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.     

Generalized triangular norm and conorm aggregation operators on ordinal spaces

We describe the basic features of the t-norm operator and then introduce a family of t-norm operators that are defined on an ordinal space. We then do the same for the t-conorms. We note the strong limitation that the requirement of associativity places on the t-norm and t-conorm operators. We particularly note how it limits our ability to model different types of reinforcement. We then define a generalization of the t-conorm aggregation operator, which relaxes the requirement of associativity, we denote these operators as GENOR operators. We show that these operators have the same functionality as the t-conorm. We provide some examples of GENOR operators which allow us to control the reinforcement process. We define a related extension for the t-norm, the GENAND operator and provide some examples.     

Parameterized additive neat OWA operators with different orness levels

The article proposes an extension of the BADD OWA operator—ANOWA (additive neat OWA) operator—and defines its orness measure. Some properties of the weighting function associated with orness level are analyzed. Then two special classes of ANOWA operator with maximum entropy and minimum variance are proposed, and the orness of the BADD OWA operator is discussed. For a given orness level, these ANOWA operators can be uniquely determined. Their aggregation values for any aggregation elements set always monotonically increase with their orness levels. Therefore they can be used as a parameterized aggregation method with orness as its control parameter and to represent the decision maker's preference. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 1045–1072, 2006.     

On Obtaining Piled OWA Operators

In this study, we propose the concept of piled ordered weighted averaging (OWA) operators, which generalize the centered OWA operators and also connect the step OWA operators with the Hurwicz OWA operators with given the orness degree. We propose a controllable algorithm to generate the family of piled OWA operators depending on their predefined three parameters: orness degree, step‐like or Hurwicz‐like degree, and the numbers of “supporting” vectors. By these preferences, we can generate infinite more piled OWA operators with miscellaneous forms, and each of them is similar to the well‐known binomial OWA operator, which is very useful but only has one form corresponding to one given orness degree.     

On the dispersion measure of OWA operators

We describe some basic features of the OWA operator. We turn to the problem of determining the weights associated with this operator and particularly the maximal dispersion (entropy) approach. We consider the possibility of using minimization of dispersion. After discussing concerns with both maximization and minimization of dispersion we investigate the possibility of finding an optimal solution intermediate to these extremes. We next consider alternative measures of dispersion. We introduce a fundamental requirement for a measure of dispersion called the Preference for Equal Division. A number of general classes of dispersion measures are provided notable among these are those based on t-norm and t-conorm operators.     

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.     

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.     

Interval-valued intuitionistic fuzzy matrix games based on Archimedean t-conorm and t-norm

Fuzzy game theory has been applied in many decision-making problems. The matrix game with interval-valued intuitionistic fuzzy numbers (IVIFNs) is investigated based on Archimedean t-conorm and t-norm. The existing matrix games with IVIFNs are all based on Algebraic t-conorm and t-norm, which are special cases of Archimedean t-conorm and t-norm. In this paper, the intuitionistic fuzzy aggregation operators based on Archimedean t-conorm and t-norm are employed to aggregate the payoffs of players. To derive the solution of the matrix game with IVIFNs, several mathematical programming models are developed based on Archimedean t-conorm and t-norm. The proposed models can be transformed into a pair of primal–dual linear programming models, based on which, the solution of the matrix game with IVIFNs is obtained. It is proved that the theorems being valid in the exiting matrix game with IVIFNs are still true when the general aggregation operator is used in the proposed matrix game with IVIFNs. The proposed method is an extension of the existing ones and can provide more choices for players. An example is given to illustrate the validity and the applicability of the proposed method.     

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.     

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.     

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.     

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.     

New decision-making techniques and their application in the selection of financial products

We develop a new approach that uses the ordered weighted averaging (OWA) operator in the selection of financial products. In doing so, we introduce the ordered weighted averaging distance (OWAD) operator and the ordered weighted averaging adequacy coefficient (OWAAC) operator. These aggregation operators are very useful for decision-making problems because they establish a comparison between an ideal alternative and available options in order to find the optimal choice. The objective of this new model is to manipulate the attitudinal character of previous methods based on distance measures, so that the decision maker can select financial products according to his or her degree of optimism, which is also known as the orness measure. The main advantage of using the OWA operator is that we can generate a parameterized family of aggregation operators between the maximum and the minimum. Thus, the analysis developed in the decision process by the decision maker is much more complete, because he or she is able to select the particular case in accordance with his or her interests in the aggregation process. The paper ends with an illustrative example that shows results obtained by using different types of aggregation operators in the selection of financial products.     

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.     

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.     

New triangular operator generators for fuzzy systems

Triangular operators (t-operators) form an integral part in the design and analysis of fuzzy systems. Simple monotonic, continuous, nonconditional functions are used in an operator generator to generate t-operators. Depending on the operator generator and the function that it uses, it becomes easier to characterize and classify the families of t-operators. In this paper, the author proposes two operator generators that will extend the domain of triangular operators in the realm of fuzzy set theory. The conventional operator generators generate a t-norm and a t-conorm by using a decreasing function and an increasing function, respectively. In contrast, in this study, increasing functions generate t-norms, while decreasing functions generate t-conorms, respectively     

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.     

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.     

The uncertain OWA aggregation with weighting functions having a constant level of orness

Since the ordered weighted averaging (OWA) operator was introduced by Yager [IEEE Trans Syst Man Cybern 1988;18:183–190], numerous aggregation operators have been presented in academic journals. Apart from a setting where exact numerical assessments on weights and input arguments can be obtained, the issue of generalizing the OWA to take into account uncertainties in weights and/or input arguments has been considered. Recently, Xu and Da [Int J Intell Syst 2002;17:569–575] proposed an uncertain OWA operator in which input arguments are given in the form of interval numbers. The interval numbers within the interval sometimes do not have the same meaning for the decision maker as is implied by the use of interval ranges. Thus, we present a way of prioritizing interval numbers, taking into account the strength of preference based on the probabilistic measure. Further, rank‐based weighting functions having constant values of orness irrespective of the number of objectives aggregated are presented and a final rank ordering of courses of action is performed by the use of those weighing functions. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 469–483, 2006.