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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
RANKING FUZZY NUMBERS IN THE SETTING OF POSSIBILITY THEORY
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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,
RANKING FUZZY NUMBERS IN THE SETTING OF POSSIBILITY THEORY
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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
RANKING FUZZY NUMBERS IN THE SETTING OF POSSIBILITY THEORY
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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
FUZZY LINEAR PROGRAMMING PROBLEMS WITH FUZZY NUMBERS
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FUZZY LINEAR PROGRAMMING PROBLEMS WITH FUZZY NUMBERS
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FUZZY LINEAR PROGRAMMING PROBLEMS WITH FUZZY NUMBERS
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FUZZY LINEAR PROGRAMMING PROBLEMS WITH FUZZY NUMBERS
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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
FUZZY MATHEMATICAL PROGRAMMINGTHE BEGINNING
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FUZZY MATHEMATICAL PROGRAMMINGTHE BEGINNING
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M . K. Luhandjula
“Fuzzy Optimization: An Appraisal” Fuzzy Sets and Systems
Vol. 30, pp. 257-282, 1989.
FUZZY MATHEMATICAL PROGRAMMINGTHE BEGINNING
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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 …
20
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 …
21
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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FUZZY QUADRATIC PROGRAMMING
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FUZZY QUADRATIC PROGRAMMING
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FUZZY QUADRATIC PROGRAMMING
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FUZZY QUADRATIC 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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1 2 3 4 5 6 7 8 9 10 11 12 13 14 151 2 3 4 5 6 7 8 9 10 11 12 13 14 15
NUMERICAL EXAMPLEPRICE MECHANISM OF PREFABRICATED HOUSES
NUMERICAL EXAMPLEPRICE MECHANISM OF PREFABRICATED HOUSES
500
1000
1500
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2500
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 151 2 3 4 5 6 7 8 9 10 11 12 13 14 15
In memorian to the professorsBellman, Tanaka, Asai, and Chanas
Thanks for your attention!
Ricardo [email protected]
Weldon [email protected]
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