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An invitation to Study Fuzzy andGeneralized Uncertainty Optimization

A Historical and Contemporary View

RICARDO COELHODepartment of Statistics and Applied MathematicsFederal University of Ceará

WELDON LODWICKDepartment of Mathematical and Statistical SciencesUniversity of Colorado Denver

MOTIVATION AND OBJECTIVE

S One of the most significant features of humanbeings is the decision-making of everydayproblems.

S The database of the practical problems, in manycases, have approximate and/or imprecise values.

S The goal of this course is to present a briefdescription how to use fuzzy set and possibilitytheories in optimization methods. 2

OUTLINE

S Generalized Uncertain Mathematical Programming – the beginning

S Approaches in uncertain environmentS costs in the objective function and/or;

S coefficients of the set of constraints

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WHERE EVERYTHING BEGAN

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Lotfi A. Zadeh

“Fuzzy Sets as a basis for a Theory of Possibility”

Fuzzy Sets and Systems. 1 3–28. 1978

RANKING FUZZY NUMBERS IN THE SETTING OF POSSIBILITY THEORY

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Didier DuboisHenri Prade

“Ranking Fuzzy Numbers in the Setting of Possibility Theory”

Information Sciences. 30 183–224. 1983


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S Let U be a set of elementary event and an event A, whichis contained. The possibility measure is defined by

S Given a normalized fuzzy set F,


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S When both A and F are fuzzy,

S An extension is interpreted in terms of the intersection ofthe level cuts of F and A

S when


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S The necessity measure is a set function defined by

S Let the complementary set of A, and a possibilitymeasure, the necessary measure can be defined by

S If the possibility measure derived from a normalizedmembership function.

POSSIBILITY THEORY

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Didier DuboisHenri Prade

“Possibility Theory”Plemun Press, 1988.

FUZZY LINEAR PROGRAMMING PROBLEMS WITH FUZZY NUMBERS

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Hideo TanakaKiyoji Asai

“Fuzzy Linear Programming Problems with Fuzzy Numbers”

Fuzzy Sets and Systems. Vol. 13, 1-10, 1984


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FUZZY MATHEMATICAL PROGRAMMINGTHE BEGINNING

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Hideo Tanaka

“Fuzzy Data Analysis by Possibilistic Linear Models”

Fuzzy Sets and Systems. Vol. 24, 363-375, 1987


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M . K. Luhandjula

“Fuzzy Optimization: An Appraisal” Fuzzy Sets and Systems

Vol. 30, pp. 257-282, 1989.


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J. J. Buckley

“Solving Possibilistic Linear Programming Problems”

Fuzzy Sets and SystemsVol. 31, pp. 329-341, 1989.

LINEAR PROGRAMMING WITH FUZZY OBJECTIVES

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H. RommelfangerR. Hanuscheck

J. Wolf

“Linear Programming with Fuzzy Objectives”

Fuzzy Sets and SystemsVol 29, pp. 31-48, 1989.

POSSIBILISTIC LINEAR PROGRAMMINGA BRIEF REVIEW …

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M. InuiguchiJ. Ramik

“Possilistic linear programming: a brief review offuzzy mathematical programming and

a comparison with stochastic programming inportfolio selection problem.”

Fuzzy Sets and SystemsVol 111, pp. 3-28, 2000.

POSSIBILISTIC LINEAR PROGRAMMINGA BRIEF REVIEW …

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RELATING DIFFERENT APPROACHES

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Miguel DelgadoJ.L. Verdegay

Amparo Vila

“Relating Different Approaches to SolveLinear Programming Problems with

Imprecise Costs.”Fuzzy Sets and Systems,

Vol. 37, pp 33-42, 1990.

FUZZY MATHEMATICAL PROGRAMMING

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FUZZY QUADRATIC PROGRAMMING

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NUMERICAL EXAMPLESUNCONSTRAINTS PROBLEMS

where c11 = c12 = c22 = (1,1,1)LR and c12 (2,1,1)LR.

NUMERICAL EXAMPLESUNCONSTRAINTS PROBLEMS

where c1 = c2 = (1,1,1)LR.

NUMERICAL EXAMPLESCONSTRAINTS PROBLEMS

where c11 = c12 = (4,1,1)LR and C21 = c22 = (5,1,1)LR.

NUMERICAL EXAMPLESCONSTRAINTS PROBLEMS

where c1 = c2 = (1,1,1)LR.

NUMERICAL EXAMPLEPRICE MECHANISM OF PREFABRICATED HOUSES

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In memorian to the professorsBellman, Tanaka, Asai, and Chanas

Thanks for your attention!

Ricardo [email protected]

Weldon [email protected]