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.     

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 study of the origin and uses of the ordered weighted geometric operator in multicriteria decision making

The ordered weighted geometric (OWG) operator is an aggregation operator that is based on the ordered weighted averaging (OWA) operator and the geometric mean. Its application in multicriteria decision making (MCDM) under multiplicative preference relations has been presented. Some families of OWG operators have been defined. In this article, we present the origin of the OWG operator and we review its relationship to the OWA operator in MCDM models. We show a study of its use in multiplicative decision‐making models by providing the conditions under which reciprocity and consistency properties are maintained in the aggregation of multiplicative preference relations performed in the selection process. © 2003 Wiley Periodicals, Inc.     

Induced choquet ordered averaging operator and its application to group decision making

Yager (Fuzzy Sets Syst 2003;137:59–69) extended the idea of order‐induced aggregation to the Choquet aggregation and defined a more general type of Choquet integral operator called the induced Choquet ordered averaging (I‐COA) operator, which take as their argument pairs, in which one component called order‐inducing variable is used to induce an ordering over the second components called argument variable and then aggregated. The aim of this paper is to develop the I‐COA operator. Some of its properties are investigated. We show its relationship to the induced‐ordered weighted averaging operator. Finally, we provide some I‐COA operators to aggregate fuzzy preference relations in group decision‐making problems. © 2009 Wiley Periodicals, Inc.     

Power-Geometric Operators and Their Use in Group Decision Making

The power-average (PA) operator and the power-ordered-weighted-average (POWA) operator are the two nonlinear weighted-average aggregation tools whose weighting vectors depend on the input arguments. In this paper, we develop a power-geometric (PG) operator and its weighted form, which are on the basis of the PA operator and the geometric mean, and develop a power-ordered-geometric (POG) operator and a power-ordered-weighted-geometric (POWG) operator, which are on the basis of the POWA operator and the geometric mean, and study some of their properties. We also discuss the relationship between the PA and PG operators and the relationship between the POWA and POWG operators. Then, we extend the PG and POWG operators to uncertain environments, i.e., develop an uncertain PG (UPG) operator and its weighted form, and an uncertain power-ordered-weighted-geometric (UPOWG) operator to aggregate the input arguments taking the form of interval of numerical values. Furthermore, we utilize the weighted PG and POWG operators, respectively, to develop an approach to group decision making based on multiplicative preference relations and utilize the weighted UPG and UPOWG operators, respectively, to develop an approach to group decision making based on uncertain multiplicative preference relations. Finally, we apply both the developed approaches to broadband Internet-service selection.      

The induced generalized OWA operator

We present the induced generalized ordered weighted averaging (IGOWA) operator. It is a new aggregation operator that generalizes the OWA operator, including the main characteristics of both the generalized OWA and the induced OWA operator. This operator uses generalized means and order-inducing variables in the reordering process. It provides a very general formulation that includes as special cases a wide range of aggregation operators, including all the particular cases of the IOWA and the GOWA operator, the induced ordered weighted geometric (IOWG) operator and the induced ordered weighted quadratic averaging (IOWQA) operator. We further generalize the IGOWA operator via quasi-arithmetic means. The result is the Quasi-IOWA operator. Finally, we present a numerical example to illustrate the new approach in a financial decision-making problem.     

诱导连续区间有序加权平均算子及其在区间数群决策中的应用

基于诱导有序加权平均(IOWA)算子和连续区间有序加权平均(C-OWA)算子,提出一种诱导连续区间有序加权平均(IC-OWA)算子,并讨论了该算子的优良性质.针对区间数互补判断矩阵提出了连续偏好矩阵的概念,定义了基于专家评判水平偏差的诱导连续区间有序加权平均(DIC-OWA)算子,并给出一种基于该算子的区间数群决策方法.最后通过算例说明了该方法的可行性.     

INGIOWGA算子的多属性决策方法及其应用

研究了方案的属性评估信息以模糊语言形式给出的多属性群决策问题，在导出的有序加权几何平均（IOWGA）算子理论的基础上，给出了一种区间数广义导出有序加权几何平均（INGIOWGA）算子，利用广义的导出有序加权平均（GIOWA）算子，对专家所给出的对应于各方案的属性评估信息进行了集结，并提出了一种基于模糊语言评估和GIOWA算子的多属性群决策方法。利用该算法对X射线实时成像检测系统方案选择中的判断信息进行集结，并且通过算例说明了该方法的有效性和实用性。     

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.     

Pythagorean hesitant fuzzy Hamacher aggregation operators and their application to multiple attribute decision making

Hamacher product is a t‐norm and Hamacher sum is a t‐conorm. They are good alternatives to algebraic product and algebraic sum, respectively. Nevertheless, it seems that most of the existing hesitant fuzzy aggregation operators are based on the algebraic operations. In this paper, we utilize Hamacher operations to develop some Pythagorean hesitant fuzzy aggregation operators: Pythagorean hesitant fuzzy Hamacher weighted average (PHFHWA) operator, Pythagorean hesitant fuzzy Hamacher weighted geometric (PHFHWG) operator, Pythagorean hesitant fuzzy Hamacher ordered weighted average (PHFHOWA) operator, Pythagorean hesitant fuzzy Hamacher ordered weighted geometric (PHFHOWG) operator, Pythagorean hesitant fuzzy Hamacher induced ordered weighted average (PHFHIOWA) operator, Pythagorean hesitant fuzzy Hamacher induced ordered weighted geometric (PHFHIOWG) operator, Pythagorean hesitant fuzzy Hamacher induced correlated aggregation operators, Pythagorean hesitant fuzzy Hamacher prioritized aggregation operators, and Pythagorean hesitant fuzzy Hamacher power aggregation operators. The special cases of these proposed operators are studied. Then, we have utilized these operators to develop some approaches to solve the Pythagorean hesitant fuzzy multiple attribute decision making problems. Finally, a practical example for green supplier selections in green supply chain management is given to verify the developed approach and to demonstrate its practicality and effectiveness.     

Density‐induced ordered weighted averaging operators

We provide a special type of induced ordered weighted averaging (OWA) operator called density‐induced OWA (DIOWA) operator, which takes the density around the arguments as the inducing variables to reorder the arguments. The density around the argument, which can measure the degree of similarity between the argument and its nearest neighbors, is associated with both the number of its nearest neighbors and its weighted average distance to these neighbors. To determine the DIOWA weights, we redefine the orness measure, and propose a new maximum orness model under a dispersion constraint. The DIOWA weights generated by the traditional maximum orness model depend upon the order of the arguments and the dispersion degree. Differently, the DIOWA weights generated by the new maximum orness model also depend upon the specific values of the density around the arguments. Finally, we illustrate how the DIOWA operator is used in the decision making, and prove the effectiveness of the DIOWA operator through comparing the DIOWA operator with other operators, i.e., the centered OWA operator, the Olympic OWA operator, the majority additive‐OWA (MA‐OWA) operator, and the kNN‐DOWA operator. © 2011 Wiley Periodicals, Inc.     

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.     

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

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

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.     

Generalized aggregation operators for intuitionistic fuzzy sets

The generalized ordered weighted averaging (GOWA) operators are a new class of operators, which were introduced by Yager (Fuzzy Optim Decision Making 2004;3:93–107). However, it seems that there is no investigation on these aggregation operators to deal with intuitionistic fuzzy or interval‐valued intuitionistic fuzzy information. In this paper, we first develop some new generalized aggregation operators, such as generalized intuitionistic fuzzy weighted averaging operator, generalized intuitionistic fuzzy ordered weighted averaging operator, generalized intuitionistic fuzzy hybrid averaging operator, generalized interval‐valued intuitionistic fuzzy weighted averaging operator, generalized interval‐valued intuitionistic fuzzy ordered weighted averaging operator, generalized interval‐valued intuitionistic fuzzy hybrid average operator, which extend the GOWA operators to accommodate the environment in which the given arguments are both intuitionistic fuzzy sets that are characterized by a membership function and a nonmembership function, and interval‐valued intuitionistic fuzzy sets, whose fundamental characteristic is that the values of its membership function and nonmembership function are intervals rather than exact numbers, and study their properties. Then, we apply them to multiple attribute decision making with intuitionistic fuzzy or interval‐valued intuitionistic fuzzy information. © 2009 Wiley Periodicals, Inc.     

A C‐OWA operator‐based approach to decision making with interval fuzzy preference relation

Yager [IEEE Trans Syst Man Cybern B 2004;34:1952–1963] introduced a continuous interval argument OWA (C‐OWA) operator, which extends the ordered weighted averaging (OWA) operator, introduced by Yager [IEEE Trans Syst Man Cybern B 1988;18:183–190], to the case in which the given argument is a continuous valued interval rather than a finite set of values. In this article, we utilize the C‐OWA operator to derive the priority vector of an interval fuzzy preference relation and then develop a practical approach to solving the decision‐making problem with interval fuzzy preference relation. Finally, a numerical example is provided to demonstrate the practicability and efficiency of the developed approach. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 1289–1298, 2006.     

Quantile induced heavy ordered weighted averaging operators and its application in incentive decision making

To integrate incentives into the information aggregation process in decision making, we propose a new type of aggregation operator, denominated as the quantile induced heavy ordered weighted averaging (QI‐HOWA) operator in this paper. A primary characteristic of this type of operator is that the quantile variable can be used not only to measure the relative performance of alternatives but also to facilitate the incentive preference expression of the decision maker. We further provide a calculation technology of the QI‐HOWA weights, in which various incentive preferences of the decision maker can be considered through parameter adjustment. In addition, we discuss certain properties of the QI‐HOWA operator and note the extent to which they are effective. Finally, a numerical example regarding the selection of optimal alternatives by incentive measures is provided, and the aggregations are compared with those of ordered weighted averaging and unweighted averaging operators to illustrate the validity of the QI‐HOWA operator.     

Induced aggregation operators in decision making with the Dempster‐Shafer belief structure

In this study, we analyze the induced aggregation operators. The analysis begins with a revision of some basic concepts such as the induced ordered weighted averaging operator and the induced ordered weighted geometric operator. We then analyze the problem of decision making with Dempster‐Shafer (D‐S) theory of evidence. We suggest the use of induced aggregation operators in decision making with the D‐S theory. We focus on the aggregation step and examine some of its main properties, including the distinction between descending and ascending orders and different families of induced operators. Finally, we present an illustrative example in which the results obtained with different types of aggregation operators can be seen. © 2009 Wiley Periodicals, Inc.     

Cardinal Consistency of Reciprocal Preference Relations: A Characterization of Multiplicative Transitivity

Consistency of preferences is related to rationality, which is associated with the transitivity property. Many properties suggested to model transitivity of preferences are inappropriate for reciprocal preference relations. In this paper, a functional equation is put forward to model the “cardinal consistency in the strength of preferences” of reciprocal preference relations. We show that under the assumptions of continuity and monotonicity properties, the set of representable uninorm operators is characterized as the solution to this functional equation. Cardinal consistency with the conjunctive representable cross ratio uninorm is equivalent to Tanino's multiplicative transitivity property. Because any two representable uninorms are order isomorphic, we conclude that multiplicative transitivity is the most appropriate property for modeling cardinal consistency of reciprocal preference relations. Results toward the characterization of this uninorm consistency property based on a restricted set of $(n-1)$ preference values, which can be used in practical cases to construct perfect consistent preference relations, are also presented.