Algebraic Product is the Definitions Main Results Only t-Norm for - PowerPoint PPT Presentation

Need for Optimization . . . Bellman-Zadeh . . . Problem: the Value . . . What We Show in . . . Algebraic Product is the Definitions Main Results Only t-Norm for Which Proof of Proposition 1 Optimization Under Fuzzy Proof of Proposition 2 Acknowledgments Constraints is Scale-Invariant Home Page Juan Carlos Figueroa Garcia 1 , Martine Ceberio 2 , and Title Page Vladik Kreinovich 2 ◭◭ ◮◮ 1 Universidad Distrital, Departamento de Ingenieria Industrial ◭ ◮ Bogota, Colombia, filthed@gmail.com Page 1 of 11 2 University of Texas at El Paso, El Paso, Texas 79968, USA mceberio@utep.edu, vladik@utep.edu Go Back Full Screen Close Quit

Need for Optimization . . . Bellman-Zadeh . . . 1. Need for Optimization under Fuzzy Constraints Problem: the Value . . . • Example: we need to build a chemical plant for pro- What We Show in . . . ducing chemicals needed for space exploration. Definitions Main Results • Among all designs x with small effect on environment, Proof of Proposition 1 we need to select the most profitable one. Proof of Proposition 2 • Here, for each alternative x , we can compute the value Acknowledgments f ( x ) of the objective function. Home Page • Constraints are formulated by using imprecise words Title Page from a natural language (like “small”). ◭◭ ◮◮ • In fuzzy logic, to each alternative x , we assign degree ◭ ◮ µ c ( x ) to which x is, e.g., small. Page 2 of 11 • E.g., if a user marks smallness of x by 7 on a scale 0 to 10, we take µ c ( x ) = 7 / 10. Go Back Full Screen • Problem: find x such that f ( x ) → max under con- straint µ c ( x ). Close Quit

Need for Optimization . . . Bellman-Zadeh . . . 2. Bellman-Zadeh Approach to Optimization un- Problem: the Value . . . der Fuzzy Constraints What We Show in . . . • First, we find the smallest value m of the objective Definitions function f ( x ) among all possible solutions x . Main Results Proof of Proposition 1 • Then, we find the largest possible value M of the ob- Proof of Proposition 2 jective function over all possible constraints. Acknowledgments • We form the degree to which x is maximal: Home Page = f ( x ) − m def µ m ( x ) M − m . Title Page ◭◭ ◮◮ • We want to find an alternative which satisfies the con- ◭ ◮ straints and maximizes the objective function. Page 3 of 11 • In fuzzy techniques, the degree of truth in “and”-statement is described by an appropriate t-norm f ( a, b ). Go Back Full Screen • So, we select x for which the degree µ s ( x ) = f & ( µ c ( x ) , µ m ( x )) is the largest. Close Quit

Need for Optimization . . . Bellman-Zadeh . . . Problem: the Value M is Not Well Defined 3. Problem: the Value . . . • Usually, we have experience with similar problems, so What We Show in . . . we know previously selected alternative(s) x . Definitions Main Results • The value f ( x ) for such “status quo” alternatives can Proof of Proposition 1 be used as the desired minimum m . Proof of Proposition 2 • Finding M is much more complicated, we do not know Acknowledgments which alternatives to include and which not to include. Home Page • If we replace the original value M with a new value Title Page M ′ > M , then the maximizing degree changes: ◭◭ ◮◮ µ m ( x ) = f ( x ) − m m ( x ) = f ( x ) − m M − m → µ ′ M ′ − m . ◭ ◮ Page 4 of 11 = M − m def • Here, µ ′ m ( x ) = λ · µ m ( x ) for λ M ′ − m < 1. Go Back • In general, diff. alternatives max µ s ( x ) = f & ( µ c ( x ) , µ m ( x )) Full Screen and µ ′ s ( x ) = f & ( µ c ( x ) , µ ′ m ( x )) = f & ( µ c ( x ) , λ · µ m ( x )). Close Quit

Need for Optimization . . . Bellman-Zadeh . . . 4. What We Show in This Talk Problem: the Value . . . In this paper: What We Show in . . . Definitions • we show that the dependence on M disappears if we Main Results use algebraic product t-norm f & ( a, b ) = a · b . Proof of Proposition 1 • We also show that this is the only t-norm for which Proof of Proposition 2 decisions do not depend on M . Acknowledgments Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 5 of 11 Go Back Full Screen Close Quit

Need for Optimization . . . Bellman-Zadeh . . . 5. Definitions Problem: the Value . . . • By a t-norm , we mean a f-n f & : [0 , 1] × [0 , 1] → [0 , 1] What We Show in . . . s.t. f & ( a, b ) = f & ( b, a ) and f & (1 , a ) = a for all a , b . Definitions Main Results • We say that optimization under fuzzy constraints is Proof of Proposition 1 scale-invariant for the t-norm f & ( a, b ) if Proof of Proposition 2 – for every set X , for every two functions Acknowledgments µ c : X → [0 , 1] and µ m : X → [0 , 1] , and Home Page – for every real number λ ∈ (0 , 1) , Title Page we have S = S ′ , where: ◭◭ ◮◮ • S is the set of all x ∈ X for which the function ◭ ◮ µ s ( x ) = f & ( µ c ( x ) , µ m ( x )) attains its maximum; Page 6 of 11 • S ′ is the set of all x ∈ X for which the function Go Back µ ′ s ( x ) = f & ( µ c ( x ) , λ · µ m ( x )) attains its maximum. Full Screen Close Quit

Need for Optimization . . . Bellman-Zadeh . . . 6. Main Results Problem: the Value . . . • Proposition 1. For the algebraic product t-norm f & ( a, b ) = What We Show in . . . a · b , optimization under fuzzy constraints is scale-invariant. Definitions Main Results • Proposition 2. a · b is the only t-norm for which Proof of Proposition 1 optimization under fuzzy constraints is scale-invariant. Proof of Proposition 2 • It is usually required that the t-norm is associative. Acknowledgments Home Page • Our result does not require associativity, so we can apply to non-associative and-operations. Title Page • Such operations sometimes more adequately represent ◭◭ ◮◮ human reasoning (Zimmermann, Zysno). ◭ ◮ Page 7 of 11 Go Back Full Screen Close Quit

Need for Optimization . . . Bellman-Zadeh . . . 7. Proof of Proposition 1 Problem: the Value . . . • For the algebraic product t-norm f & ( a, b ) = a · b : What We Show in . . . Definitions • S is the set of all x ∈ X for which the function Main Results µ s ( x ) = µ c ( x ) · µ m ( x ) attains its maximum, and Proof of Proposition 1 • S ′ is the set of all x ∈ X for which the function Proof of Proposition 2 µ ′ s ( x ) = µ c ( x ) · λ · µ m ( x ) attains its maximum. Acknowledgments • Here, µ ′ s ( x ) = λ · µ s ( x ) for a positive number λ . Home Page • Clearly, µ s ( x ) ≥ µ s ( y ) if and only if λ · µ s ( x ) ≥ λ · µ s ( y ). Title Page • So the optimizing sets S and S ′ for µ s ( x ) and µ ′ ◭◭ ◮◮ s ( x ) = λ · µ s ( x ) indeed coincide. ◭ ◮ Page 8 of 11 Go Back Full Screen Close Quit

Need for Optimization . . . Bellman-Zadeh . . . 8. Proof of Proposition 2 Problem: the Value . . . • Let f & ( a, b ) be a t-norm for which optimization under What We Show in . . . fuzzy constraints is scale-invariant. Definitions Main Results • Let a, b ∈ [0 , 1]; let us prove that f & ( a, b ) = a · b . Proof of Proposition 1 • Let us consider X = { x 1 , x 2 } with µ c ( x 1 ) = µ m ( x 2 ) = a Proof of Proposition 2 and µ c ( x 2 ) = µ m ( x 1 ) = 1. Acknowledgments • Here, µ s ( x 1 ) = f & ( µ c ( x 1 ) , µ m ( x 1 )) = f & ( a, 1) = a . Home Page • Similarly, µ s ( x 2 ) = f & ( µ c ( x 2 ) , µ m ( x 2 )) = f & (1 , a ) = a . Title Page ◭◭ ◮◮ • Since µ s ( x 1 ) = µ s ( x 2 ), the optimizing set S consists of both elements x 1 and x 2 . ◭ ◮ • Due to scale-invariance, for λ = b , S ′ = S = { x 1 , x 2 } Page 9 of 11 is optimizing for µ ′ s ( x ) = f & ( µ c ( x ) , λ · µ m ( x )). Go Back • Thus, µ ′ s ( x 1 ) = µ ′ s ( x 2 ), i.e., f & ( a, b · 1) = f & (1 , b · a ). Full Screen • So, f & ( a, b ) = f & (1 , b · a ) = a · b . Q.E.D. Close Quit

Need for Optimization . . . Bellman-Zadeh . . . 9. Acknowledgments Problem: the Value . . . • This work was supported in part: What We Show in . . . Definitions – by the National Science Foundation grants 0953339, Main Results HRD-0734825, HRD-1242122, DUE-0926721, Proof of Proposition 1 – by Grants 1 T36 GM078000-01 and 1R43TR000173- Proof of Proposition 2 01 from the National Institutes of Health, and Acknowledgments – by grant N62909-12-1-7039 from the Office of Naval Home Page Research. Title Page • This work was performed when J. C. Figueroa Garcia ◭◭ ◮◮ was a visiting researcher at Univ. of Texas at El Paso. ◭ ◮ Page 10 of 11 Go Back Full Screen Close Quit

10. Bibliography Need for Optimization . . . Bellman-Zadeh . . . Problem: the Value . . . • R. E. Bellman and L. A. Zadeh, “Decision making What We Show in . . . in a fuzzy environment”, Management Science , 1970, Definitions Main Results Vol. 47, No. 4, pp. B141–B145. Proof of Proposition 1 Proof of Proposition 2 • L. A. Zadeh, “Fuzzy sets”, Information and Control , Acknowledgments 1965, Vol. 8, pp. 338–353. Home Page • H. H. Zimmerman and P. Zysno, “Latent connectives Title Page in human decision making”, Fuzzy Sets and Systems , 1980, Vol. 4, pp. 37–51. ◭◭ ◮◮ ◭ ◮ Page 11 of 11 Go Back Full Screen Close Quit