Xinwang Liu

On the properties of parametric geometric OWA operator

Based on the properties of ordered weighted averaging (OWA) operator weights, the parametric geometric OWA (PGOWA) operator and parametric maximum entropy OWA (PMEOWA) operator are proposed. The properties of PGOWA operator are analyzed. The consistence of the orness level and the aggregation value for any arbitrary aggregated elements with PGOWA weights is proved. The equivalence of PGOWA and PMEOWA is proved. With PGOWA operator, we cannot only generate maximum entropy OWA (MEOWA) weights with given orness degree more easily than the methods of Filev and Yager [Inform. Sci. 85 (1995) 11] and Fuller and Majlender [Fuzzy Sets Syst. 124 (1) (2001) 53], but also get the MEOWA weights with given aggregation results for a specific aggregated set.

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 makers 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.

Orness and parameterized RIM quantifier aggregation with OWA operators

A necessary and sufficient condition for the ordered weighted average (OWA) aggregation value of an arbitrary aggregated set to consistently increase with the orness level is proposed. The OWA operator properties associated with the orness level are extended. Then, with the generating function representation of Regular Increasing Monotone (RIM) quantifier, all these conditions and properties are extended to the case of RIM quantifiers, which can be seen as the continuous OWA operator with free dimension. Some families of consistency RIM quantifiers and their corresponding OWA operators are summarized. Some existing linguistic term RIM quantifiers are collected and two parameterized generalization forms of them are proposed, which can be useful for the selection and comparison of the linguistic quantifier in theory and applications.

Two new models for determining OWA operator weights

The determination of ordered weighted averaging (OWA) operator weights is a very crucial issue of applying the OWA operator for decision making. This paper proposes two new models for determining the OWA operator weights. The weights determined by the new models do not follow a regular distribution and therefore make more sense than those obtained by other methods.

The solution equivalence of minimax disparity and minimum variance problems for OWA operators

The aim of this paper is to answer the open question of Wang and Parkan (Information Sciences 175 (2005), 20-29) that the solutions of maximum entropy OWA operator problem under given orness level and the minimax disparity OWA operator problem under given orness level are equivalent. They both have the same equidifferent form, which composes a weighting vector of nonnegative arithmetic progression and zeros. This equidifferent OWA weighting vector generating method can also be seen as an improved solution method for the minimum variance and minimax dispersion problems respectively.

A general model of parameterized OWA aggregation with given orness level

The paper proposes a general optimization model with separable strictly convex objective function to obtain the consistent OWA (ordered weighted averaging) operator family. The consistency means that the aggregation value of the operator monotonically changes with the given orness level. Some properties of the problem are discussed with its analytical solution. The model includes the two most commonly used maximum entropy OWA operator and minimum variance OWA operator determination methods as its special cases. The solution equivalence to the general minimax problem is proved. Then, with the conclusion that the RIM (regular increasing monotone quantifier) can be seen as the continuous case of OWA operator with infinite dimension, the paper further proposes a general RIM quantifier determination model, and analytically solves it with the optimal control technique. Some properties of the optimal solution and the solution equivalence to the minimax problem for RIM quantifier are also proved. Comparing with that of the OWA operator problem, the RIM quantifier solutions are usually more simple, intuitive, dimension free and can be connected to the linguistic terms in natural language. With the solutions of these general problems, we not only can use the OWA operator or RIM quantifier to obtain aggregation value that monotonically changes with the orness level for any aggregated set, but also can obtain the parameterized OWA or RIM quantifier families in some specific function forms, which can incorporate the background knowledge or the required characteristic of the aggregation problems.

On the properties of equidifferent OWA operator

Getting OWA weights under given orness level is an active topic in the OWA operator research. The paper proposes a series of weights generating methods in equidifferent forms. Similar to the geometric (maximum entropy) OWA operator, we propose a parameterized OWA operator called equidifferent OWA operator, which consist the adjacent weighes with a common difference. The maximum spread equidifferent OWA (MSEOWA) operator is equivalent to the minimum variance OWA operator, but is more computational efficient. Some properties associated with the orness level are discussed. One of them is that the aggregation value for any elements set is always increasing with the orness level, which can used as a parameterized aggregation method with orness as its control parameter. These properties similar to that of the geometric (maximum entropy) OWA operator, which can also be seen as the discrete case of equidifferent RIM (regular increasing monotone) quantifiers. The general forms of equidifferent OWA operator are proposed, and the weights generating methods are also extended in a similar way.

A group decision making model considering both the additive consistency and group consensus of intuitionistic fuzzy preference relations

An approach is proposed to improve the consistency level of IFPR.One algorithm is developed to achieve acceptable level of consensus about IFPRs.A framework is developed for GDM with IFPRs. Group decision making with intuitionistic fuzzy preference information involves three steps: the consistency checking process, the consensus checking process and the selection process. In this paper, we investigate the above three steps with respect to additive intuitionistic fuzzy preference relation (IFPR). Firstly, a consistency index is introduced to measure the additive consistency level of IFPR and an automated approach is developed to improve the consistency level with respect to IFPR to an acceptable level. Meanwhile, the group consensus index of IFPR is defined and one algorithm for reaching acceptable level of consensus is developed. Secondly, after implementing this modified algorithm, consistency index and group consensus index are investigated. And we prove that they can hold on, even receive better levels. Thirdly, a framework for group decision making is developed based on IFPR with additive consistency and group consensus. Finally, an illustrative example is presented to demonstrate the effectiveness and applicability of the proposed approach.

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 makers preference.

On the Stress Function-Based OWA Determination Method With Optimization Criteria

The ordered weighted averaging (OWA) determination method with stress function was proposed by Yager, and it makes the OWA operator elements scatter in the shape of the stress function. In this paper, we extend the OWA determination with the stress function method using an optimization model. The proposed method transforms the OWA optimal solution elements into the interpolation points of the stress function. The proposed method extends the basic form of the stress function method with both scale and vertical shift transformations. We also explore a number of properties of this optimization-based stress function method. The OWA operator optimal solution elements can distribute as the shape of the given stress function in a parameterized way, in which case, the solution always possesses the arithmetic average operator as a special case.