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2011 | Buch

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Recent Developments in the Ordered Weighted Averaging Operators: Theory and Practice

herausgegeben von: Ronald R. Yager, Janusz Kacprzyk, Gleb Beliakov

Verlag: Springer Berlin Heidelberg

Buchreihe : Studies in Fuzziness and Soft Computing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This volume presents the state of the art of new developments, and some interesting and relevant applications of the OWA (ordered weighted averaging) operators. The OWA operators were introduced in the early 1980s by Ronald R. Yager as a conceptually and numerically simple, easily implementable, yet extremely powerful general aggregation operator. That simplicity, generality and implementability of the OWA operators, combined with their intuitive appeal, have triggered much research both in the foundations and extensions of the OWA operators, and in their applications to a wide variety of problems in various fields of science and technology.

Part I: Methods includes papers on theoretical foundations of OWA operators and their extensions. The papers in Part II: Applications show some more relevant applications of the OWA operators, mostly means, as powerful yet general aggregation operators. The application areas are exemplified by environmental modeling, social networks, image analysis, financial decision making and water resource management.

Inhaltsverzeichnis

Frontmatter

Methods

Frontmatter

OWA Operators and Nonadditive Integrals

Abstract

We give a survey on the relations between nonadditive integrals (Choquet integral, Sugeno integral) and the OWA operator and its variants. We give also some behavioral indices for the OWA operator, as orness, veto and favor indices, etc. Finally, we propose the use of p-symmetric capacities for a natural generalization of the OWA operator.

Michel Grabisch

The WOWA Operator: A Review

Abstract

The WOWA operator (Weighted OWA) was proposed as a generalization of both the OWA and the Weighted mean. Formally, it is an aggregation operator that permits the aggregation of a set of numerical data with respect to two weighting vectors: one corresponding to the one of the weighted mean and the other corresponding to the one of the OWA. In this chapter we review this operator as well as some of its main results.

Vicenç Torra

Induced Ordered Weighted Averaging Operators

Abstract

Since the introduction of the ordered weighted averaging operator [18], the OWA has received great attention with applications in fields including decision making, recommender systems [8, 21], classification [10] and data mining [16] among others. The most important step in the calculation of the OWA is the permutation of the input vector according to the size of its arguments. In some applications, it makes sense that the inputs be reordered by values different to those used in calculation. For instance, if we have a number of mobile sensor readings, we may wish to allocate more importance to the reading taken from the sensor closest to us at a given point in time, rather than the largest reading.

Gleb Beliakov, Simon James

A Review of the OWA Determination Methods: Classification and Some Extensions

Abstract

The OWA operator determination is an important prerequisite step for OWA operator applications. With the application of OWA operator in various areas, the OWA operator determination becomes an active topic in recent years. Based on recent developments, the paper give a summary on the OWA determination methods in classification way: the optimization criteria methods, the sample learning methods, the function based methods, the argument dependent methods and the preference methods. Some relationships between the methods in the same kind and the relationships between different kinds are provided. An uniform framework to connect these OWA determination methods together is also attempted. Some extensions, problems and future research directions are given with discussions.

Xinwang Liu

Fuzzification of the OWA Operators for Aggregating Uncertain Information with Uncertain Weights

Abstract

Yager’s ordered weighted averaging (OWA) operator has been widely applied in various domains. Yager’s traditional OWA operator focuses exclusively on the aggregation of crisp numbers with crisp weights. However, uncertainty prevails in almost every process of real world decision making, and so there is a need to find OWA mechanisms to aggregate uncertain information. In this chapter, we generalise Yager’s OWA operator and describe two novel uncertain operators, namely the type-1 OWA operator and type-2 OWA operator. The type-1 OWA operator is able to aggregate type-1 fuzzy sets, whilst the type-2 OWA operator is able to aggregate type-2 fuzzy sets. Therefore, the two new operators are capable of aggregating uncertain opinions or preferences with uncertain weights in soft decision making. This chapter also indicates that not only Yager’s OWA operator but also some existing operators of fuzzy sets, including the join and meet of type-1 fuzzy sets, are special cases of the type-1 OWA operators. We further suggest the concepts of joinness and meetness of type-1 OWA operators, which can be considered as the extensions of the concepts-orness and andness in Yager’s OWA operator, respectively. Given the high computing overhead involved in aggregating general type-2 fuzzy sets using the type-2 OWA operator, an interval type-2 fuzzy sets oriented OWA operator is also defined. Some examples are provided to illustrate the proposed concepts.

Shang-Ming Zhou, Francisco Chiclana, Robert I. John, Jonathan M. Garibaldi

A Majority Guided Aggregation Operator in Group Decision Making

Abstract

This chapter is about majority modelling in the context of group (multi-expert) decision making, to the aim of defining a decision strategy which takes into account the individual opinions of the decision makers. The concept of majority plays in this context a key role: what is often needed is an overall opinion which synthesizes the opinions of the majority of the experts. The reduction of the individual experts’ opinions into a representative value (which we call the majority opinion) is usually performed through an aggregation process. In this chapter we describe two distinct approaches to the definition and consequent computation of a majority opinion within fuzzy set theory, where majority can be expressed by a linguistic quantifier (such as most). We first consider the case where linguistic quantifiers are associated with aggregation operators; in this case a majority opinion is computed by aggregating the individual opinions. To model this semantics of linguistic quantifiers the Induced Ordered Weighted Averaging operators (IOWA) are used with a modified definition of their weighting vector. We then consider a second case where the concept of majority is modelled as a vague concept. Based on this interpretation a formalization of a fuzzy majority opinion as a fuzzy subset is described.

Gabriella Pasi, Ronald R. Yager

Generating OWA Weights from Individual Assessments

Abstract

In this contribution we propose a method for generating OWA weighting vectors from the individual assessments on a set of alternatives in such a way that these weights minimize the disagreement among individual assessments and the outcome provided by the OWA operator. For measuring that disagreement we have aggregated distances between individual and collective assessments by using a metric and an aggregation function. We have paid attention to Manhattan and Chebyshev metrics and arithmetic mean and maximum as aggregation functions. In this setting, we have proven that medians and the mid-range are the solutions for some cases. When a general solution is not available, we have provided some mathematical programs for solving the problem.

José Luis García-Lapresta, Bonifacio Llamazares, Teresa Peña

The Role of the OWA Operators as a Unification Tool for the Representation of Collective Choice Sets

Abstract

We consider various group decision making and voting procedures presented in the perspective of two kinds of aggregation of partial scores related to the individuals’ (group’s) testimonies with respect to alternatives and individuals. We show that the ordered weighthed averaging (OWA) operators can be viewed as a unique aggregation tool that – via the change of the order of aggregation, type of aggregation, etc. – can be used for a uniform and elegant formalization of basic group decision making, social choice and voting rules under fuzzy and nonfuzzy preference relations and fuzzy and nonfuzzy majority.

Janusz Kacprzyk, Hannu Nurmi, Sławomir Zadrożny

Applying Linguistic OWA Operators in Consensus Models under Unbalanced Linguistic Information

Abstract

In Group Decision Making (GDM) the automatic consensus models are guided by different consensus measures which usually are obtained by aggregating similarities observed among experts’ opinions. Most GDM problems based on linguistic approaches use symmetrically and uniformly distributed linguistic term sets to express experts’ opinions.However, there exist problemswhose assessments need to be represented by means of unbalanced linguistic term sets, i.e., using term sets which are not uniformly and symmetrically distributed. The aim of this paper is to present different Linguistic OWA Operators to compute the consensus measures in consensusmodels for GDMproblems with unbalanced fuzzy linguistic information.

E. Herrera-Viedma, F. J. Cabrerizo, I. J. Pérez, M. J. Cobo, S. Alonso, F. Herrera

Applications

Frontmatter

Fusion Strategies Based on the OWA Operator in Environmental Applications

Abstract

Ill-known environmental phenomena are often modeled by means of multisource spatial data fusion. Generally, these fusion strategies have to cope with distinct kinds of uncertainty, related to the ill-defined knowledge of the phenomenon, the lack of classified data, the distinct trust of the information sources, the imprecision of the observed variables. In this chapter we discuss the advantage of modeling multisource spatial data fusion in the environmental field based on the OWA operator, and overview two applications. The first application is aimed at defining an environmental indicator of anomaly at continental scale based on a fusion of partial hints of evidence of anomaly. The second application computes seismic hazard maps based on a consensual fusion strategy defined by an extended OWA operator that accounts for data imprecision, and reliability of the data sources. In particular, the proposed fusion function models a consensual dynamics and is parameterized so as to consider a varying spatial neighborhood of the data to fuse.

G. Bordogna, M. Boschetti, A. Brivio, P. Carrara, M. Pagani, D. Stroppiana

Decision Making with Dempster-Shafer Theory Using Fuzzy Induced Aggregation Operators

Abstract

We develop a new approach for decision making with Dempster-Shafer theory of evidence where the available information is uncertain and it can be assessed with fuzzy numbers. With this approach, we are able to represent the problem without losing relevant information, so the decision maker knows exactly which are the different alternatives and their consequences. For doing so, we suggest the use of different types of fuzzy induced aggregation operators in the problem. Then, we can aggregate the information considering all the different scenarios that could happen in the analysis. As a result, we get new types of fuzzy induced aggregation operators such as the belief structure – fuzzy induced ordered weighted averaging (BS-FIOWA) and the belief structure – fuzzy induced hybrid averaging (BS-FIHA) operator. We study some of their main properties. We further generalize this approach by using fuzzy induced generalized aggregation operators. We also develop an application of the new approach in a financial decision making problem about selection of financial strategies.

José M. Merigó, Montserrat Casanovas

Two Methods for Image Compression/Reconstruction Using OWA Operators

Abstract

In this chapter we address image compression by means of two alternative algorithms. In the first algorithm, we associate to each image an interval-valued fuzzy relation, and we build an image which is n times smaller than the original one, by using two-dimensional OWA operators. The experimental results show that, in this case, best results are obtained with ME-OWA operators. In the second part of the work, we describe a reduction algorithm that replaces the image by several eigen fuzzy sets associated with it. We obtain these eigen fuzzy sets by means of an equation that relates the OWA operators we use and the relation (image) we consider. Finally, we present a reconstruction method based on an algorithm which minimizes a cost function, with this cost function built by means of two-dimensional OWA operators.

H. Bustince, D. Paternain, B. De Baets, T. Calvo, J. Fodor, R. Mesiar, J. Montero, A. Pradera

OWA-Based Fuzzy m-ary Adjacency Relations in Social Network Analysis

Abstract

In this paper we propose an approach to Social Network Analysis (SNA) based on fuzzy m-ary adjacency relations. In particular, we show that the dimension of the analysis can naturally be increased starting from the traditional two–dimensional case and interesting results can be derived. Therefore, fuzzy m-ary adjacency relations can be computed starting from fuzzy binary relations and introducing OWA-based aggregations. The behavioral assumptions derived from the measure and the exam of individual propensity to connect with other suggest that OWA operators can be considered particularly appealing tools in characterizing such relations.

Matteo Brunelli, Mario Fedrizzi, Michele Fedrizzi

Soft Computing in Water Resources Management by Using OWA Operator

Abstract

The Ordered Weighted Averaging (OWA) operator is an efficient multi criteria decision making (MCDM) method. This study introduces a new method to obtain the order weights of this operator. The new method is based on the combination of fuzzy quantifiers and neat OWA operators. Fuzzy quantifiers are applied for soft computing in modeling the social preferences (optimism degree of the decision maker, DM). In using neat operators, the ordering of the inputs is not needed resulting in better computation efficiency.

One of the frequently-used ways to control water shortages is inter-basin water transfer (IBWT). Efficient decision making on this subject is however a real challenge for the water institutions. These decisions should include multiple criteria, model uncertainty, and also the optimistic/pessimistic view of the decision makers. The theoretical results are illustrated by ranking four IBWT projects for the Zayanderud basin, Iran. The results demonstrate that by using the new method, more sensitive decisions can be obtained to deal with limited water resources.

The results of this study also show that this new method is more appropriate than the other traditional MCDM methods in systems engineering since it takes the optimism/pessimism nature of the DM into account in a quantifiable way. The comparison of the computational results with the current state of the projects shows the optimistic character of the DM. A sensitivity analysis illustrates how the rankings of the water projects depend on the optimism degree of the DMs.

Mahdi Zarghami, Ferenc Szidarovszky

Combination of Similarity Measures in Ontology Matching Using the OWA Operator

Abstract

In this paper, we provide a novel solution for ontology matching by using the ordered weighted average (OWA) operator to aggregate multiple values obtained from different similarity measures. We have implemented the solution in the ontology matching system FOAM. Using the similarity measures in FOAM, we analyze the way to choose different OWA operators and compare our system with others.

Qiu Ji, Peter Haase, Guilin Qi

Backmatter

Titel: Recent Developments in the Ordered Weighted Averaging Operators: Theory and Practice
herausgegeben von: Ronald R. Yager
Janusz Kacprzyk
Gleb Beliakov
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-17910-5
Print ISBN: 978-3-642-17909-9
DOI: https://doi.org/10.1007/978-3-642-17910-5

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