The relationship between the minimum-variance and minimax disparity RIM quantifier problems

Recently, Liu and Lou [On the equivalence of some approaches to the OWA operator and RIM quantifier determination, Fuzzy Sets and Systems 159 (2007) 1673-1688] investigated the equivalence of solutions to the minimum-variance and minimax disparity RIM quantifier problems. However, their proofs are very sensitive to the assumption, and some are mathematically incomplete. In this regard, this paper provides a counterexample of the minimax disparity RIM quantifier problem for the case in which generating functions are continuous. The paper also provides a correct proof of the minimax disparity RIM quantifier problem for the case in which generating functions are absolutely continuous and a generalized result for the minimum-variance RIM quantifier problem for the case in which generating functions are Lebesgue integrable. Based on the results, the paper provides a correct relationship between the minimum-variance and minimax disparity RIM quantifier problems.     

The orness measures for two compound quasi-arithmetic mean aggregation operators

The paper first summarizes the orness measures and their common characteristics of some averaging operators: the quasi-arithmetic mean, the ordered weighted averaging (OWA) operator, the regular increasing monotone (RIM) quantifier and the weighted function average operator, respectively. Then it focuses on the aggregation properties and operator determination methods for two kinds of quasi-arithmetic mean-based compound aggregation operators: the quasi-OWA (ordered weighted averaging) operator and the Bajraktarevi? mean. The former is the combination of the quasi-arithmetic mean and the OWA operator, while the latter is the combination of the quasi-arithmetic mean and the weighted function average operator. Two quasi-OWA operator forms are given, where the OWA operator is assigned directly or generated from a RIM (regular increasing monotone) quantifier indirectly. The orness indexes to reflect the or-like level of the quasi-OWA operator and Bajraktarevi? mean are proposed. With generating function techniques, the properties of the quasi-OWA operator and Bajraktarevi? mean are discussed to show the rationality of these orness definitions. Based on these properties, two families of parameterized quasi-OWA operator and Bajraktarevi? mean with exponential and power function generators are proposed and compared. It shows that the method of this paper can also be applied to other function-based aggregation operators.     

Parametric aggregation in ordered weighted averaging

Incorporating further information into the ordered weighted averaging (OWA) operator weights is investigated in this paper. We first prove that for a constant orness the minimax disparity model [13] has unique optimal solution while the modified minimax disparity model [16] has alternative optimal OWA weights. Multiple optimal solutions in modified minimax disparity model provide us opportunity to define a parametric aggregation OWA which gives flexibility to decision makers in the process of aggregation and selecting the best alternative. Finally, the usefulness of the proposed parametric aggregation method is illustrated with an application in metasearch engine.     

A combination approach for obtaining the minimize disparity OWA operator weights

One of the key issues in the theory of ordered weighted averaging operator is the determination of OWA operator weights. In this paper, a simple combination approach for obtaining minimal disparity OWA operator weights is proposed. The proposed approach generates the OWA operator weights by minimizing the combination disparity between any two adjacent weights and its expectation. This involves the formulation and solution of a linear programming model and a quadratic programming model for a given degree of orness. A numerical example demonstrated simpleness and effectiveness of the methods proposed in this paper.     

Generalized weighted exponential proportional aggregation operators and their applications to group decision making

We present a new aggregation operator called the generalized ordered weighted exponential proportional averaging (GOWEPA) operator, which is based on an optimal model. We study some properties and different families of the GOWEPA operator. We also generalize the GOWEPA operator. The key advantage of the GOWEPA operator is that it is an aggregation operator with theoretic basis on aggregation. Moreover, we propose an orness measure of the GOWEPA operator and indicate some properties of this orness measure. Furthermore, we introduce the least exponential squares method (LESM) to determine the GOWEPA operator weights based on its orness measure. In the end, we develop an application of the new approach in a case of group decision making in investment selection.     

Ordered median problem with demand distribution weights

The ordered median function unifies and generalizes most common objective functions used in location theory. It is based on the ordered weighted averaging (OWA) operator with the preference weights allocated to the ordered distances. Demand weights are used in location problems to express the client demand for a service thus defining the location decision output as distances distributed according to measures defined by the demand weights. Typical ordered median model allows weighting of several clients only by straightforward rescaling of the distance values. However, the OWA aggregation of distances enables us to introduce demand weights by rescaling accordingly clients measure within the distribution of distances. It is equivalent to the so-called weighted OWA (WOWA) aggregation of distances covering as special cases both the weighted median solution concept defined with the demand weights (in the case of equal all the preference weights), as well as the ordered median solution concept defined with the preference weights (in the case of equal all the demand weights). This paper studies basic models and properties of the weighted ordered median problem (WOMP) taking into account the demand weights following the WOWA aggregation rules. Linear programming formulations were introduced for optimization of the WOWA objective with monotonic preference weights thus representing the equitable preferences in the WOMP. We show MILP models for general WOWA optimization.     

Probability density functions based weights for ordered weighted averaging (OWA) operators: An example of water quality indices

This paper explores the application of ordered weighted averaging (OWA) operators to develop water quality index, which incorporates an attitudinal dimension in the aggregation process. The major thrust behind selecting the OWA operator for aggregation of multi-criteria is its capability to encompass a range of operators bounded between minimum and maximum. A new approach for generating OWA weight distributions using probability density functions (PDFs) is proposed in this paper. The basic parameters (mean and standard deviation) of the probability density functions can be determined using the number of criteria (e.g., water quality indicators) in the aggregation process.     

On characterizing features of OWA aggregation operators

We introduce the ordered weighted averaging (OWA) operator and emphasize how the choice of the weights, the weighting vector, allows us to implement different types of aggregation. We describe two important characterizing features associated with OWA weights. The first of these is the attitudinal character and the second is measure of dispersion. We discuss some methods for generating the weights and the role that these characterizing features can play in the determination of the OWA weights. We note that while in many cases these two features can help provide a clear distinction between different types of OWA operators there are some important cases in which these two characterizing features do not distinguish between OWA aggregations. In an attempt to address this we introduce a third characterizing feature associated with an OWA aggregation called the focus. We look at the calculation of this feature in a number of different situations.     

Bottleneck combinatorial optimization problems with uncertain costs and the OWA criterion

In this paper a class of bottleneck combinatorial optimization problems with uncertain costs is discussed. The uncertainty is modeled by specifying a discrete scenario set containing a finite number of cost vectors, called scenarios. In order to choose a solution the Ordered Weighted Averaging aggregation operator (OWA for short) is applied. The OWA operator generalizes traditional criteria in decision making under uncertainty such as the maximum, minimum, average, median, or Hurwicz criterion. New complexity and approximation results in this area are provided. These results are general and remain valid for many problems, in particular for a wide class of network problems.     

Distance measures with heavy aggregation operators

The use of distance measures and heavy aggregations in the ordered weighted averaging (OWA) operator is studied. We present the heavy ordered weighted averaging distance (HOWAD) operator. It is a new aggregation operator that provides a parameterized family of aggregation operators between the minimum distance and the total distance operator. Thus, it permits to analyze an aggregation from its usual average (normalized distance) to the sum of all distances available in the aggregation process. We analyze some of its main properties and particular cases such as the normalized Hamming distance, the weighted Hamming distance and the OWA distance (OWAD) operator. This approach is generalized by using quasi-arithmetic means obtaining the quasi-arithmetic HOWAD (Quasi-HOWAD) operator and with norms obtaining the heavy OWA norm (HOWAN). Further extensions to this approach are presented by using moving averages forming the moving HOWAD (HOWMAD) and the moving Quasi-HOWAN (Quasi-HOWMAN) operator. The applicability of the new approach is studied in a decision making model regarding the selection of national policies. We focus on the selection of monetary policies. The key advantage of this approach is that we can consider several sources of information that are independent between them.     

Robust hypothesis testing for asymmetric nominal densities under a relative entropy tolerance

In this paper, we address an open problem raised by Levy (2009) regarding the design of a binary minimax test without the symmetry assumption on the nominal conditional probability densities of observations. In the binary minimax test, the nominal likelihood ratio is a monotonically increasing function and the probability densities of the observations are located in neighborhoods characterized by placing a bound on the relative entropy between the actual and nominal densities. The general minimax testing problem at hand is an infinite-dimensional optimization problem, which is quite difficult to solve. In this paper, we prove that the complicated minimax testing problem can be substantially reduced to solve a nonlinear system of two equations having only two unknown variables, which provides an efficient numerical solution.     

Determining attribute weights to improve solution reliability and its application to selecting leading industries

In multiple attribute decision analysis, many methods have been proposed to determine attribute weights. However, solution reliability is rarely considered in those methods. This paper develops an objective method in the context of the evidential reasoning approach to determine attribute weights which achieve high solution reliability. Firstly, the minimal satisfaction indicator of each alternative on each attribute is constructed using the performance data of each alternative. Secondly, the concept of superior intensity of an alternative is introduced and constructed using the minimal satisfaction of each alternative. Thirdly, the concept of solution reliability on each attribute is defined as the ordered weighted averaging (OWA) of the superior intensity of each alternative. Fourthly, to calculate the solution reliability on each attribute, the methods for determining the weights of the OWA operator are developed based on the minimax disparity method. Then, each attribute weight is calculated by letting it be proportional to the solution reliability on that attribute. A problem of selecting leading industries is investigated to demonstrate the applicability and validity of the proposed method. Finally, the proposed method is compared with other four methods using the problem, which demonstrates the high solution reliability of the proposed method.     

Induced aggregation operators in the VIKOR method and its application in material selection

In this paper, a hybrid decision making approach integrating induced aggregation operators into VIKOR is proposed for tackling multicriteria problems with conflicting and noncommensurable (different units) criteria. For doing so, we develop a new distance aggregation operator called the induced ordered weighted averaging standardized distance (IOWASD) operator. It is an aggregation operator that provides a wide range of standardized distance measures between the maximum and the minimum by using the induced OWA (IOWA) operator. The main advantage of the IOWA-based VIKOR (IOWA-VIKOR) is that it is able to reflect the complex attitudinal character of the decision maker by using order inducing variables and provide much more complete information for decision making. We also studied some of the IOWASD’s main properties and different particular cases and further generalized it by using the induced generalized OWA (IGOWA) operator. Finally, we apply the integrated IOWA-VIKOR method in a multi-criteria decision making problem regarding the selection of materials and the results are compared for different types of standardized distance aggregation operators.     

一种模糊有序加权(FOWA)算子及其应用

针对多个三角模糊数的集结问题，提出一种新的模糊有序加权(FOWA)算子。该算子是对传统OWA算子的扩展，它使三角模糊数可根据其所在排序位置进行集结。分析FOWA算子所具有的性质，给出在群决策中模糊信息集结的一个应用算例。     

The uncertain probabilistic OWA distance operator and its application in group decision making

In this paper, we present the uncertain probabilistic ordered weighted averaging distance (UPOWAD) operator. Its main advantage is that it uses distance measures in a unified framework between the probability and the OWA operator that considers the degree of importance of each concept in the aggregation. Moreover, it is able to deal with uncertain environments represented in the form of interval numbers. We study some of its main properties and particular cases such as the uncertain probabilistic distance (UPD) and the uncertain OWA distance (UOWAD) operator. We end the paper by presenting an application to a group decision making problem regarding the selection of robots.     

Parameterized OWA operator weights: An extreme point approach

Since Yager first presented the ordered weighted averaging (OWA) operator to aggregate multiple input arguments, it has received much attention from the fields of decision science and computer science. A critical issue when selecting an OWA operator is the determination of the associated weights. For this reason, numerous weight generating methods, including rogramming-based approaches, have appeared in the literature. In this paper, we develop a general method for obtaining OWA operator weights via an extreme point approach. The extreme points are represented by the intersection of an attitudinal character constraint and a fundamental ordered weight simplex. The extreme points are completely identified using the proposed formula, and the OWA operator weights can then be expressed by a convex combination of the identified extreme points. With those identified extreme points, some new OWA operator weights can be generated by a centroid or a user-directed method, which reflects the decision-maker’s incomplete preferences. This line of reasoning is further extended to encompass situations in which the attitudinal character is specified in the form of interval with an aim to relieve the burden of specifying the precise attitudinal character, thus obtaining less-specific expressions that render human judgments readily available. All extreme points corresponding to the uncertain attitudinal character are also obtained by a proposed formula and then used to prioritize the multitude of alternatives. Meanwhile, two dominance rules are effectively used for prioritization of alternatives.     

Minimax regret solution to multiobjective linear programming problems with interval objective functions coefficients

The current paper focuses on a multiobjective linear programming problem with interval objective functions coefficients. Taking into account the minimax regret criterion, an attempt is being made to propose a new solution i.e. minimax regret solution. With respect to its properties, a minimax regret solution is necessarily ideal when a necessarily ideal solution exists; otherwise it is still considered a possibly weak efficient solution. In order to obtain a minimax regret solution, an algorithm based on a relaxation procedure is suggested. A numerical example demonstrates the validity and strengths of the proposed algorithm. Finally, two special cases are investigated: the minimax regret solution for fixed objective functions coefficients as well as the minimax regret solution with a reference point. Some of the characteristic features of both cases are highlighted thereafter.     

Aggregation of ordinal data using ordered weighted averaging operator weights

In multi-criteria decision-making problems, ordinal data themselves provide a convenient instrument for articulating preferences but they impose some difficulty on the aggregation process since ambiguity prevails in the preference structure inherent in the ordinal data. One of the key concerns in the aggregation of ordinal data is to differentiate among the rank positions by reflecting decision-maker??s preferences. Since individual attitude is fairly different, it is presumable that each ranking position has different importance. In other words, the quantification schemes among the rank positions could vary depending on the individual preference structure. We find that, among others, the ordered weighted averaging (OWA) operator can help to take this concept into effect on several reasons. First, the OWA operator provides a means to take into account a discriminating factor by introducing the measure of attitudinal character. Second, it can produce appropriate ranking weights corresponding to each rank position by solving a mathematical program subject to the constraint of attitudinal character. To better understand the attitudinal character playing a role as a discriminating factor, we develop centered ranking weights from ordinal weak relations among the ranking positions and then investigate their properties to relate them with the OWA operator weights having the maximum entropy. Finally, we present a method for generating the OWA operator weights via rank-based weighting functions.     

Belief structures,weight generating functions and decision-making

We describe the Dempster–Shafer belief structure and provide some of its basic properties. We introduce the plausibility and belief measures associated with a belief structure. We note that these are not the only measures that can be associated with a belief structure. We describe a general approach for generating a class of measures that can be associated with a belief structure using a monotonic function on the unit interval, called a weight generating function. We study a number of these functions and the measures that result. We show how to use weight-generating functions to obtain dual measures from a belief structure. We show the role of belief structures in representing imprecise probability distributions. We describe the use of dual measures, other then plausibility and belief, to provide alternative bounding intervals for the imprecise probabilities associated with a belief structure. We investigate the problem of decision making under belief structure type uncertain. We discuss two approaches to this decision problem. One of which is based on an expected value of the OWA aggregation of the payoffs associated with the focal elements. The second approach is based on using the Choquet integral of a measure generated from the belief structure. We show the equivalence of these approaches.     

A Lower Bound for the Penalty Parameter in the Exact Minimax Penalty Function Method for Solving Nondifferentiable Extremum Problems

In the paper, we consider the exact minimax penalty function method used for solving a general nondifferentiable extremum problem with both inequality and equality constraints. We analyze the relationship between an optimal solution in the given constrained extremum problem and a minimizer in its associated penalized optimization problem with the exact minimax penalty function under the assumption of convexity of the functions constituting the considered optimization problem (with the exception of those equality constraint functions for which the associated Lagrange multipliers are negative—these functions should be assumed to be concave). The lower bound of the penalty parameter is given such that, for every value of the penalty parameter above the threshold, the equivalence holds between the set of optimal solutions in the given extremum problem and the set of minimizers in its associated penalized optimization problem with the exact minimax penalty function.