Probabilities in the OWA operator

We analyze the use of the probability in the ordered weighted average (OWA). We introduce the probabilistic OWA (POWA) operator. It is an aggregation operator that provides a parameterized family of aggregation operators between the minimum and the maximum that considers the degree of importance that the probability and the OWA operator have in the aggregation. We study some of its main properties and particular cases. We also study the construction of interval and fuzzy numbers with POWA operators. We study the applicability of the POWA operator and we see that it is very broad because all the previous studies that use the probability can be revised with this new approach. We develop an application in a group decision making problem regarding investment selection.     

Generalized power aggregation operators and their applications in group decision making

In this paper, we investigate a generalized power average (GPA) operator and its weighted form, which are on the basis of the power average (PA) operator and the generalized mean, and develop a generalized power ordered weighted average (GPOWA) operator based on the power ordered weighted average (POWA) operator. Then, we extend these operators to uncertain environments and present an uncertain generalized power average (UGPA) operator and its weighted form, and an uncertain generalized power ordered weighted average (UGPOWA) operator to aggregate the input arguments taking the form of interval of numerical values. We also extend the GPA operator and the GPOWA operator to intuitionistic fuzzy environment, and obtain the generalized intuitionistic fuzzy power averaging (GIFPA) operator and the generalized intuitionistic fuzzy power ordered weighted averaging (GIFPOWA) operator. Moreover, some properties of these operators are studied. We also present new approaches on the basis of the proposed operators in an example of strategic decision making.     

Some properties of the weighted OWA operator.

Based on the researches on ordered weighted average (OWA) operator, the weighted OWA operator (WOWA) and especially the quantifier guided aggregation method, with the generating function representation of regular increasing monotone (RIM) quantifier technique, we discuss the properties of WOWA operator with RIM quantifier in the respect of orness. With the continuous OWA and WOWA ideas recently proposed by Yager, an improvement on the continuous OWA and WOWA operator is proposed. The properties of WOWA are also extended from discrete to the continuous case. Based on these properties, two families of parameterized RIM quantifiers for WOWA operator are proposed, which have exponential generating function and piecewise linear generating function respectively. One interesting property of these two kinds of RIM quantifiers is that for any aggregated set (or variable) under any weighted (distribution) function, the aggregation values are always consistent with the orness (optimistic) levels, so they can be used to represent the decision maker's preference, and we can get the preference value of fuzzy sets or random variables with the orness level of RIM quantifier as their control parameter.     

A new soft likelihood function based on power ordered weighted average operator

The likelihood function is widely used in data processing but the classical likelihood function is too strict to deal with data with extreme values in real applications. To address this issue, a soft likelihood function based on power ordered weighted average (POWA) operator for probability distribution is proposed. Compared with the existing soft likelihood functions, one of the main features of the proposed likelihood function is that the weighted mean is displayed by the geometric mean of the weighted probability products. As a result, the effect on final result caused by extreme data can be efficiently decreased. Numerical examples are used to illustrate the efficiency of the proposed soft likelihood function.     

Pythagorean Fuzzy Multiattribute Group Decision Making with Probabilistic Information and OWA Approach

The aim of this paper is to develop a Pythagorean fuzzy multiattribute group decision making (MAGDM) method based on probabilistic information and the ordered weighted averaging (OWA) approach. The Pythagorean fuzzy probabilistic ordered weighted averaging (PFPOWA) operator is presented. It is a new aggregation operator that considers the probabilities and the OWA in the same formulation. Therefore, it is able to take into account the degree of importance that each concept has in the particular problem considered. Some main properties and different particular cases of the PFPOWA operators are studied. Moreover, a method based on the proposed operator for multiattribute group decision making is put forward. Finally, an example showing analysis of a supplier selection is given to verify the effectiveness and practicability of the proposed method.     

Induced and uncertain heavy OWA operators

In this paper, we analyse in detail the ordered weighted averaging (OWA) operator and some of the extensions developed about it. We specially focus on the heavy aggregation operators. We suggest some new extensions about the OWA operator such as the induced heavy OWA (IHOWA) operator, the uncertain heavy OWA (UHOWA) operator and the uncertain induced heavy OWA (UIHOWA) operator. For these three new extensions, we consider some of their main properties and a wide range of special cases found in the weighting vector such as the heavy weighted average (HWA) and the uncertain heavy weighted average (UHWA). We further generalize these models by using generalized and quasi-arithmetic means obtaining the generalized heavy weighted average (GHWA), the induced generalized HOWA (IGHOWA) and the uncertain IGHOWA (UIGHOWA) operator. Finally, we develop an application of the new approach in a decision-making problem.     

On the properties of equidifferent RIM quantifier with generating function

Comparing the large number of research papers on the ordered weighted averaging (OWA) operator, the researches on relative quantifier are relatively rare so far. In the present paper, based on the quantifier guided aggregation method with OWA operator which was proposed by Yager [“Quantifier guided aggregation using OWA operators”, Int. J. Intell. Syst., 11, pp. 49–73, 1996], a generating function representation method for regular increasing monotone (RIM) quantifiers is proposed. We extend the the properties of OWA operator to the RIM quantifier which is represented with a monotone function instead of the OWA weighting vector. A class of parameterized equidifferent RIM quantifier which has minimum variance generating function is proposed and its properties are also analyzed. The equidifferent RIM quantifier is consistent with its orness level for any aggregated elements, which can be used to represent the decision maker's preference.     

基于语言型混合算子的模糊信息聚合方法

在多属性决策过程中经常会用到聚合算子,有序加权平均聚合(OWA)算子是最常用的聚合算子之一,通常用于聚合确切的数值.然而,现实世界部分信息的不确定性以及决策者对一些信息的模糊性,使得部分信息不能用确切的数值表示,从而导致OWA算子及其扩展算子向着多元化发展.对此,给出一种语言型混合有序加权平均聚合(LHOWA)算子,同时研究该算子所应具备的一些基本性质,并给出一种基于该算子的语言型信息聚合方法,用于多属性决策过程中模糊信息的聚合.最后,通过一个煤矿安全评价的算例对所提出方法的优越性进行了验证.     

基于OWA算子的模糊n-cell数排序方法

针对基于模糊n-cell数的多属性排序问题,提出了一种基于有序加权平均算子（OWA算子）的模糊n-cell数排序方法。该方法首先根据样本数据对评估对象的属性构造模糊n-cell数,其次根据均值将属性按照从大到小排列,然后选取合适的权重向量,应用OWA算子进行信息聚合得到综合模糊n-cell数,接着根据各分量均值得到排序结果。最后,将该方法运用到实例中,并与传统的均值方法进行了比较。结果表明该方法不仅灵活有效,可根据具体情况选择不同的OWA权重来消除部分不合理的情况,使结果更有说服力,还弥补了传统均值方法的不足。     

On the methods of decision making under uncertainty with probability information

By considering the decision maker's attitude of profit and risk, we propose an alternative selection method that can include the methods of decision making under ignorance and decision making under risk as special cases. An index to measure the decision maker's risk‐averse degree is proposed. With a given optimistic level of profit and risk, the evaluation results of the alternatives can be obtained with a geometric ordered weighted average (OWA) operator and a basic defuzzification distribution (BADD) neat OWA operator. Some properties of these two kinds of OWA operator in the problem of decision making under uncertainty are also proposed. © 2004 Wiley Periodicals, Inc. Int J Int Syst 19: 1217–1238, 2004.     

Monotonic argument‐dependent OWA operators

The ordered weighted averaging (OWA) operator introduced by Yager is one of the most popular aggregation technique. In this paper, we develop two kinds of argument‐dependent OWA (DOWA) operators including the pessimistic‐dependent OWA (PE‐DOWA) operator and optimistic‐dependent OWA (OP‐DOWA) operator, that point out that the PE‐DOWA operator is decreasing and the OP‐DOWA operator is increasing, and investigate some properties of our proposed monotonic DOWA operators in detail. Furthermore, we introduce the concept of original function in which a gradient vector generates the weights of the PE‐DOWA and OP‐DOWA operators. Meanwhile, we propose two classes of original functions including summing‐type original function and multiplying‐type original function and investigate the sufficient monotonic conditions for the DOWA operators generated by the original functions. Finally, we discuss the characteristics and properties of our proposed DOWA operators in detail and use a numerical example to illustrate the flexibility of our proposed operators.     

Comment on “Some hybrid weighted averaging operators and their application to decision making”

The weighted averaging (WA) operator and the ordered weighted averaging (OWA) operator are the basic aggregation operators. Recently, a new hybrid weighted arithmetical averaging (HWAA) operator is proposed by Lin and Jiang to provide a unified framework between the WA and OWA operators. In this paper, I have some comments on their results. The major one concerns the monotonicity of the HWAA operator.     

模糊决策表中基于OWA算子的三支属性约简

在模糊决策表中,基于有序加权平均（Ordered weighted averaging,OWA）算子建立的相容关系中,OWA算子是一个信息集成工具。本文利用OWA算子及其诱导的截集相容关系,分别讨论基于正域、负域与边界域的属性约简。首先,利用OWA算子中的权重给出属性区分对象的贡献度;然后,根据属性贡献度定义每个属性被约简的可能性;从而给出模糊决策表的一种启发式三支属性约简方法,该方法可以减少属性约简的搜索空间并避免属性约简的盲目性;最后,利用实例来分析影响该三支区域属性约简方法的主要因素,并说明该启发式属性约简方法的合理性和有效性。本文提出的属性贡献度度量及启发式三支属性约简方法将减少属性约简的搜索空间,有益于模糊决策表特征选取的应用。     

Exploring the Ordered Weighted Averaging Operator Knowledge Domain: A Bibliometric Analysis

Ordered weighted averaging (OWA) operator has been received increasingly widespread interest since its appearance in 1988. Recently, a topic search with the keywords “ordered weighted averaging operator” or “OWA operator” on Web of Science (WOS) found 1231 documents. As the publications about OWA operator increase rapidly, thus a scientometric analysis of this research field and discovery of its knowledge domain becomes very important and necessary. This paper studies the publications about OWA operator between 1988 and 2015, and it is based on 1213 bibliographic records obtained by using topic search from WOS. The disciplinary distribution, most cited papers, influential journals, as well as influential authors are analyzed through citation and cocitation analysis. The emerging trends in OWA operator research are explored by keywords and references burst detection analysis. The research methods and results in this paper are meaningful for researchers associated with OWA operator field to understand the knowledge domain and establish their own future research direction.     

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.     

Quantitative orness for lattice OWA operators

This paper deals with OWA (ordered weighted average) operators defined on any complete lattice endowed with a t-norm and a t-conorm and satisfying a certain finiteness local condition. A parametrization of these operators is suggested by introducing a quantitative orness measure for each OWA operator, based on its proximity to the OR operator. The meaning of this measure is analyzed for some concrete OWA operators used in color image reduction, as well as for some OWA operators used in a medical decision making process.     

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.     

结合OWA算子和模糊DEMATEL的风险评估方法

针对传统故障模式与影响分析(FMEA)方法在实际应用中的不足,提出一种基于有序加权平均(OWA)算子和决策试行与评价实验法(DEMATEL)的风险排序方法。FMEA专家对故障模式的3个风险因子给出模糊评价信息,应用OWA算子对评估信息进行集结,得到各故障原因对故障模式的影响强度。采用模糊DEMATEL法构建FMEA系统要素间的初始直接影响矩阵,经过运算可得综合影响矩阵,并计算各故障原因的原因度,据此进行产品或系统的失效风险评估。运用该方法对地铁车门系统的基础部件进行安全性分析,并将所得结果与传统RPN方法的结果做对比,验证了该方法的可行性和有效性。     

A unified model between the weighted average and the induced OWA operator

We present a new model that uses the weighted average (WA) and the induced ordered weighted averaging (IOWA) operator in the same formulation. We call it the induced ordered weighted averaging–weighted average (IOWAWA) operator. We study some of its main properties and we see that it has a lot of particular cases such as the WA and the OWA operator. The main advantage of the IOWAWA operator is that it unifies the IOWA operator with the WA in the same formulation considering the degree of importance that each concept has in the aggregation. We analyze the applicability of this new approach and we see that it is very broad because it can be applied in a wide range of fields such as statistics, economics, decision theory and engineering. Theoretically, we could state that all the previous models and applications based on the WA and the IOWA can be revised and improved with this new approach because they will be included in this framework as a particular case. We focus on an application in a multi-person decision-making in political management.     

Similarity classifier with ordered weighted averaging operators

In this article we extend the similarity classifier to cover also ordered weighted averaging (OWA) operators. Earlier, similarity classifier was mainly used with generalized mean operator, but in this article we extend this aggregation process to cover more general OWA operators. With OWA operators we concentrate on linguistic quantifier guided aggregation where several different quantifiers are studied and on how they best suite for the similarity classifier. Our proposed method is applied to real world medical data sets which are new thyroid, hypothyroid, lymphography and hepatitis data sets. Results are very promising and show improvement compared to the earlier used generalized mean operator. In this article we will show that by using OWA operators instead of generalized mean, we can improve classification accuracy with chosen data sets.