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.     

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.     

Group decision-making procedure based on incomplete reciprocal relations

Xu (Int J Approx Reason 36:261–270, 2004) introduced the concepts of incomplete reciprocal relation and additive consistent incomplete reciprocal relation. The aim of this paper is to develop a novel procedure for group decision making with incomplete reciprocal relations. The procedure utilizes each given incomplete reciprocal relation to construct an auxiliary reciprocal relation based on additive transitivity, and then aggregates directly these auxiliary reciprocal relations into a collective auxiliary reciprocal relation. After that, based on the collective auxiliary reciprocal relation, a simple linear system of equations is established for ranking alternatives. Finally, a numerical example is given to illustrate the developed procedure.     

基于不同类型残缺判断矩阵的群决策方法

定义了一系列残缺不确定判断矩阵的概念,并且定义了一种连续区间数据有序加权几何（C-OWG）算子．利用连续区间数据有序加权平均（C-OWA）子和C-OWG算予等转化途径把不同类型的残缺不确定判断矩阵均一致化为残缺期望值互补判断矩阵,进而提出一种基于不同类型残缺判断矩阵的群决策方法．最后通过实例验证了方法的可行性和有效性．     

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.     

Some properties of the induced continuous ordered weighted geometric operators in group decision making

In Computers and Industrial Engineering 56 (2009) 1545–1552, an induced continuous ordered weighted geometric (ICOWG) operator is presented to deal with group decision making (GDM) problems with interval multiplicative preference relations. But, we still do not know whether the ICOWG operator can improve the consensus among a group of decision makers. The aim of this paper is to study some desired properties of the ICOWG operator in GDM problems. Firstly, the concept of Compatibility Degree and Compatibility Index (CI) is defined. We then present the Compatibility Index induced COWG (CI-ICOWG) operator to aggregate interval multiplicative preference relations, which induces the order of argument values based on the Compatibility Index of decision makers (DMs). The main novelty of the CI-ICOWG operator is that it aggregates individual preference relation in such a way that more importance is placed on the most compatibility one. Thus, the CI-ICOWG operator can guarantee that the Compatibility Degree is at least as good as the arithmetic mean of all the individual Compatibility Degrees. Additionally, if the leading decision maker’s interval multiplicative preference relation P   and each of interval multiplicative preference relations R⁽¹⁾,R⁽²⁾,…,R^(m)$R^{(1)}\text{,}{R}^{(2)}\text{,}\dots \text{,}{R}^{(m)}$ are of acceptable compatibility, then P   and the collective judgement matrix (CJM) of R⁽¹⁾,R⁽²⁾,…,R^(m)$R^{(1)}\text{,}{R}^{(2)}\text{,}\dots \text{,}{R}^{(m)}$ are of acceptable compatibility. Finally, an illustrative numerical example is used to verify the developed approaches.     

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.     

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.     

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.     

The induced continuous ordered weighted geometric operators and their application in group decision making

In [IEEE Trans. Syst., Man, Cybernet.––Part B 29 (1999) 141], a more general class of OWA operators called the induced ordered weighted averaging (IOWA) operators is developed. Later, Yager and Xu [Fuzzy Sets and Syst, 157 (2006) 1393–1402.] introduced the continuous ordered weighted geometric operator(COWG), which is suitable for individual decision making problems taking the form of interval multiplicative preference relation. The aim of this paper is to develop some induced continuous ordered weighted geometric (ICOWG) operators. In particular, we present the reliability induced COWG (R-ICOWG) operator, which applies the ordering of the argument values based upon the reliability of the information sources; and the relative consensus degree induced COWG (RCD-ICOWG) operator, which applies the ordering of the argument values based upon the relative consensus degree of the information sources. Some desirable properties of the ICOWG operators are studied, and then, the ICOWG operators are applied to group decision making with interval multiplicative preference relations.     

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.     

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.     

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.     

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.