recentes avan¸cos em sistemas...

Recentes Avancos emSistemas “Fuzzy”

Editores:Benjamın BedregalFernando A. C. GomideFlaulles BergamaschiLaecio C. de Barros

Regivan H. N. SantiagoRonei M. de Moraes

Wladimir Seixas

ISBN: 978-85-8215-064-1

Joao Pessoa-PB, Agosto de 2014

Sociedade Brasileira de Matematica Aplicada e Computacional.

Prefacio

Este livro contem alguns dos recentes avancos na area de Sistemas Difusos.“Sistemas Fuzzy” e uma terminologia cunhada nesta obra com o objetivo dedesignar Sistemas Computacionais ou Sistemas Teoricos baseados na Teoria dosConjuntos Difusos de Zadeh.

Desde o seu surgimento nos anos 60 o crescimento desta area tem sidoconsideravel. Ela alargou sua influencia desde a area de Sistemas de Cont-role ate campos que inicialmente nao se imaginava; como e o caso de areasda matematica, sistemas de apoio a decisao, inteligencia artificial e outros.

Essa obra visa apresentar parte desta influencia. Ela e um texto bilingue,algumas contribuicoes encontram-se em portugues e outras em lıngua inglesa.Todas essas contribuicoes passaram pela analise de um comite que atestou arelevancia das mesmas. Alem disso, elas foram apresentadas e debatidas du-rante o Terceiro Congresso Brasileiro de Sistemas Fuzzy (III CBSF) realizadoem Agosto, na cidade de Joao Pessoa (Paraıba).

Agosto de 2014 Regivan Hugo N. SantiagoCoordenador de Programa do III CBSF


Organizacao

As contribuicoes aqui contidas foram apresentadas e debatidas no III CBSF.Ele foi organizado pela Universidade Federal da Paraıba e realizado gracas asseguintes instituicoes e pessoas:

Instituicoes Apoiadoras

North American Fuzzy Information Processing Society (NAFIPS)European Society for Fuzzy Logic and Technology (EUSFLAT)International Fuzzy Systems Association (IFSA)Sociedade Brasileira de Automatica (SBA)Sociedade Brasileira de Inteligencia Computacional (SBIC)Sociedade Brasileira de Matematica Aplicada e Computacional (SBMAC)

Comite Executivo

Ronei M. de Moraes (DE-UFPB) ChairLiliane S. Machado (DI-UFPB) Co-ChairBenjamın Rene Callejas Bedregal (UFRN)Fernando A. C. Gomide (UNICAMP)Laecio C. de Barros (UNICAMP)Jose Arnaldo Roveda (UNESP - Sorocaba)

Comite de Programa

Coordenadores: Regivan H. N. Santiago (UFRN)Wladimir Seixas (UFSCar)

Avaliadores

Adriao D. D. Neto — UFRNAnderson P. Cruz — UFRNAndre P. Lemos — UFMGAndre Carvalho — USPAnne M. Canuto — UFRNAurora Pozo — UFPRBenjamın Bedregal — UFRNDavid C. Martins Jr — UFABC

Eduardo Palmeira — UESCFabiana Santana — UFRNFagner Santana — UFRNFernando Gomide — UnicampFlaulles B. Bergamaschi — UESBFrancisco De A. Carvalho — UFPE


Gracaliz Dimuro — FURGGuilherme Barreto — UFCHelida Santos — UFRNHeloisa Camargo — UFSCarHeriberto R.-Flores — U. de TarapacaHumberto Bustince — UPNAIng Ren — UFPEIvan Mezzono — UFERSAJavier Fernandez — UPNAJoao Alcantara — UFCJoao Marcos — UFRNJose A. F. Costa — UFRNJose A. Roveda — Unesp-SorocabaLaecio C. Barros — UNICAMPLuciana Gomes — UFSCarMagda Peixoto — UFSCarMarcelo Ferreira — UFPBMarcos E. Valle — UNICAMPMaria J. Castanho — UnicentroMarie J. Lesot — LIP6 - UPMCMarilton S. de Aguiar — UFPELMarina T. Mizukoshi — UFGMario Benevides — UFRJMarjory C. Abreu — UFRNMarley Vellasco — PUC-RioMichal Baczynski — University of Sile-siaMoiseis Cecconello — UFMT

Myriam Delgado — UTFPRNeli Ortega — USPPatricia Melin — Tijuana Institute ofTechnologyPaulo Almeida — CEFET-MGPetrucio Viana — UFFRegivan H. N. Santiago — UFRNRenata Reiser — UFPelRicardo Tanscheit — PUC-RioRicardo C. Silva — USPRonei Moraes — UFPBRonildo Moura — UFRNRosana Jafelice —UFURoseli Romero — USPSandra Sandri — INPESandra Roveda — UNESPVilem Novak — University of OstravaViviane Mattos — FURGVladik Kreinovich — University ofTexas at El PasoWeldon Lodwick — University of Col-orado at DenverWilson Oliveira — UFRPEWladimir Seixas — UFSCarYurilev Chalco Cano — Universid deTarapaca

Avaliadores Ad-hoc

Roxana ContrerasDaniel LeiteCesar ValenciaGiancarlo LuccaAdenilton J. Silva

Rogerio R. De VargasAlisson Marques Da SilvaMarcus Rocha


Instituicoes Financiadoras

Coordenacao de Aperfeicoamento de Pessoal de Nıvel Superior — CAPESConselho Nacional de Desenvolvimento Cientıfico e Tecnologico — CNPQUniversidade Federal da Paraıba — UFPB

Organizacao Local

Ronei M. de Moraes (DE-UFPB)Liliane S. Machado (DI-UFPB)Danielly C. S. C. Holmes (PPGMDS-UFPB)Frederico F. Ribeiro (PPGMDS-UFPB)Elaine A. M. G. Soares (DI-UFPB)Douglas S. Ferreira (PPGI-UFPB)Edviges F. C. Lima (PPGI-UFPB)Jose T. D. Segundo (DI-UFPB)Laisa R. de Sa (PPGMDS-UFPB)Luana R. Almeida (PPGMDS-UFPB)Thiago V. V. Batista (DI-UFPB)Vicente R. S. Neto (DI-UFPB)


Recent Advances onFuzzy Systems

Editors:Benjamın BedregalFernando A. C. GomideFlaulles BergamaschiLaecio C. de Barros

Regivan H. N. SantiagoRonei M. de Moraes

Wladimir Seixas

ISBN: 978-85-8215-064-1

Joao Pessoa-PB, August 2014


Foreword

This book contains some recent trends in the Fuzzy Systems field. Fuzzy Sys-tems is terminology coined in this book to designate both Computational andTheoretic Systems inspired on Zadeh’s Fuzzy Set Theory.

Since its creation in the 60’s the field of fuzzy systems enjoyed a consid-erably growth. It expanded its influence from the field of Control Systems tounimaginable other fields, like: mathematics, artificial inteligence, etc.

This book aims to present such influence. It is a bilingual text. Some contri-butions are in Portuguese and some in English. All of them were submitted to anevaluation committee. After such evaluation, they were presented and discussedduring the Third Congress of Fuzzy Systems (III CBSF), which was held from17-20 August 2014 in Joao Pessoa, Brazil.

August 2014 Regivan Hugo N. SantiagoProgram Chair of III CBSF


Organization

The contributions were submitted to the III CBSF, which was organized by theFederal University of Paraıba and was realized thanks to the following peopleand institutions:

Support

North American Fuzzy Information Processing Society (NAFIPS)European Society for Fuzzy Logic and Technology (EUSFLAT)International Fuzzy Systems Association (IFSA)Sociedade Brasileira de Automatica (SBA)Sociedade Brasileira de Inteligencia Computacional (SBIC)Sociedade Brasileira de Matematica Aplicada e Computacional (SBMAC)

Steering Committee

Ronei M. de Moraes (DE-UFPB) ChairLiliane S. Machado (DI-UFPB) Co-ChairBenjamın Rene Callejas Bedregal (UFRN)Fernando A. C. Gomide (UNICAMP)Laecio C. de Barros (UNICAMP)Jose Arnaldo Roveda (UNESP - Sorocaba)

Program Committee

Chairs: Regivan H. N. Santiago (UFRN)Wladimir Seixas (UFSCar)

Referees

Adriao D. D. Neto — UFRNAnderson P. Cruz — UFRNAndre P. Lemos — UFMGAndre Carvalho — USPAnne M. Canuto — UFRNAurora Pozo — UFPRBenjamın Bedregal — UFRN

David C. Martins Jr — UFABCEduardo Palmeira — UESCFabiana Santana — UFRNFagner Santana — UFRNFernando Gomide — UnicampFlaulles B. Bergamaschi — UESB


Francisco De A. Carvalho — UFPEGracaliz Dimuro — FURGGuilherme Barreto — UFCHelida Santos — UFRNHeloisa Camargo — UFSCarHeriberto R.-Flores — U. de TarapacaHumberto Bustince — UPNAIng Ren — UFPEIvan Mezzono — UFERSAJavier Fernandez — UPNAJoao Alcantara — UFCJoao Marcos — UFRNJose A. F. Costa — UFRNJose A. Roveda — Unesp-SorocabaLaecio C. Barros — UNICAMPLuciana Gomes — UFSCarMagda Peixoto — UFSCarMarcelo Ferreira — UFPBMarcos E. Valle — UNICAMPMaria J. Castanho — UnicentroMarie J. Lesot — LIP6 - UPMCMarilton S. de Aguiar — UFPELMarina T. Mizukoshi — UFGMario Benevides — UFRJMarjory C. Abreu — UFRNMarley Vellasco — PUC-RioMichal Baczynski — University of Sile-sia

Moiseis Cecconello — UFMTMyriam Delgado — UTFPRNeli Ortega — USPPatricia Melin — Tijuana Institute ofTechnologyPaulo Almeida — CEFET-MGPetrucio Viana — UFFRegivan H. N. Santiago — UFRNRenata Reiser — UFPelRicardo Tanscheit — PUC-RioRicardo C. Silva — USPRonei Moraes — UFPBRonildo Moura — UFRNRosana Jafelice —UFURoseli Romero — USPSandra Sandri — INPESandra Roveda — UNESPVilem Novak — University of OstravaViviane Mattos — FURGVladik Kreinovich — University ofTexas at El PasoWeldon Lodwick — University of Col-orado at DenverWilson Oliveira — UFRPEWladimir Seixas — UFSCarYurilev Chalco Cano — Universid deTarapaca

Ad-hoc Referees

Roxana ContrerasDaniel LeiteCesar ValenciaGiancarlo LuccaAdenilton J. Silva

Rogerio R. De VargasAlisson Marques Da SilvaMarcus Rocha


Financial Support

Coordenacao de Aperfeicoamento de Pessoal de Nıvel Superior — CAPESConselho Nacional de Desenvolvimento Cientıfico e Tecnologico — CNPQUniversidade Federal da Paraıba — UFPB

Local Organization

Ronei M. de Moraes (DE-UFPB)Liliane S. Machado (DI-UFPB)Danielly C. S. C. Holmes (PPGMDS-UFPB)Frederico F. Ribeiro (PPGMDS-UFPB)Elaine A. M. G. Soares (DI-UFPB)Douglas S. Ferreira (PPGI-UFPB)Edviges F. C. Lima (PPGI-UFPB)Jose T. D. Segundo (DI-UFPB)Laisa R. de Sa (PPGMDS-UFPB)Luana R. Almeida (PPGMDS-UFPB)Thiago V. V. Batista (DI-UFPB)Vicente R. S. Neto (DI-UFPB)


Sumario/Table of Contents

Fuzzy Exponential Recurrent Neural Networks forGray-scale Image Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 001

Rede Neuro-Fuzzy Evolutiva com Selecao deEntradas Aplicada na Modelagem de Sistemas . . . . . . . 013

Sufficient Conditions for Interval-valued OptimalControl Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 025

Adaptacao do Paradigma Orientado a Notificacoespara desenvolvimento de sistemas fuzzy . . . . . . . . . . . . . . . 036

Traffic classification and decision scheme for hybridoptical networks based on Fuzzy logic . . . . . . . . . . . . . . . . . 048

Sistema P-Fuzzy Aplicado na Modelagem daVelocidade de um corpo em queda livre . . . . . . . . . . . . . . . 054

Toward Automatic Rule-base Design in ProbabilisticFuzzy Classifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 063

Numeros fuzzy f-correlacionados . . . . . . . . . . . . . . . . . . . . . . . . . . 073

Modelo presa-predador do tipo Lotka-Volterra comencontros baseados em t-normas . . . . . . . . . . . . . . . . . . . . . . . 083

Sistema Inteligente para Apoio a Decisao naOperacao de uma Malha de Escoamento de Petroleo 091

Um estudo sobre pontos crıticos subjetivos . . . . . . . . . . . . . . 104

Introducao a aritmetica das algebras relacionais fuzzy . 114

Introduzindo Funcoes Migrativas Intervalares . . . . . . . . . . . 117

Analise dos arranjos de coordenacao em cadeiasprodutivas agroindustriais com base na abordagemfuzzy: um estudo de caso . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Neural Networks for Forward Models Applied inFollowing a Moving Target Task with a Puma 560Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Deteccao de Falhas em Motores Eletricos Utilizandoa Rede Neural RBF com os Seguintes Algoritmosde Treinamento, K-Means, C-Means e Ck-Means . . . . 148

Implementation of fuzzy Mamdani controller on twoAxis of six degrees of Freedom robot . . . . . . . . . . . . . . . . . . 157

A proposal for the reduction of examples in datasetsto optimize the extraction of classification rules . . . . . 169

Um Algoritmo para o Problema do Caminho MınimoFuzzy Utilizando Dominancia e Similaridade . . . . . . . . . 181

Uma Nova Nocao de Metrica Fuzzy e sua Topologianao Metrizavel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

Uma abordagem via controle fuzzy para um modelopresa-predador . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

Abordagens de Problemas de Valor Inicial Fuzzy . . . . . . . 215

Modelagem via Teoria Fuzzy: aplicacao do Modelode Takagi-Sugeno . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

Utilizacao de Matriz de Relacoes Fuzzy em Avaliacaode Impacto Ambiental: Estudo de Caso daEmpresa Automobilıstica Toyota Do Brasil . . . . . . . . . . . 231

GPFIS: Um Sistema Fuzzy-Genetico para Classicacao . 233

Um Sistema Fuzzy-Genetico Aplicado a Previsao dePrecos de Etanol e Acucar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

Controle pid-fuzzy de sistema resfriador de leite . . . . . . . . 257

New Types of Strongly Prime Fuzzy Ideals . . . . . . . . . . . . . . 269

Comparacao entre logica fuzzy e arvore de decisaona identificacao de hipertensos nao aderentes aotratamento . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

Sintonia de controlador hıbrido PID/Fuzzy emtempo real via computacao evolutiva . . . . . . . . . . . . . . . . . . 293

Redes Neuro-Fuzzy Evolutivas Embarcadas Aplicadasao Gerenciamento de Motores a Combustao . . . . . . . . . . 302

Sistema fuzzy de apoio a recuperacao de areasdestinadas a conservacao ambiental: estudo decaso do CEA/IAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

Indice de percepcao da qualidade ambientalponderado por sistemas fuzzy: uma propostametodologica para diagnostico participativo . . . . . . . . . . 327

Series Temporais Fuzzy: Um Modelo de PrevisaoBaseado no Intervalo Temporal das Amostras . . . . . . . . 329

Actor Critic Reinforcement Learning for the FuzzyPI Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

O uso de Sistemas Fuzzy para o Ensino deEspectroscopia no Infravermelho . . . . . . . . . . . . . . . . . . . . . . . 353

Modelagem Fuzzy do processo de incubacao de ovosde codorna japonesa (Coturnix coturnix japonica) . . 355

Using t-Norms for Generalizing Choquet Integralswith an Application to Fuzzy Rule-BasedClassification Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

Deteccao de Consumos Irregulares Utilizando LogicaNebulosa para a Reducao de Perdas Nao Tecnicasem Concessionarias de Distribuicao de EnergiaEletrica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

Algoritmo de Evolucao Diferencial com ParametrosAjustaveis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

Assessment of Fuzzy Gaussian Naive Bayes ClassifierUsing Data with Different Statistical Distributions . . 401

Transformacao de Valores crisp em Valores fuzzy:Comparacao dos Resultados . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

Robust and level-continuous functions via optimalpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

Toward an intuitionistic extension of G-implicationoperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

Uma Nova Forma de Inicializar os Centros dosGrupos em Algoritmos Tipo Fuzzy C-Means . . . . . . . . . 432

Otimizacao do projeto de um parque eolico utilizandoabordagem hıbrida de Algoritmos Evolucionariose estrategias de busca local . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

Controlador fuzzy para impressora 3D de baixo custo . 454

Sobre a conjugada de uma funcao Sobrepostaquase-homogenea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466

Numeros Complexos Graduados . . . . . . . . . . . . . . . . . . . . . . . . . . 474

A Review of Membership Functions TuningApproaches in Multi-Objective EvolutionaryFuzzy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486

Comparacao de Desempenho entre Logica Binaria eLogica Fuzzy no Controle de Sistemas Dinamicos . . . 498

New Results for Strong Typical Hesitant FuzzyNegations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

Fuzzy graph solution of heat equations with fuzzydiffusion paramenters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522

Fuzzy Ideals of Product Operator on Bounded FuzzyLattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529

Controle Fuzzy Aplicado a Sistemas AnticolisaoFrontal e Distribuicao Eletronica de FrenagemAutomotiva . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537

Controlador Fuzzy Aplicado a Veıculos Eletricos comTransmissao Variavel Contınua . . . . . . . . . . . . . . . . . . . . . . . . . 548

Formulacao matematica do problema de otimizacao,restrito a linhas de transmissao compactas de230KV e 138 KV, com multiplos objetivos usandoalgoritmo genetico . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560

Sıntese de Arvores Fuzzy de Padroes atraves deProgramacao Genetica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572

Um Modelo de Suporte a Decisao Baseado emRegras Fuzzy para Predicao da Seguranca Alimentar 584

Modelagem e Desenvolvimento de Simulador deTurbina Eolica Utilizando uma Estrategia deControle Fuzzy Do Tipo Takagi-Sugeno . . . . . . . . . . . . . . . 595

Sistema de Suporte Decisao Espacial Aplicado aoGerenciamento de Acidentes de Transito . . . . . . . . . . . . . 605

Projeto de Sistemas de Controle com ModelagemTakagi-Sugeno e Implementacao de ControladoresFuzzy com Retroacao de Estados . . . . . . . . . . . . . . . . . . . . . . . 618

Avaliacao de um Sistema de Controle Fuzzy em umRobo Movel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628

Aplicacao de Teoria Fuzzy na Delimitacao de Zonasde Manejo em Agricultura de Precisao . . . . . . . . . . . . . . . . 636

Analysis of the Consumption with AggregationOperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638

A Bibliometric Overview of Financial Studies . . . . . . . . . . . 650

On homogeneous aggregation functions . . . . . . . . . . . . . . . . . . 657

On a generalization of the notion of monotonicity . . . . . . 663

On the definition of capacities from overlap functionsand overlap indexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668

Generalized Hukuhara differentiability and intervaloptimization problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675

Fuzzy Exponential Recurrent Neural Networks forGray-scale Image Retrieval�

Marcos Eduardo Valle

University of Campinas, Department of Applied Mathematics,CEP 13083-859. Campinas – SP, BrazilE-mail: [email protected]

Abstract. Associative memories (AMs) are mathematical models inspired bythe human brain ability to store and recall information. This paper introduces thefuzzy exponential recurrent neural networks (FERNNs), which can implementan AM for the storage and recall of fuzzy sets. The novel models are obtained bymodifying the multivalued exponential recurrent neural network of Chiueh andTsai. Briefly, a FERNN defines recursively a sequence of fuzzy sets obtained byaveraging the stored fuzzy sets weighted by an exponential of a fuzzy comparisonmeasure between the current fuzzy set and the stored items. Computational exper-iments reveal that FERNNs can be effectively used for the retrieval of gray-scaleimages corrupted by either Gaussian noise or salt-and-pepper noise.

Keywords: Associative memory, recurrent neural network, fuzzy system, gray-scale image processing.

1 Introduction

Associative memories (AMs) are mathematical constructs motivated by the human brainability to store and recall information [1]. Such as the biological neural network, an AMshould be able to retrieve a memorized information from a possibly incomplete or cor-rupted item. Formally, an AM is designed for the storage of a finite set {a1,a2, . . . ,ap},called the fundamental memory set. Afterwards, the AM model is expect to retrieve amemorized concept aξ in response to the presentation of a partial or noisy version aξ

of aξ. Applications of AMs cover, for instance, pattern classification and recognition[2–4], optimization [5], computer vision and image retrieval [6–8], prediction [9, 10],control [11, 12], and language understanding [13].

The Hopfield neural network is one of the most widely known neural network usedto realize an AM [14]. In spite of its attractive features, including a characterization interms of an energy function and a variety of applications [15], the Hopfield networksuffers from a low absolute storage capacity. Specifically, the asymptotic number ofitems that can be stored and subsequently recovered exactly by the Hopfield network isproportional to n/log(n), where n is the length of the vectors aξ, for ξ = 1, . . . , p [16].A simple but significant improvement in storage capacity of the Hopfield network is

� This work was supported in part by by CNPq under grant no. 304240/2011-7, FAPESP undergrant no. 2013/12310-4, and FAEPEX/Unicamp under grant no. 519.292.

Proceedings of III Brazilian Congress on Fuzzy Systems (III CBSF)João Pessoa-PB 17-20 Aug. 2014 001

Generalized Hukuhara differentiability and interval optimization problems

8. W. A. Lodwick, Interval and Fuzzy Analysis: A Unified Approach, Advances inimaging and electronic physics 148, edited by Peter W. Hawkes, Academic Press76–192 (2007).

9. S. Markov, Calculus for interval functions of a real variable, Computing 22 (1979)325-377.

10. R. Osuna-Gomez, Y. Chalco-Cano, A. Rufian-Lizana, Hernandez-Jimenez, Neces-sary and sufficient conditions for fuzzy optimality problems, submitted to publica-tion.

11. L. Stefanini and B. Bede, Generalized Hukuhara differentiability of interval-valuedfunctions and interval differential equations, Nonlinear Analysis: Theory, Methods& Applications 71, 1311–1328 (2009).

12. L. Stefanini, A generalization of Hukuhara difference and division for interval andfuzzy arithmetic, Fuzzy Sets and Systems 161, 1564–1584, (2010).

13. H-C. Wu, The Karush-Kuhn-Tucker optimality conditions in an optimization prob-lem with interval-valued objective function, European Journal of Operational Re-search 176, 46–59 (2007).

14. H-C. Wu, The Karush-Kuhn-Tucker optimality conditions in multiobjective pro-gramming problems with interval-valued objective functions, European Journal ofOperational Research 196, 49–60 (2009).

Proceedings of III Brazilian Congress on Fuzzy Systems (III CBSF)João Pessoa-PB 17-20 Aug. 2014 683

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