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.     

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。     

Learning the stress function pattern of ordered weighted average aggregation using DBSCAN clustering

Ordered weighted average (OWA) operator provides a parameterized class of mean type operators between the minimum and the maximum. It is an important tool that can reflect the strategy of a decision maker for decision-making problems. In this study, the idea of obtaining the stress function from OWA weights has been put forward to generalize and characterize OWA weights. The main idea in this paper is mainly constructed on the basis that, generally, stress functions can be constructed using a mixture of constant and linear components. So, we can consider the stress function as a piecewise linear function. For obtaining stress functions as piecewise linear functions, we present a clustering-based approach for OWA weight generalization. This generalization is made using the DBSCAN algorithm as the learning method of a stress function associated with known OWA weights. In the learning process, the whole data set is divided into clusters, and then linear functions are obtained via a least squares estimator.     

Quantile-induced uncertain heavy ordered weighted averaging operator and the application in incentive evaluation problems

To uncertain evaluation problems, we integrate incentive management into the aggregation process and propose an aggregation operator called quantile-induced uncertain heavy ordered weighted averaging (QI-UHOWA) operator, which is an extension of the quantile-induced heavy ordered weighted averaging (QI-HOWA) operator. We provide an approach for determining the quantile order-inducing variables by using the technique for order preference by similarity to ideal solution method and the Hamming distance. In this case, the quantile values are measurements of relative developments of alternatives. Furthermore, we analyze the main properties of the operator including commutativity, boundedness, and monotonicity with uniform development space. The QI-UHOWA weighting vector is calculated using the maximum entropy measure with a given level of incentive attitude. We further expand the weighting method to the case of hierarchical stimulation. Moreover, the QI-UHOWA operator is generalized using the quasi-arithmetic mean. Finally, a numerical example regarding the selection of the optimal candidate(s) is given. The aggregation results are compared with those of the UOWA and QI-UOWA operator to illustrate the validity of the QI-UHOWA operator.     

Measuring global prosperity using data envelopment analysis and OWA operator

Prosperity is one of the key economic indicators of a nation's success. The measure of a country's true prosperity is best achieved by considering a set of criteria and identifying the optimal weights associated with each criterion. This study introduces a novel method for measuring global prosperity by employing a combination of variables that characterize economic wealth and social wellbeing using data envelopment analysis (DEA) and ordered weighted averaging (OWA) operator. It extends the existing global prosperity assessment approach proposed by the Legatum Institute, an international organization that produces a global prosperity index every year. The current Legatum Prosperity Index is obtained by averaging a set of distinct variables, but it fails to identify the optimal variable weights for each country. This is a significant drawback that we address in this study. Using DEA, each country can freely assign optimal weights that are most favorable to achieving maximum prosperity. It provides a flexible and competitive environment in which all countries can present their strengths, thereby creating a level playing field. This study also uses multilevel DEA efficiency frontiers for classifying countries into different groups based on their levels of prosperity score. Additionally, we apply the OWA operator to distinguish further between the countries within each cluster.     

Embedding OWA under preference ranking for DEA cross-efficiency aggregation: Issues and procedures

Cross-efficiency (CE) evaluation is an extension of data envelopment analysis (DEA) used for fully ranking decision-making units (DMUs). The ranking process is normally performed on the matrix of CE scores. An ultimate efficiency score is computed for each DMU through an adequate amalgamation process. The preference ranking approach can be seen as an amalgamation technique based on the rank orders of the CE scores. In this paper, we review this approach by putting more emphasis on the aggregation aspect. We highlight the zero vote issue and we show that the latter has been neglected in the extant aggregation procedures. Consequently, we develop two ordered weighted averaging (OWA)-based procedures that attempt to meet effectively the requirements of an aggregation mechanism while exploiting the positive properties of the preference-ranking approach. The merits of the proposed procedures are evaluated on a sample of manufacturing systems by considering, for OWA weights generation, different OWA models with different orness degrees.     

The use of ordered weighted averaging method for decision making under uncertainty

A decision making under uncertainty (DMUU) prevails at the outset and often evolves into a decision making under partial uncertainty as information on the states of nature, for example, a probability distribution, is advanced. Many methods have emerged for solving the DMUU problems, which includes the classical decision criteria and the domain criterion. Yager (1988) introduced a new approach, the so‐called ordered weighted averaging (OWA) as a viable method for solving the DMUU problems. The OWA weights to be used in the aggregation are generated under the degree of optimism provided by a decision maker and then combined with the reordered payoffs to produce aggregated payoffs for each strategy. The reordering process, one of the characterizing features of the OWA method, enables us to perform various types of aggregations including maximax, maximin, and Hurwicz‐α index in conjunction with the generated weights. The OWA method obviously extends the Hurwicz approach by taking into account the tradeoffs among the entire payoffs while the Hurwicz approach considers a tradeoff only between the two extremes, the maximum and the minimum payoffs. In this paper, we examine the features of the OWA method in light of Milnor's set of requirements for reasonable decision criteria, thus providing a solid methodological foundation for the DMUU. The OWA method can also be used to solve a group DMUU problem by exploiting individual decision results in the situation when the use of a fuzzy majority is advocated.     

Induced cluster-based OWA operators with reliability measures and the application in group decision-making

Considering the distributed structural characteristics of arguments to be aggregated, we propose a new type of aggregation operator, called induced cluster-based ordered weighted averaging (OWA; abbreviated as cluster-IOWA) operator, in this article. The main characteristic of the cluster-IOWA operator is that the arguments are aggregated by local clusters, and the order-inducing variable is used for representing a particular characteristic with respect to a local cluster. The cluster-OWA operator is commutativity, idempotence, and boundedness. We then discuss two important issues with respect to the cluster-IOWA operator. The order-inducing variables are determined by considering the overall reliability of the local cluster. Based on this, the position weighting vector of the local clusters is designed by taking into account both the reliability measures and the decision maker's preference. Finally, a numerical example, regarding the performance evaluation of middle managers carried out by a group of participants, is developed to illustrate the application and validity of the cluster-IOWA operator.     

Dynamic weights allocation according to uncertain evaluation information

Weights allocation methods are critical in Multi-Criteria Decision Making. Given numerical importances for each involved criterion, direct normalizing those numerical importances to obtain weights for those criteria is plain, lack of flexibility, and thus cannot well model some more types of subjective preferences of different decision makers like Dominance Strength as defined in this study. We show that concave RIM quantifier Q based OWA weights allocation method can well handle and model such preference. However, in real decision making those numerical importances are very often embodied by uncertain information such as independent random variables with discrete or continuous distributions, statistic information and interval numbers. In any of those circumstances, simple RIM quantifier Q based OWA weights allocation cannot work. Therefore, in this study, we will propose some special dynamic weights allocation methods to gradually allocate weights and accumulate allocated parts to each criterion, and finally, obtain a total weights collection. When the uncertain numerical importances become equivalent to general real numbers, the method automatically degenerates into general RIM quantifier based OWA weights allocation. The innovative weight allocations have discrete and continuous versions: the former can be well programmed while the latter has neat and succinct mathematical expression. The method can also be widely used in many other applications like some economic problems including investment quota allocation for one’s favorite stocks, and the dynamic OWA aggregation for interval numbers.     

On intuitionistic fuzzy decision-making using soft likelihood functions

Inspired by Yager, in this paper, we present the concept of likelihood for intuitionistic fuzzy sets (IFSs), and propose an approach for flexible computation of likelihood functions of IFSs for multicriteria decision-making (MCDM). We employ ordered weighted average (OWA) aggregation method to soften the strong likelihood constraint condition. The OWA measure can be considered as the attitudinal character, which determines OWA weights, including optimistic or pessimistic likelihood values. Then the reliability-based soft likelihood function is developed by considering the reliability of intuitionistic fuzzy information. Some examples are conducted by using the proposed (reliability-based) soft likelihood functions in intuitionistic fuzzy environment for MCDM problem, and the results are analyzed in detail.     

A priori identification of preferred alternatives of OWA operators by relational analysis of arguments

We present a reverse decision-aiding method that is distinct from previously reported ordered weighted averaging (OWA) aggregation methods. The proposed method is implemented in two phases. In the first phase, potentially best alternative, defined as one having any weighting vector that enables it to be at least as good as the others, is identified. In the second phase, the maximum and the minimum attitudinal characters for such an alternative are computed as the highest and the lowest values which it can attain under the weights-set identified in the first phase. These two phases are governed only by the relational analysis of input arguments, without soliciting the decision-maker to supply a specific attitudinal character. The proposed method can be applied to cases when it is difficult to obtain a precise attitudinal character and when, even if a precise one is obtained, the OWA operator weights are different, depending on the weights generating methods adopted. Further, if uncertain attitudinal character in the form of the interval number is available, its projection into the results of the proposed method yields less alternatives of consideration, in some cases, a single best alternative. The proposed method also allows for a priori identification of alternatives prone to change at a particular range of attitudinal character.     

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.     

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.     

基于梯形模糊隶属函数的复合语言多目标决策

对于一些复杂的决策问题, 使用比较语言比单一语言更能准确地表达专家的看法. 据此, 提出一种同时使用单一语言和比较语言的新算法. 根据上下文无关文法将比较语言表达转换为犹豫模糊语言术语集(HFLTS), 并应用有序加权算子(OWA) 计算出由梯形隶属函数表示的模糊语言术语集的模糊包络, 有效地简化了基于HFLTS 的词计算过程. 最后应用逼近理想解排序(TOPSIS) 方法进行决策.

     

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.     

Maximum Bayesian entropy method for determining ordered weighted averaging operator weights

Determination of the ordered weighted averaging (OWA) operators is an important issue in the theory of the OWA operator weights. In this paper, the main existing models for determining the OWA operator weights are outlined and the concept of the Bayesian entropy is introduced. Based upon the Bayesian entropy the maximum Bayesian entropy approach for obtaining the OWA operator weights is proposed. In this model it is assumed, according to previous experiences or from theoretical considerations that a decision maker may have reasons to consider a given prior OWA vector. Finally the new model is solved according to the prior OWA vector with specific level of orness comparing the results with other methods. The results demonstrate the efficiency of our model in generating the OWA operator weights. An applied example is also presented to illustrate the applications of the proposed model.     

Behavioral ordered weighted averaging operator and the application in multiattribute decision making

This paper introduces a new type of behavioral ordered weighted averaging (BOWA) operator, to incorporate decision maker’s gains and losses behavior tendency into the information aggregation process. The main characteristic of this BOWA operator is that it considers behavioral weights and ordered weights in the same formulation. We further provide a calculation method of the behavioral weights, in which various psychological preferences of different attribute types of the decision maker can be expressed intuitively. In addition, we discuss some particular cases of BOWA operator and its main properties. Finally, a numerical example is used to illustrate the use of the proposed method.     

大数据分割式极限学习机算法

在海量数据输入背景下，为提升极限学习机算法的学习速度，降低计算机内存消耗，提出一种分割式极限学习机算法。将海量数据分割成[K]等份，分别训练极限学习机并获得单一外权，基于算术平均算子得到分割式极限学习机的综合外权；为避免异常数据对极限学习机输出结果的影响，采用有序加权平均算子融合单一极限学习机的输出信息，使分割式极限学习机的输出结果更为稳定。数值对比仿真显示：分割式极限学习机比传统极限学习机的学习速度、拟合精度和内存消耗都高，验证了该方法的有效性和可行性。     

Using robust optimization to play against an imperfect opponent

 Traditional game theory is based on the assumption that the opponent is a perfect reasoner and all payoff information is available. Based on this assumption, game theory recommends to estimate the quality of each possible strategy by its worst possible consequences. In real-life, opponents are often not perfect and payoff information is often not exact. If the only disadvantage of some action is that an unusually clever opponent can find a complicated way to counter it, then this action may be a perfect recommendation for a play against a normal (not unusually clever) opponent. In other words, to estimate the quality of each move, instead of a normal minimum of possible consequences, we must consider the robust minimum that takes into consideration the fact that some of the consequences will never occur to the normal opponent. We show that in a reasonable statistical setting, this idea leads to the class of OWA operators. It turns out that playing against an imperfect opponent is not only a more realistic strategy, it is also often a simpler one: e.g., for the simplest game for which playing against a perfect opponent is computationally intractable (NP-hard), playing against an imperfect opponent is computationally feasible.     

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.