On the properties of OWA operator weights functions with constant level of orness

The result of aggregation performed by the ordered weighted averaging (OWA) operator heavily depends upon the weighting vector used. A number of methods have been presented for obtaining the associated weights. In this paper, we present analytic forms of OWA operator weighting functions, each of which has properties of rank-based weights and a constant level of orness, irrespective of the number of objectives considered. These analytic forms provide significant advantages for generating the OWA weights over previously reported methods. First, the OWA weights can be efficiently generated by using proposed weighting functions without solving a complicated mathematical program. Moreover, convex combinations of these specific OWA operators can be used to generate the OWA operators with any predefined values of orness once specific values of orness are a priori stated by the decision maker. Those weights have a property of constant level of orness as well. Finally, the OWA weights generated at a predefined value of orness make almost no numerical difference with maximum entropy OWA weights in terms of dispersion.     

A linear programming model for determining ordered weighted averaging operator weights with maximal Yager’s entropy

It has a wide attention about the methods for determining OWA operator weights. At the beginning of this dissertation, we provide a briefly overview of the main approaches for obtaining the OWA weights with a predefined degree of orness. Along this line, we next make an important generalization of these approaches as a special case of the well-known and more general problem of calculation of the probability distribution in the presence of uncertainty. All these existed methods for dealing these kinds of problems are quite complex. In order to simplify the process of computation, we introduce Yager’s entropy based on Minkowski metric. By analyzing its desirable properties and utilizing this measure of entropy, a linear programming (LP) model for the problem of OWA weight calculation with a predefined degree of orness has been built and can be calculated much easier. Then, this result is further extended to the more realistic case of only having partial information on the range of OWA weights except a predefined degree of orness. In the end, two numerical examples are provided to illustrate the application of the proposed approach.     

Improving minimax disparity model to determine the OWA operator weights

Determining the Ordered Weighted Averaging (OWA) operator weights is important in decision making applications. Several approaches have been proposed in the literature to obtain the associated weights. This paper provides an alternative disparity model to identify the OWA operator weights. The proposed mathematical model extends the existing disparity approaches by minimizing the sum of the deviation between two distinct OWA weights. The proposed disparity model can be used for a preference ranking aggregation. A numerical example in preference ranking and an application in search engines prove the usefulness of the generated OWA weights.     

A novel ordered weighted averaging weight determination based on ordinal dispersion

One of the most common techniques to find the adequate weights in ordered weighted averaging (OWA) operators is based on the orness concept, where the weights are determined by maximizing the entropy (variation) for a fixed orness value. But such an entropy represents a dispersion measure for nominal variables, while weights in an OWA operator are essentially ordinal rather than nominal. Hence, in this paper, we propose a novel way to determine OWA weights based upon ordinal dispersion measures instead of an standard entropy measure. From this approach, we find an explicit formula for the weights, and we illustrate differences by means some multicriteria decision-making examples.     

Least‐squared ordered weighted averaging operator weights

The ordered weighted averaging (OWA) operator by Yager (IEEE Trans Syst Man Cybern 1988; 18; 183–190) has received much more attention since its appearance. One key point in the OWA operator is to determine its associated weights. Among numerous methods that have appeared in the literature, we notice the maximum entropy OWA (MEOWA) weights that are determined by taking into account two appealing measures characterizing the OWA weights. Instead of maximizing the entropy in the formulation for determining the MEOWA weights, a new method in the paper tries to obtain the OWA weights that are evenly spread out around equal weights as much as possible while strictly satisfying the orness value provided in the program. This consideration leads to the least‐squared OWA (LSOWA) weighting method in which the program is to obtain the weights that minimize the sum of deviations from the equal weights since entropy is maximized when all the weights are equal. Above all, the LSOWA method allocates the positive and negative portions to the equal weights that are identical but opposite in sign from the middle point in the number of criteria. Furthermore, interval LSOWA weights can be constructed when a decision maker specifies his or her orness value in uncertain numerical bounds and we present a method, with those uncertain interval LSOWA weights, for prioritizing alternatives that are evaluated by multiple criteria. © 2008 Wiley Periodicals, Inc.     

Weighted Maximum Entropy OWA Aggregation With Applications to Decision Making Under Risk

We introduce the basic features of the ordered weighted averaging (OWA) operator. Particular emphasis is put on the task of obtaining the associated weights. We discuss the maximal entropy OWA (MEOWA) approach to obtaining the weights. This approach is based upon the specification of a parameter characterizing the desired type of aggregation and then the solving of a mathematical programming problem whose objective is to maximize the entropy of the weights subject to this parameter. Here, we provide an alternative way of getting these MEOWA weights based upon the use of a weight-generating function. The introduction of this function allows us to obtain the MEOWA weights for the case in which each argument has a distinct degree of importance. The development of this approach allows us to use the OWA operator in decision making under risk. Here, we are able include probabilistic information as well as decision attitude to construct customized decision functions.     

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.     

Choosing OWA operator weights in the field of Social Choice

One of the most important issues in the theory of OWA operators is the determination of associated weights. This matter is essential in order to use the best-suited OWA operator in each aggregation process. Given that some aggregation processes can be seen as extensions of majority rules to the field of gradual preferences, it is possible to determine the OWA operator weights by taking into account the class of majority rule that we want to obtain when individuals do not grade their pairwise preferences. However, a difficulty with this approach is that the same majority rule can be obtained through a wide variety of OWA operators. For this reason, a model for selecting the best-suited OWA operators is proposed in this paper.     

An argument‐dependent approach to determining OWA operator weights based on the rule of maximum entropy

The methods for determining OWA operator weights have aroused wide attention. We first review the main existing methods for determining OWA operator weights. We next introduce the principle of maximum entropy for setting up probability distributions on the basis of partial knowledge and prove that Xu's normal distribution‐based method obeys the principle of maximum entropy. Finally, we propose an argument‐dependent approach based on normal distribution, which assigns very low weights to these “false” or “biased” opinions and can relieve the influence of the unfair arguments. A numerical example is provided to illustrate the application of the proposed approach. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 209–221, 2007.     

Determination of Ordered Weighted Averaging Operator Weights Based on the M‐Entropy Measures

In this paper, based upon the M‐Entropy measures, two new models for obtaining the ordered weighted averaging (OWA) operators are propoosed. In these models, it is assumed, according to available information, that the OWA weights are in a decreasing or increasing order. Some properties of the models are analyzed, and the method of Lagrange multipliers is used to provide a direct way to find these weights. The models are solved with a specific level of orness comparing the results with some other related models. The results demonstrate the efficiency of the M‐Entropy models in generating the OWA operator weights.     

Some remarks on the LSOWA approach for obtaining OWA operator weights

One of the key issues in the theory of ordered‐weighted averaging (OWA) operators is the determination of their associated weights. To this end, numerous weighting methods have appeared in the literature, with their main difference occurring in the objective function used to determine the weights. A minimax disparity approach for obtaining OWA operator weights is one particular case, which involves the formulation and solution of a linear programming model subject to a given value of orness and the adjacent weight constraints. It is clearly easier for obtaining the OWA operator weights than from previously reported OWA weighting methods. However, this approach still requires solving linear programs by a conventional linear program package. Here, we revisit the least‐squared OWA method, which intends to produce spread‐out weights as much as possible while strictly satisfying a predefined value of orness, and we show that it is an equivalent of the minimax disparity approach. The proposed solution takes a closed form and thus can be easily used for simple calculations. © 2009 Wiley Periodicals, Inc.     

A New Ordered Weighted Averaging Operator to Obtain the Associated Weights Based on the Principle of Least Mean Square Errors

Determining OWA (ordered weighted averaging) weights has received more and more attention since the appearance of the OWA operator. Based on the principle of least mean squared errors, a new parametric OWA operator is proposed to obtain its associated weights. In coordination with fuzzy inference and a few of judgments on weights provided by decision makers (DMs), the new operator is carefully designed to avoid some problems of the existing ones, such as uncertainty in determining an objective function and the measure of orness, etc. Some properties of the problem are discussed to guarantee reliability in theory. A real‐life problem and two simulation experiments are performed to investigate its efficiency. All results show that the proposed operator can be a useful tool to express DMs’ preference information flexibly and objectively.     

基于OWA算子的区间值加权模糊推理

针对如何对区间值模糊产生式规则赋予合理权值的问题,将OWA算子引入到区间值模糊推理中。介绍一种基于OWA算子的区间值赋权方法,根据此方法给出区间值模糊集上的加权模糊产生式规则的推理算法。在采用该算法的过程中,为合理地计算输入事实与规则前件的匹配程度,引入基于OWA算子的区间值模糊匹配函数值和总体贴近度的计算方法。实例分析表明了所给出的区间值模糊推理算法的有效性和可行性。     

Generating Z‐number based on OWA weights using maximum entropy

In the application of Z‐number, how to generate Z‐number is a significant and open issue. In this paper, we proposed a method of generating Z‐number based on the OWA weights using maximum entropy considering the attitude (preference) of the decision maker. Some numerical examples are used to illustrate the effectiveness of the proposed method. Results show that the attitude (preference) of the decision maker can give an optimal possibility distribution of the reliability for Z‐number using maximum entropy.     

Sensitivity Analysis of Multiple Criteria Decision Making Method Based on the OWA Operator

Ordered weighted averaging (OWA) operator's weights and orness measure play important roles in the application of the OWA operator to decision‐making problems because the decision result may be different owing to the change in either of them. The aim of this paper is to investigate the influence that the change of OWA operator's weights or orness measure exerts on the decision result. We first give the range of the OWA operator's weights to keep the ranking order of alternatives or the optimal alternative unchanged. Then we make a sensitivity analysis to the orness measure to explore the dependency of the decision result on the orness measure. The results of analysis may provide a decision basis according to which decision makers are able to make a reasonable decision. Finally, a practical example is provided to illustrate the proposed sensitivity analysis methods.     

基于贝叶斯的改进WSNs信任评估模型

基于贝叶斯和熵,提出一种改进的WSNs信任评估模型。考虑到非入侵因素带来的网络异常行为,引入异常衰减因子,利用修正后的贝叶斯方程估算直接信任,并利用滑窗和自适应遗忘因子进行更新。根据直接信任的置信水平确定其是否足够可信来作为综合信任,减少网络能耗,并降低恶意反馈的影响。如果直接信任不足够可信,计算间接信任来获得综合信任,利用熵来对不同的推荐赋予权重,克服主观分配权重带来的局限性,加强模型的适应性。仿真实验表明,该模型能够有效检测恶意节点,具有较高的检测率和较低的误检率,同时在很大程度上降低了网络的能量消耗。     

Maximum expert consensus models with linear cost function and aggregation operators

In group decision making problems, consensus is a very important issue for the aggregation of individual opinions. Based on the concept of maximum expert consensus model (MECM), this paper incorporates aggregation operators into the MECM, and proposes a novel framework of MECM. When the aggregation operator is set to be the weighted averaging operator or the ordered weighed averaging (OWA) operator, this paper equivalently transforms the MECM into mixed 0–1 linear programming problems. Additionally, this paper also shows that the minimum cost consensus model under the OWA operator with any weights can be similarly transformed into a mixed 0–1 linear programming problem. Numerical examples and a comparison analysis are used to demonstrate the validity of the proposed model.     

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。     

An overview of methods for determining OWA weights

The ordered weighted aggregation (OWA) operator has received more and more attention since its appearance. One key point in the OWA operator is to determine its associated weights. In this article, I first briefly review existing main methods for determining the weights associated with the OWA operator, and then, motivated by the idea of normal distribution, I develop a novel practical method for obtaining the OWA weights, which is distinctly different from the existing ones. The method can relieve the influence of unfair arguments on the decision results by weighting these arguments with small values. Some of its desirable properties have also been investigated. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 843–865, 2005.     

Notes on properties of the OWA weights determination model

This paper solves the recently open problem related to the OWA weights determination minimax model presented by Amin and Emrouznejad [Amin G. R., & Emrouznejad, A. (2006). An extended minimax disparity to determine the OWA operator weights. Computers & Industrial Engineering, 50, pp. 312–316]. So the contribution of this work is that it explains further the properties of the proposed OWA weights determination minimax model.