Gleb Beliakov, Ana Pradera, Tomasa Calvo

Aggregation Functions: A Guide for Practitioners

Studies in Fuzziness and Soft Computing, Volume 221

Editor-in-chiefProf. Janusz KacprzykSystems Research InstitutePolish Academy of Sciencesul. Newelska 601-447 WarsawPolandE-mail: kacprzyk@ibspan.waw.pl

Further volumes of this series can befound on our homepage: springer.com

Vol. 208. Oscar Castillo, Patricia Melin,Janusz Kacprzyk, Witold Pedrycz (Eds.)Hybrid Intelligent Systems, 2007ISBN 978-3-540-37419-0

Vol. 209. Alexander Mehler,Reinhard KöhlerAspects of Automatic Text Analysis, 2007ISBN 978-3-540-37520-3

Vol. 210. Mike Nachtegael, DietrichVan der Weken, Etienne E. Kerre,Wilfried Philips (Eds.)Soft Computing in Image Processing, 2007ISBN 978-3-540-38232-4

Vol. 211. Alexander GegovComplexity Management in FuzzySystems, 2007ISBN 978-3-540-38883-8

Vol. 212. Elisabeth Rakus-AnderssonFuzzy and Rough Techniques in MedicalDiagnosis and Medication, 2007ISBN 978-3-540-49707-3

Vol. 213. Peter Lucas, José A. Gámez,Antonio Salmerón (Eds.)Advances in Probabilistic GraphicalModels, 2007ISBN 978-3-540-68994-2

Vol. 214. Irina GeorgescuFuzzy Choice Functions, 2007ISBN 978-3-540-68997-3

Vol. 215. Paul P. Wang, Da Ruan,Etienne E. Kerre (Eds.)Fuzzy Logic, 2007ISBN 978-3-540-71257-2

Vol. 216. Rudolf SeisingThe Fuzzification of Systems, 2007ISBN 978-3-540-71794-2

Vol. 217. Masoud Nikravesh, Janusz Kacprzyk,Lofti A. Zadeh (Eds.)Forging New Frontiers: FuzzyPioneers I, 2007ISBN 978-3-540-73181-8

Vol. 218. Masoud Nikravesh, Janusz Kacprzyk,Lofti A. Zadeh (Eds.)Forging New Frontiers: FuzzyPioneers II, 2007ISBN 978-3-540-73184-9

Vol. 219. Roland R. Yager, Liping Liu (Eds.)Classic Works of the Dempster-Shafer Theoryof Belief Functions, 2007ISBN 978-3-540-25381-5

Vol. 220. Humberto Bustince,Francisco Herrera, Javier Montero (Eds.)Fuzzy Sets and Their Extensions:Representation, Aggregation and Models, 2007ISBN 978-3-540-73722-3

Vol. 221. Gleb Beliakov, Tomasa Calvoand Ana PraderaAggregation Functions: A Guidefor Practitioners, 2007ISBN 978-3-540-73720-9

Gleb BeliakovAna PraderaTomasa Calvo

Aggregation Functions:A Guide for Practitioners

With 181 Figures and 7 Tables

Dr. Gleb Beliakov Dr. Ana PraderaSchool of Engineering and IT Escuela Superior de CienciasDeakin University Experimentales y Tecnología221 Burwood Hwy Universidad Rey Juan Carlos. c/ Tulipan s/nBurwood 3125 28933 Móstoles, MadridAustralia Spaingbeliako@gmail.com ana.pradera@urjc.es

Professor Tomasa CalvoEscuela Técnica Superior de Ingeniería InformáticaUniversidad de AlcaláCrta de Madrid-Barcelona Km 33, 629971Alcalá de Henares, MadridSpaintomasa.calvo@uah.es

Library of Congress Control Number: 2007932777

ISSN print edition: 1434-9922ISSN electronic edition: 1860-0808ISBN 978-3-540-73720-9 Springer Berlin Heidelberg New York

This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publicationor parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violationsare liable for prosecution under the German Copyright Law.

Springer is a part of Springer Science+Business Mediaspringer.com

c© Springer-Verlag Berlin Heidelberg 2007

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply,even in the absence of a specific statement, that such names are exempt from the relevant protective lawsand regulations and therefore free for general use.

Typesetting: Integra Software Services Pvt. Ltd., IndiaCover design: WMX Design, Heidelberg

Printed on acid-free paper SPIN: 12047963 89/3180/Integra 5 4 3 2 1 0

To

Gelui Patricia, Chaquen and SofiaG.B.

UltanoA.P.

Raquel, Carmina, Adela and CarlosT.C.

Preface

The target audience of this book are computer scientists, system architects,knowledge engineers and programmers, who face a problem of combining var-ious inputs into a single output. Our intent is to provide these people with aneasy-to-use guide about possible ways of aggregating input values given on anumerical scale, and ways of choosing/constructing aggregation functions fortheir specific applications.

A prototypical user of this guide is a software engineer who works onbuilding an expert or decision support system, and is interested in how tocombine information coming from different sources into a single numericalvalue, which will be used to rank the alternative decisions. The complexity ofbuilding such a system is so high, that one cannot undertake a detailed studyof the relevant mathematical literature, and is rather interested in a simpleoff-the-shelf solution to one of the many problems in this work.

We present the material in such a way that its understanding does not re-quire specific mathematical background. All the relevant notions are explainedin the book (in the introduction or as footnotes), and we avoid referring toadvanced topics (such as algebraic structures) or using pathological examples(such as discontinuous functions). While mathematically speaking these top-ics are important, they are well explained in a number of other publications(some are listed at the end of the introduction). Our focus is on practicalapplications, and our aims are conciseness, relevance and quick applicability.

We treat aggregation functions which map several inputs from the inter-val [0, 1] to a single output in the same interval. By no means this is theonly possible framework for aggregating the inputs or performing informationfusion – in many cases the inputs are in fact discrete or binary. However itis often possible and useful to map them into the unit interval, for exampleusing degrees of membership in fuzzy sets. As we shall see, even in this sim-plified framework, the theory of aggregation is very rich, so choosing the rightoperation is still a challenge.

As we mentioned, we present only the most important mathematical prop-erties, which can be easily interpreted by the practitioners. Thus effectively

VIII Preface

this book is an introduction to the subject. Yet we try to cover a very broadrange of aggregation functions, and present some state-of-the-art techniques,typically at the end of each section.

Chapter 1 gives a broad introduction to the topic of aggregation func-tions. It covers important general properties and lists the most importantprototypical examples: means, ordered weighted averaging (OWA) functions,Choquet integrals, triangular norms and conorms, uninorms and nullnorms.It addresses the problem of choosing the right aggregation function, and alsointroduces a number of basic numerical tools: methods of interpolation andsmoothing, linear and nonlinear optimization, which will be used to constructaggregation functions from empirical data.

Chapters 2 – 4 give a detailed discussion of the four broad classes of aggre-gation functions: averaging functions (Chapter 2), conjunctive and disjunctivefunctions (Chapter 3) and mixed functions (Chapter 4). Each class has manydistinct families, and each family is treated in a separate section. We give aformal definition, discuss important properties and their interpretation, andalso present specific methods for fitting a particular family to empirically col-lected data. We also provide examples of computer code (in C++ language)for calculating the value of an aggregation function, various generalizations,advanced constructions and pointers to specific literature.

In Chapter 5 we discuss the general problem of fitting chosen aggregationfunctions to empirical data. We formulate a number of mathematical program-ming problems, whose solution provides the best aggregation function from agiven class which fits the data. We also discuss how to evaluate suitability ofsuch functions and measure consistency with the data.

In Chapter 6 we present a new type of interpolatory aggregation functions.These functions are constructed based on empirical data and some generalmathematical properties, by using interpolation or approximation processes.The aggregation functions are general (i.e., they typically do not belong toany specific family), and are not expressed via an algebraic formula but rathera computational algorithm. While they may lack certain interpretability, theyare much more flexible in modeling the desired behavior of a system, andnumerically as efficient as an algebraic formula. These aggregation functionsare suitable for computer applications (e.g., expert systems) where one caneasily specify input–output pairs and a few generic properties (e.g., symmetry,disjunctive behavior) and let the algorithm build the aggregation functionsautomatically.

The final Chapter 7 outlines a few classes of aggregation functions notcovered elsewhere in this book, and presents various additional propertiesthat may be useful for specific applications. It also provides pointers to theliterature where these issues are discussed in detail.

Appendix A outlines some of the methods of numerical approximation andoptimization that are used in the construction of aggregation functions, andprovides references to their implementation. Appendix B contains a numberof problems that can be given to undergraduate and graduate students.

Preface IX

This book comes with a software package AOTool, which can be freelydownloaded from http://www.deakin.edu.au/∼gleb/aotool.html. AOToolimplements a large number of methods for fitting aggregation functions (ei-ther general or from a specific class) to empirical data. These methods aredescribed in the relevant sections of the book. AOTool allows the user to loadempirical data (in spreadsheet format), to calculate the parameters of the bestaggregation function which fits these data, and save these parameters for fu-ture use. It also allows the user to visualize some two-dimensional aggregationfunctions.

We reiterate that this book is oriented towards practitioners. While basicunderstanding of aggregation functions and their properties is required fortheir successful usage, the examples of computer code and the software pack-age for building these functions from data allow the reader to implement mostaggregation functions in no time. It takes the complexity of implementationoff the users, and allows them to concentrate on building their specific system.

Melbourne, Mostoles, Alcala de Henares Gleb BeliakovMay 2007 Ana Pradera

Tomasa Calvo

Notations used in this book

� the set of real numbers;N the set {1, 2, . . . , n};2X the power set (i.e., the set of all subsets of the set X);Ac the complement of the set A;x n-dimensional real vector, usually from [0, 1]n;< x,y > scalar (or dot) product of vectors x and y;x↘ permutation of the vector x which arranges its components in

non-increasing order;x↗ permutation of the vector x which arranges its components in

non-decreasing order;fn a function of n variables, usually fn : [0, 1]n → [0, 1];F an extended function F :

⋃

n∈{1,2,...}[0, 1]n → [0, 1];

f ◦ g the composition of functions f and g;g−1 the inverse of the function g;g(−1) the pseudo-inverse of the function g;N a strong negation function;v a fuzzy measure;w a weighting vector;log the natural logarithm;

Averaging functionsM(x) arithmetic mean of x;Mw(x) weighted arithmetic mean of x with the weighting vector w;Mw,[r](x) weighted power mean of x with the weighting vector w and

exponent r;Mw,g(x) weighted quasi-arithmetic mean of x with the weighting vector

w and generator g;Med(x) median of x;Meda(x) a-median of x;Q(x) quadratic mean of x;Qw(x) weighted quadratic mean of x with the weighting vector w;

XII Notations

H(x) harmonic mean of x;Hw(x) weighted harmonic mean of x with the weighting vector w;G(x) geometric mean of x;Gw(x) weighted geometric mean of x with the weighting vector w;OWAw(x) ordered weighted average of x with the weighting vector w;OWQw(x) quadratic ordered weighted average of x with the weighting

vector w;OWGw(x) geometric ordered weighted average of x with the weighting

vector w;Cv(x) discrete Choquet integral of x with respect to the fuzzy measure

v;Sv(x) discrete Sugeno integral of x with respect to the fuzzy measure

v;Conjunctive and disjunctive functions

T triangular norm;S triangular conorm;TP , TL, TD the basic triangular norms (product, �Lukasiewicz and drastic

product);SP , SL, SD the basic triangular conorms (dual product, �Lukasiewicz and

drastic sum);C copula;

Mixed functionsU uninorm;V nullnorm;UT,S,e uninorm with underlying t-norm T , t-conorm S and neutral

element e;VT,S,a nullnorm with underlying t-norm T , t-conorm S and absorbent

element a;Eγ,T,S exponential convex T-S function with parameter γ and t-norm

and t-conorm T and S;Lγ,T,S linear convex T-S function with parameter γ and t-norm and

t-conorm T and S;Qγ,T,S,g T-S function with parameter γ, t-norm and t-conorm T and S

and generator g;OS,w S-OWA function with the weighting vector w and t-conorm S;OT,w T-OWA function with the weighting vector w and t-norm T ;OS,T,w ST-OWA function with the weighting vector w, t-conorm S and

t-norm T .

Notations XIII

Acronyms and Abbreviations

LAD least absolute deviationLP linear programmingLS least squaresOWA ordered weighted averagingQP quadratic programmingt–norm triangular normt–conorm triangular conorms.t. subject tow.r.t. with respect toWOWA weighted ordered weighted averaging

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 What is an aggregation function? . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 What are aggregation functions used for? . . . . . . . . . . . . . . . . . . . 51.3 Classification and general properties . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.1 Main classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.2 Main properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.3.4 Comparability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.3.5 Continuity and stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.4 Main families and prototypical examples . . . . . . . . . . . . . . . . . . . 231.4.1 Min and Max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.4.2 Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.4.3 Medians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.4.4 Ordered weighted averaging . . . . . . . . . . . . . . . . . . . . . . . . 251.4.5 Choquet and Sugeno integrals . . . . . . . . . . . . . . . . . . . . . . . 261.4.6 Conjunctive and disjunctive functions . . . . . . . . . . . . . . . . 271.4.7 Mixed aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.5 Composition and transformation of aggregation functions . . . . . 281.6 How to choose an aggregation function . . . . . . . . . . . . . . . . . . . . . 311.7 Numerical approximation and optimization tools . . . . . . . . . . . . 341.8 Key references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2 Averaging Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.1 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.2 Classical means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.3 Weighted quasi-arithmetic means . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.3.2 Main properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.3.4 Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.3.5 Weighting triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

XVI Contents

2.3.6 Weights dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.3.7 How to choose weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.4 Other means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622.4.1 Gini means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622.4.2 Bonferroni means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.4.3 Heronian mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642.4.4 Generalized logarithmic means . . . . . . . . . . . . . . . . . . . . . . 652.4.5 Mean of Bajraktarevic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.5 Ordered Weighted Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.5.2 Main properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.5.3 Other types of OWA functions . . . . . . . . . . . . . . . . . . . . . . 712.5.4 Generalized OWA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732.5.5 How to choose weights in OWA . . . . . . . . . . . . . . . . . . . . . 772.5.6 Choosing parameters of generalized OWA . . . . . . . . . . . . 87

2.6 Choquet Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 902.6.1 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 902.6.2 Definitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . . 922.6.3 Types of fuzzy measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 982.6.4 Interaction, importance and other indices . . . . . . . . . . . . . 1052.6.5 Special cases of the Choquet integral . . . . . . . . . . . . . . . . . 1092.6.6 Fitting fuzzy measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102.6.7 Generalized Choquet integral . . . . . . . . . . . . . . . . . . . . . . . 115

2.7 Sugeno Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1162.7.1 Definition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

2.8 Medians and order statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1192.8.1 Median . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1192.8.2 Order statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

2.9 Key references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

3 Conjunctive and Disjunctive Functions . . . . . . . . . . . . . . . . . . . . 1233.1 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1233.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1243.3 Generalized OR and AND – functions . . . . . . . . . . . . . . . . . . . . . . 1243.4 Triangular norms and conorms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

3.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1263.4.2 Main properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1293.4.3 Strict and nilpotent t–norms and t–conorms . . . . . . . . . . 1323.4.4 Archimedean t–norms and t–conorms . . . . . . . . . . . . . . . . 1343.4.5 Additive and multiplicative generators . . . . . . . . . . . . . . . 1353.4.6 Pseudo-inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1383.4.7 Isomorphic t–norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1403.4.8 Comparison of continuous Archimedean t–norms . . . . . . 1433.4.9 Ordinal sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1443.4.10 Approximation of continuous t–norms . . . . . . . . . . . . . . . . 148

Contents XVII

3.4.11 Families of t–norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1493.4.12 Lipschitz–continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1673.4.13 Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1693.4.14 How to choose a triangular norm/conorm . . . . . . . . . . . . 1703.4.15 How to fit additive generators . . . . . . . . . . . . . . . . . . . . . . . 1723.4.16 Introduction of weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

3.5 Copulas and dual copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1823.6 Other conjunctive and disjunctive functions . . . . . . . . . . . . . . . . . 1863.7 Noble reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1903.8 Key references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

4 Mixed Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1974.1 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1974.2 Uninorms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

4.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2004.2.2 Main properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2004.2.3 Main classes of uninorms . . . . . . . . . . . . . . . . . . . . . . . . . . . 2024.2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2064.2.5 Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2114.2.6 Fitting to the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

4.3 Nullnorms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2144.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2144.3.2 Main properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2154.3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2174.3.4 Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2184.3.5 Fitting to the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

4.4 Generated functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2204.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2214.4.2 Main properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2234.4.3 Classes of generated functions . . . . . . . . . . . . . . . . . . . . . . . 2244.4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2274.4.5 Fitting to the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

4.5 T-S functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2304.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2314.5.2 Main properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2334.5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2344.5.4 Fitting to the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

4.6 Symmetric sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2394.6.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2404.6.2 Main properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2414.6.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2444.6.4 Fitting to the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

4.7 ST-OWA functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2494.7.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2494.7.2 Main properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

XVIII Contents

4.7.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2544.7.4 U-OWA functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2564.7.5 Fitting to the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

4.8 Key references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

5 Choice and Construction of Aggregation Functions . . . . . . . . 2615.1 Problem formalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2615.2 Fitting empirical data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2625.3 General problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2635.4 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

5.4.1 Linearization of outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2685.5 Assessment of suitability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

6 Interpolatory Type Aggregation Functions . . . . . . . . . . . . . . . . . 2716.1 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2716.2 Construction based on spline functions . . . . . . . . . . . . . . . . . . . . . 272

6.2.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2726.2.2 Preservation of specific properties . . . . . . . . . . . . . . . . . . . 273

6.3 Construction based on Lipschitz interpolation . . . . . . . . . . . . . . . 2766.4 Preservation of specific properties . . . . . . . . . . . . . . . . . . . . . . . . . 278

6.4.1 Conjunctive, disjunctive and averaging behavior . . . . . . . 2786.4.2 Absorbing element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2796.4.3 Neutral element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2796.4.4 Mixed behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2816.4.5 Given diagonal and opposite diagonal . . . . . . . . . . . . . . . . 2846.4.6 Given marginals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2866.4.7 Noble reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

7 Other Types of Aggregation and Additional Properties . . . . 2977.1 Other types of aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2977.2 Some additional properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

A Tools for Approximation and Optimization . . . . . . . . . . . . . . . . 305A.1 Univariate interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305A.2 Univariate approximation and smoothing . . . . . . . . . . . . . . . . . . . 307A.3 Approximation with constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 309A.4 Multivariate approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312A.5 Convex and non-convex optimization . . . . . . . . . . . . . . . . . . . . . . . 316

A.5.1 Univariate optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318A.5.2 Multivariate constrained optimization . . . . . . . . . . . . . . . . 319A.5.3 Multilevel optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322A.5.4 Global optimization: stochastic methods . . . . . . . . . . . . . . 323A.5.5 Global optimization: deterministic methods . . . . . . . . . . . 324

A.6 Main tools and libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

Contents XIX

B Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355