore, we propose a simple and straightforward approach in ranking and selecting the given alternatives. It is worth pointing out that the developed models can also be used to deal with the situations where the three distinct uncertain preference structures are reduced to the traditional ones, i.e., utility values, fuzzy preference relations, and multiplicative preference relations. Finally, we use a practical example to illustrate in detail the calculation process of the developed approach.      

A consensus model for group decision making problems with linguistic interval fuzzy preference relations

Sometimes, we find decision situations in which it is difficult to express some preferences by means of concrete preference degrees. In this paper, we present a consensus model for group decision making problems in which the experts use linguistic interval fuzzy preference relations to represent their preferences. This model is based on two consensus criteria, a consensus measure and a proximity measure, and on the concept of coincidence among preferences. We compute both consensus criteria in the three representation levels of a preference relation and design an automatic feedback mechanism to guide experts in the consensus reaching process.     

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.     

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.