Need for Optimization . . . Need for Fuzzy . . . How to Optimize . . . A Problem Which t-Norm Case When This . . . Is Most Appropriate for Our Answers First Result: Product . . . Bellman-Zadeh Optimization What If We Use a . . . Third Result: It Is Not . . . Olga Kosheleva 1 , Vladik Kreinovich 1 , and Home Page Shahnaz Shahbazova 2 Title Page 1 University of Texas at El Paso ◭◭ ◮◮ El Paso, TX 79968, USA ◭ ◮ olgak@utep.edu, vladik@utep.edu 2 Azerbaijan Technical University Page 1 of 100 Baku, Azerbaijan shahbazova@gmail.com Go Back Full Screen Close Quit
Need for Optimization . . . Need for Fuzzy . . . 1. Need for Optimization Under Constraints How to Optimize . . . • In many practical problems: A Problem Case When This . . . – we need to find an optimal alternative a opt Our Answers – among all alternatives from the set P of all possible First Result: Product . . . ones. What If We Use a . . . • Optimal means that the value of the corresponding ob- Third Result: It Is Not . . . jective function f ( x ) is the largest possible: Home Page Title Page f ( a opt ) = max a ∈ P f ( a ) . ◭◭ ◮◮ ◭ ◮ Page 2 of 100 Go Back Full Screen Close Quit
Need for Optimization . . . Need for Fuzzy . . . 2. Need for Fuzzy Constraints How to Optimize . . . • The above formulation works well if we know the set P . A Problem Case When This . . . • In practice, for some alternatives a , we are not sure Our Answers that these alternatives are possible. First Result: Product . . . • For such alternatives, an expert can describe to what What If We Use a . . . extent these alternatives are possible. Third Result: It Is Not . . . Home Page • This description is often made in terms of imprecise (“fuzzy”) words from natural language. Title Page • Zadeh invented fuzzy logic specifically: ◭◭ ◮◮ – to translate such imprecise natural-language knowl- ◭ ◮ edge Page 3 of 100 – into precise computer-understandable form. Go Back • E.g., we ask each expert to estimate, on a scale, say, 0 Full Screen to 10, to what extend each alternative is possible. Close Quit
Need for Optimization . . . Need for Fuzzy . . . 3. Need for Fuzzy Constraints (cont-d) How to Optimize . . . • If an expert marks 7 on a scale of 0 to 10, we say that A Problem the expert’s degree of confidence that a is possible is Case When This . . . Our Answers µ ( a ) = 7 / 10 = 0 . 7 . First Result: Product . . . What If We Use a . . . • This way: Third Result: It Is Not . . . – to each alternative a , Home Page – we assign a degree µ ( a ) ∈ [0 , 1] to which, according Title Page to the experts, this alternative is possible. ◭◭ ◮◮ • The corresponding function µ is known as a member- ◭ ◮ ship function or, alternatively, as a fuzzy set . Page 4 of 100 Go Back Full Screen Close Quit
Need for Optimization . . . Need for Fuzzy . . . 4. How to Optimize Under Fuzzy Constraints How to Optimize . . . • How to optimize a function f ( a ) under fuzzy con- A Problem straints – described by a membership function µ ( a )? Case When This . . . Our Answers • This question was raised in a joint paper of L. Zadeh First Result: Product . . . and Richard Bellman, a famous specialist in control. What If We Use a . . . • Their main idea is to look for an alternative which is, Third Result: It Is Not . . . to the largest extent, both possible and optimal. Home Page • To be more precise, first, we need to describe the degree Title Page µ opt ( a ) to which an alternative is optimal. ◭◭ ◮◮ • Then, for each alternative a , we need to combine: ◭ ◮ – the degree µ ( a ) to which this alternative is possible Page 5 of 100 and – the degree µ opt ( a ) to which this alternative is opti- Go Back mal Full Screen – into a single degree to which a is possible and op- Close timal. Quit
Need for Optimization . . . Need for Fuzzy . . . 5. Optimizing Under Fuzzy Constraints (cont-d) How to Optimize . . . • Finally, we select an alternative a opt for which the com- A Problem bined degree is the largest possible. Case When This . . . Our Answers • Let us start with the first step: finding out to what First Result: Product . . . extent an alternative a is optimal. What If We Use a . . . • Of course, if some alternative has 0 degree of possibil- Third Result: It Is Not . . . ity, this means that this alternative is not possible. Home Page • So, we should consider only alternatives from the set Title Page def A = { a : µ ( a ) > 0 } . ◭◭ ◮◮ • If two alternatives a and a ′ have the same value of the ◭ ◮ objective function f ( a ) = f ( a ′ ), then, intuitively, Page 6 of 100 – our degree of confidence that the alternative a is Go Back optimal Full Screen – should be the same as our degree of confidence that the alternative a ′ is possible. Close Quit
Need for Optimization . . . Need for Fuzzy . . . 6. Optimizing Under Fuzzy Constraints (cont-d) How to Optimize . . . • Thus, the degree µ opt ( a ) should only depend on the A Problem value f ( a ). Case When This . . . Our Answers • In other words, we should have µ opt ( a ) = F ( f ( a )) for First Result: Product . . . some function F ( x ). What If We Use a . . . • Here: Third Result: It Is Not . . . Home Page – when the value f ( a ) is the smallest possible, i.e., def when f ( a ) = f = min a ∈ A f ( a ) , Title Page ◭◭ ◮◮ – then we are absolutely sure that this alternative is not optimal, i.e., that µ opt ( a ) = 0. ◭ ◮ • Thus, we should have F ( f ) = 0. Page 7 of 100 Go Back Full Screen Close Quit
Need for Optimization . . . Need for Fuzzy . . . 7. Optimizing Under Fuzzy Constraints (cont-d) How to Optimize . . . • On the other hand: A Problem def Case When This . . . – if the value f ( a ) is the largest possible: f ( a ) = f = Our Answers max a ∈ A f ( a ) , First Result: Product . . . – then we are absolutely sure that this alternative is What If We Use a . . . optimal, i.e., that µ opt ( a ) = 1. Third Result: It Is Not . . . • Thus, we should have F ( f ) = 1. Home Page • So, we need to select a function F ( x ) for which F ( f ) = Title Page 0 and F ( f ) = 1. ◭◭ ◮◮ • It is also reasonable to require that the function F ( f ) ◭ ◮ increases with f . Page 8 of 100 • The simplest such function is linear: Go Back = f ( a ) − f def Full Screen F ( f ( a )) = L ( f ( a )) . f − f Close Quit
Need for Optimization . . . Need for Fuzzy . . . 8. Optimizing Under Fuzzy Constraints (cont-d) How to Optimize . . . • However, non-linear functions are also possible. A Problem Case When This . . . • We can also have F ( f ( a )) = S ( L ( F ( a ))) for some non- Our Answers linear scaling f-n S ( x ) for which S (0) = 0 and S (1) = 1. First Result: Product . . . • We need: What If We Use a . . . – to combine the degrees µ ( a ) and F ( f ( a )) of the Third Result: It Is Not . . . statements “ a is possible” and “ a is optimal” Home Page – into a single degree describing to what extent a is Title Page both possible and optimal. ◭◭ ◮◮ • For this, we can use an “and”-operation (t-norm) ◭ ◮ f & ( x, y ) . Page 9 of 100 • The most widely used “and”-operations are min( x, y ) Go Back and x · y . Full Screen • Thus, we find the alternative a for which the value Close d ( a ) = f & ( µ ( a ) , F ( f ( a ))) is the largest possible. Quit
Need for Optimization . . . Need for Fuzzy . . . 9. Optimizing Under Fuzzy Constraints (cont-d) How to Optimize . . . • If we use a linear scaling function F ( x ), then we select A Problem a for which the following value is the largest: Case When This . . . Our Answers � � µ ( a ) , f ( a ) − f d ( a ) = f & . First Result: Product . . . f − f What If We Use a . . . Third Result: It Is Not . . . • When f & ( x, y ) = min( x, y ), then we get Home Page � � µ ( a ) , f ( a ) − f Title Page d ( a ) = min . f − f ◭◭ ◮◮ ◭ ◮ • When f & ( x, y ) = x · y , then we get Page 10 of 100 d ( a ) = µ ( a ) · f ( a ) − f . f − f Go Back Full Screen Close Quit
Need for Optimization . . . Need for Fuzzy . . . 10. A Problem How to Optimize . . . • The problem with this definition is that it depends on A Problem the values f and f . Case When This . . . Our Answers • Thus, it depends on the exact shape of the set First Result: Product . . . A = { a : µ ( a ) > 0 } . What If We Use a . . . Third Result: It Is Not . . . • In practice, experts have only approximate idea of the Home Page corresponding degrees µ ( a ). Title Page • So when µ ( a ) is very small, it could be 0, or vice versa. ◭◭ ◮◮ • These seemingly minor changes in the membership ◭ ◮ function can lead to huge changes in the set A . Page 11 of 100 • Thus, they can lead to huge changes in the values f Go Back and f . Full Screen Close Quit
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