Ordered Weighted Averaging Weights Research Articles

A preference elicitation approach for the ordered weighted averaging criterion using solution choice observations

Decisions under uncertainty or with multiple objectives usually require the decision maker to formulate a preference regarding risks or trade-offs. If this preference is known, the ordered weighted averaging (OWA) criterion can be applied to aggregate scenarios or objectives into a single function. Formulating this preference, however, can be challenging, as we need to make explicit what is usually only implicit knowledge. We explore an optimization-based method of preference elicitation to identify appropriate OWA weights. We follow a data-driven approach, assuming the existence of observations, where the decision maker has chosen the preferred solution, but otherwise remains passive during the elicitation process. We then use these observations to determine the underlying preference by finding the preference vector that is at minimum distance to the polyhedra of feasible vectors for each of the observations. Using our optimization-based model, weights are determined by solving an alternating sequence of linear programs and standard OWA problems. Numerical experiments on risk-averse preference vectors for selection, assignment and knapsack problems show that our passive elicitation method compares well against having to conduct pairwise comparisons and performs particularly well when there are inconsistencies in the decision maker’s choices.

Beta-Bézier OWA operator

Determination of associated weight vectors for an ordered weighted averaging (OWA) operator is always a crucial issue, and for this purpose, several weight vector determination models have been introduced in the literature. In this article, we propose a new biparametric OWA operator, called a “beta-Bézier OWA operator”. The beta-Bézier OWA operator provides an infinite number of weight vectors for a fixed degree of orness. The orness value of this class of OWA operators is its most essential characteristic, which always depends on one of its parameters, and can be prefixed according to the decision-maker choice and also irrespective of the number of arguments aggregated. The optimum OWA weight vector can be calculated without solving a complicated optimization problem. It becomes the same as several other OWA operators at particular values of parameters. The maximum Bayesian entropy OWA operator weight vector is measured with use of the proposed OWA operator for a given level of optimism/pessimism (orness). It is also compared with many other OWA operators.

Choquet integral optimisation with constraints and the buoyancy property for fuzzy measures

This work concerns solving optimisation problems where the objective function is expressed as a Choquet integral. This objective generalises a linear objective function (with positive weights) and allows for interaction to be modeled between coalitions of the decision variables. We leverage results from optimising the ordered weighted averaging (OWA) operator and propose efficient solution approaches for the asymmetric objectives both for the simplest case of a single constraint and then for multiple comonotone constraints. To solve problems with a large number of variables, we rely on the so-called antibuoyancy property, previously applied to OWA weights, and which we extend to general fuzzy measures. This characterisation not only facilitates a restriction of the domain on which the solution lies but also allows us to relate the Choquet integral’s behavior in such cases to the Pigou-Dalton progressive transfers principle. We characterise the Choquet integrals consistent with the Pigou-Dalton principle. Theoretical results are supported by numerical experiments, which illustrate significant gains in performance. Our results offer opportunities for scalability to a much higher number of variables.

A Surrogate Water Quality Index to assess groundwater using a unified DEA-OWA framework.

In this paper, we introduce a new approach, based on a unified framework incorporating Data Envelopment Analysis (DEA) and Ordered Weighted Averaging (OWA), for assessing water quality in contextual settings that involve a large number of hydrochemical parameters. In order to enhance discrimination among water sources, the DEA model is adopted with data-driven input variables, called "surrogate optimistic closeness values," computed through an aggregation procedure that includes the observed values of the hydrochemical parameters with OWA weights. The proposed DEA-OWA methodology has been employed to assess the quality of 51 water samples, collected from irrigation wells in Sereflikochisar Basin, Turkey, by means of 19 hydrochemical parameters. Using different subjectivity levels, the Surrogate Water Quality Indices (SWQIs) that are produced are proven effective in enhancing discrimination among the water sources while enabling a more robust water quality-based ranking. The k-means analysis has been used for clustering the water quality of the wells into Excellent, Good, Permissible, and Unsuitable rather than using pre-set boundaries. Only one water source has been identified as Excellent, whereas 17.65%, 45.10%, and 35.29% of the sampled wells, respectively, are categorized with Good, Permissible, and Unsuitable water quality. Inferred from wells' location, the results suggest that the groundwater might be drastically affected by saline water intrusion from Lake Tuz. The latter conclusion has been corroborated through a Tobit regression analysis.

Some bipolar-preferences-involved aggregation methods for a sequence of OWA weight vectors

The ordered weighted averaging (OWA) operator and its associated weight vectors have been both theoretically and practically verified to be powerful and effective in modeling the optimism/pessimism preference of decision makers. When several different OWA weight vectors are offered, it is necessary to develop certain techniques to aggregate them into one OWA weight vector. This study firstly details several motivating examples to show the necessity and usefulness of merging those OWA weight vectors. Then, by applying the general method for aggregating OWA operators proposed in a recent literature, we specifically elaborate the use of OWA aggregation to merge OWA weight vectors themselves. Furthermore, we generalize the normal preference degree in the unit interval into a preference sequence and introduce subsequently the preference aggregation for OWA weight vectors with given preference sequences. Detailed steps in related aggregation procedures and corresponding numerical examples are also provided in the current study.

Extended and infinite ordered weighted averaging and sum operators with numerical examples

This study discusses some variants of Ordered WeightedAveraging (OWA) operators and related information aggregation methods. Indetail, we define the Extended Ordered Weighted Sum (EOWS) operator and theExtended Ordered Weighted Averaging (EOWA) operator, which are applied inscientometrics evaluation where the preference is over finitely manyrepresentative works. As contrast, we also define the Infinite OrderedWeighted Sum (InOWS) operator and the Infinite Ordered Weighted Averaging(InOWA) operator, which are more suitable for the correspondingscientometrics evaluation where all of works of scholars are considered. Wealso define the family of Infinite Gaussian maxitive OWA weights functionand the family of Infinite Gaussian OWA weights function, and discuss someof their mathematical properties. Some illustrative examples, comparisonsand figures are provided to better expound their applicability inscientometrics evaluation.

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.

A New Family of OWA Operators Featuring Constant Orness

Determination of the weights of an ordered weighted averaging (OWA) operator is a vital constituent of the operator's aggregation procedure. Consequently, several weight generation techniques have evolved in the literature. Here, using inverse hypergeometric distribution, we use a novel parametric function to generate the weight vector of an OWA operator. The proposed inverse hypergeometric OWA (InHyp-OWA) operator generates a unique weight vector for every choice of its parameter. Also, using the proposed weight function, OWA weight vectors can be generated without solving any complicated optimization problem. An important property of InHyp-OWA operator is that for a given parameter value, its orness value remains constant, regardless of the number of objectives aggregated. Hence, InHyp-OWA operator is a new member of the family of OWA operators with constant orness. This class of OWA operators can utilize a prejudiced preference to determine the corresponding weight vector. The proposed approach provides a number of advantages over existing weight generating methods for OWA operators. One of the remarkable advantages of the InHyp-OWA operator is the fact that it generates weight vectors with strictly monotonic components. It also generalizes the well-known Borda–Kendall OWA operator.

The General Least Square Deviation OWA Operator Problem

A crucial issue in applying the ordered weighted averaging (OWA) operator for decision making is the determination of the associated weights. This paper proposes a general least convex deviation model for OWA operators which attempts to obtain the desired OWA weight vector under a given orness level to minimize the least convex deviation after monotone convex function transformation of absolute deviation. The model includes the least square deviation (LSD) OWA operators model suggested by Wang, Luo and Liu in Computers & Industrial Engineering, 2007, as a special class. We completely prove this constrained optimization problem analytically. Using this result, we also give solution of LSD model suggested by Wang, Luo and Liu as a function of n and α completely. We reconsider two numerical examples that Wang, Luo and Liu, 2007 and Sang and Liu, Fuzzy Sets and Systems, 2014, showed and consider another different type of the model to illustrate our results.

Elliptical distribution-based weight-determining method for ordered weighted averaging operators

The ordered weighted averaging (OWA) operators play a crucial role in aggregating multiple criteria evaluations into an overall assessment supporting the decision makers’ choice. One key point steps is to determine the associated weights. In this paper, we first briefly review some main methods for determining the weights by using distribution functions. Then we propose a new approach for determining OWA weights by using the regular increasing monotone quantifier. Motivated by the idea of normal distribution-based method to determine the OWA weights, we develop a method based on elliptical distributions for determining the OWA weights, and some of its desirable properties have been investigated.

Multiple criteria decision making for linguistic judgments with importance quantifier guided ordered weighted averaging operator

In this paper, an importance quantifier-guided Ordered Weighted Averaging (OWA) Operator for linguistic decision judgments is developed. To create a ‘softer’ decision aggregation task, the decision criteria are associated with the degree of importance in aggregation. The importance quantifier-guided OWA operator is extended from Yager’s OWA using linguistic quantifiers. These interval type 2 fuzzy sets are used to map the linguistic quantifiers. The first stage of applying the importance quantifier-guided OWA operator is to rank a set of interval type 2 fuzzy sets. A new interval type 2 fuzzy set ranking methodology is proposed. Such ranking methodology is concerned with two newly defined type 2 fuzzy set characteristics, the extended level set (alpha-cut) and the parametric generalized graded mean integration representation. The partial ordering set properties of the ranking methodology are rigorously discussed. The framework of multiple criteria decision making with linguistic judgments is constructed using the importance quantifier-guided OWA operator. The optimal importance quantifier-guided OWA type 2 fuzzy weights for decision criteria, and the overall decision fuzzy rates for alternatives are obtained. The mixed integer linear programming models are created specifically for solving the optimal importance quantifier-guided type 2 fuzzy weights. The piecewise linear and the general non-linear quantifier models are developed. The overall procedure for interval type 2 fuzzy sets multiple criteria decision making with the optimal importance quantifier-guided OWA weights is formulated. A multiple criteria decision making problem, New Product Development Project Screening, is used to demonstrate the proposed methodology.

Selecting a model for generating OWA operator weights in MAGDM problems by maximum entropy membership function

Abstract In this paper, we consider the assignment of ordered weighted averaging (OWA) operator weights to solve Multi-attributes Group Decision Making (MAGDM) problems with linguistic preference information. Since there are many available programming models that can be used for determining valid weights for attributes, choosing an appropriate OWA model for practical analysis is one of the key issues in MAGDM. Therefore, this research develops a novel approach that uses the prior OWA weights vectors and identifies an appropriate model by means of maximum entropy membership function from the a priori chosen OWA models to rank and/or evaluate alternatives. To develop the approach, we derive an analytic form for the maximum entropy membership function using the principle of maximum entropy and the Lagrange multipliers method. Then we present a bank branch example to demonstrate the applicability of our method.

Generating OWA weights using truncated distributions

Ordered weighted averaging (OWA) operators have been widely used in decision making these past few years. An important issue facing the OWA operators' users is the determination of the OWA weights. This paper introduces an OWA determination method based on truncated distributions that enables intuitive generation of OWA weights according to a certain level of risk and trade-off. These two dimensions are represented by the two first moments of the truncated distribution. We illustrate our approach with the well-know normal distribution and the definition of a continuous parabolic decision-strategy space. We finally study the impact of the number of criteria on the results.

Robustifying OWA Operators for Aggregating Data With Outliers

We propose a version of ordered weighted averaging (OWA) operators, which are robust against inputs with outliers. Outliers may heavily bias the outputs of the standard OWA. The penalty-based method proposed here comprises both outlier detection and reallocation of weights of the OWA. At the first stage, the outliers are identified based on a robust criterion that can accommodate up to half the inputs being outliers, but at the same time not removing the inputs unnecessarily. Three numerical algorithms for calculating the optimal value of this criterion are proposed. At the second stage, the OWA weights are recalculated for a subset of clean data while preserving the overall character of the weighting vector. The method is numerically tested on simulated data and exemplified on aggregating a large number of online ratings where the outliers represent biased, missing, or erroneous evaluations.

Sensitivity analysis of the ordered weighted averaging operator via linear models

The efficient computation of the weights in an ordered weighted averaging (OWA) operator plays an important role in the successful design and application of the OWA operator. In some applications, an OWA weight vector with positive and distinct components for any desired orness level is preferable. This paper proposes two complementary linear OWA (CLOWA) models for determining the weights and a compact form of the optimal weights in the general case (i.e., for any level of orness and n). The proposed models produce a unique OWA weight vector with distinct and positive components for any orness level α∈(0,1). In addition to a combined goodness measure, the sensitivity analysis on the outputs of theOWA operator with respect to the optimism degree of the decision maker (DM) is important. This study obtains the combined goodness measure and sensitivity analysis models for two linear methods: the minimax disparity (MD) and new proposed models. Then, to assign reliable ranks to the alternatives in the presence of two conflicting objectives—maximizing the combined goodness measure and minimizing its sensitivity to the optimism degree of the DM—the composite measure of goodness is extended. The proposed methods are applied in a water resource management problem.

An approach to obtain the generalized mixed linear stress function for known owa weights with artificial bee colony algorithm

Abstract
 OWA (Ordered Weighted Averaging) is a flexible aggregation operator which is come up with Yager to create a decision function in multi-criteria decision making. It is possible to determine how optimistic or pessimistic the decision maker's opinion with the value obtained from the weights of this operator. The determination of OWA weights cannot provide characterization by itself. If it is desired to aggregate various sized objects in terms of generalization and reusability of OWA weights, a more general form is needed. In this study, we propose the parameterized piecewise linear stress function and the approach to characterize OWA weights. The stress function is expressed by parameters which are obtained by artificial bee colony algorithm. Also the weights are approximately found by using parameters.
 Keywords – OWA operator, aggregation, artificial bee colony algorithm.

Analytic approach on maximum Bayesian entropy ordered weighted averaging operators

In this paper the maximum Bayesian entropy model for obtaining the ordered weighted averaging (OWA) operators was analyzed. The model is based on a given prior OWA vector and the obtained weights are called the maximum Bayesian entropy OWA (MBEOWA) weights. An analytic form for the MBEOWA weights was determined by using the Lagrange multipliers method and some properties of this form were investigated. Finally, a general model for determining the OWA weights based on the Bayesian entropy was introduced and discussed which included some other models. The efficiency of the proposed model was demonstrated by some examples.

The OWA Weights of Improved Minimax Disparity Model

One of the critical and prerequisite issues in the ordered weighted averaging (OWA) operator applications is the determination of the OWA operator weights. To this end, this paper removes some of the constraints of the improved minimax disparity (MD) model and obtains its optimal simplex tableau in the general case (i.e., for any level of orness and n), and then for the desired n introduces optimal basic feasible solutions of the model. The study also presents the weights of the preference ranking aggregation system without solving any model and suggests a secondary goal model for selecting a unique preference ranking aggregation weights through the alternative optimal weights of the improved MD model. The usefulness of the proposed methods is indicated by using an application to rank Ph.D. candidates.

A Model to Determine OWA Weights and Its Application in Energy Technology Evaluation

To determine the ordered weighted averaging (OWA) weights, the latent information function is developed to analyze the likelihood of occurrence for the preference value. The more likelihood a preference value is, the bigger the weight is, and vice versa. The proposed model is further extended to the situation where the preference value is an interval number by introducing a new method for interval number comparison. An example of energy technology evaluation is provided to demonstrate that the proposed approach is reasonable and simple.

Prioritization of textile fabric defects using ordered weighted averaging operator

Prioritizing and ranking products’ defects are important decision-making problems in many manufacturing sectors. This paper proposes a methodology for ranking defects based on the ordered weighted averaging (OWA) operator. The power of the proposed approach stems from its ability to rank defects with no restriction on the application context and no limitation on the number and type of variables used. Using a case study from the textile manufacturing industry, three OWA weights generating models have been implemented and it is shown that the OWA-based ranking approach is technically robust and practically flexible as a support for decision-making.