Need for Fuzzy Knowledge Where Fuzzy Degrees . . . Centroid . . . Optimization under . . . Probability-Based Probability-Based . . . Approach Explains Let’s Improve . . . The Resulting . . . (and Even Improves) This Is Indeed Better . . . Fuzzy Optimization Heuristic Formulas of Home Page Defuzzification Title Page ◭◭ ◮◮ Christian Servin 1 , Olga Kosheleva 2 , and Vladik Kreinovich 2 ◭ ◮ 1 El Paso Community College, El Paso, TX 79915, USA, Page 1 of 36 cservin@gmail.com 2 University of Texas at El Paso, El Paso, Texas 79968, USA, Go Back olgak@utep.edu, vladik@utep.edu Full Screen Close Quit
Need for Fuzzy Knowledge Where Fuzzy Degrees . . . 1. Need for Fuzzy Knowledge Centroid . . . • In many practical situations, ranging from medicine to Optimization under . . . driving, we rely on expert knowledge of: Probability-Based . . . Let’s Improve . . . – how to cure diseases, The Resulting . . . – how to drive in a complex city environment, etc. This Is Indeed Better . . . • Some medical doctors are more qualified than others, Fuzzy Optimization some drivers are more skilled than others. Home Page • It is therefore desirable to incorporate their skills and Title Page their knowledge in a computer-based system. ◭◭ ◮◮ • This will help other experts perform better. ◭ ◮ • Ideally, the system will make expert-quality decisions Page 2 of 36 on its own, without the need for the experts. Go Back Full Screen Close Quit
Need for Fuzzy Knowledge Where Fuzzy Degrees . . . 2. Need for Fuzzy Knowledge (cont-d) Centroid . . . • One of the main obstacles to designing such a system Optimization under . . . is the fact that: Probability-Based . . . Let’s Improve . . . – experts usually formulate their knowledge by using The Resulting . . . imprecise (“fuzzy”) words from natural language, This Is Indeed Better . . . – examples: “close”, “fast”, “small”, etc., but Fuzzy Optimization – computers are not efficient in processing words, they Home Page are much more efficient in processing numbers. Title Page • It is therefore desirable to represent the natural-language ◭◭ ◮◮ fuzzy knowledge in numerical terms. ◭ ◮ • Such technique was proposed in the 1960s by Lotfi Page 3 of 36 Zadeh from Berkeley under the name of fuzzy logic . Go Back Full Screen Close Quit
Need for Fuzzy Knowledge Where Fuzzy Degrees . . . 3. Need for Fuzzy Knowledge (cont-d) Centroid . . . • In fuzzy logic, to represent each word like “small” in Optimization under . . . numerical terms, we assign: Probability-Based . . . Let’s Improve . . . – to each possible value x of the corresponding quan- The Resulting . . . tity, This Is Indeed Better . . . – a degree µ ( x ) ∈ [0 , 1] to which, in the expert’s op- Fuzzy Optimization tion, the value x can be described by this word, Home Page – e.g., to what extent x is small. Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 36 Go Back Full Screen Close Quit
Need for Fuzzy Knowledge Where Fuzzy Degrees . . . 4. Where Fuzzy Degrees Come From Centroid . . . • There are many different ways to elicit the desired de- Optimization under . . . grees. Probability-Based . . . Let’s Improve . . . • If we are just starting the analysis and we do not have The Resulting . . . any records, then we can ask an expert: This Is Indeed Better . . . – to mark, on a scale, say, from 0 to 10, Fuzzy Optimization – to what extent x is small. Home Page • If the expert marks 7, we take 7/10 as the desired de- Title Page gree. ◭◭ ◮◮ • Usually, however, we already have a reasonably large ◭ ◮ database of records in which the experts: Page 5 of 36 – used the corresponding word Go Back – to describe different values of the corresponding Full Screen quantity x . Close Quit
Need for Fuzzy Knowledge Where Fuzzy Degrees . . . 5. Where Fuzzy Degrees Come From (cont-d) Centroid . . . • For example, when we describe the meaning of the Optimization under . . . word “small”, then: Probability-Based . . . Let’s Improve . . . – for values x which are really small, we will have a The Resulting . . . large number of such records; This Is Indeed Better . . . – on the other hand, for values x which are not too Fuzzy Optimization small, we will have a few such records; Home Page – indeed, few experts will consider these values to be Title Page small. ◭◭ ◮◮ • We can estimate the frequency with which different ◭ ◮ values x appear in our records. Page 6 of 36 • This frequency can be described by a probability den- sity function (pdf) ρ ( x ). Go Back • When x is really small, the value ρ ( x ) is big. Full Screen Close Quit
Need for Fuzzy Knowledge Where Fuzzy Degrees . . . 6. Where Fuzzy Degrees Come From (cont-d) Centroid . . . • When x is not so small, fewer experts will consider this Optimization under . . . value to be small. Probability-Based . . . Let’s Improve . . . • Thus, the value ρ ( x ) will be much smaller. The Resulting . . . • Thus, in principle, we could use the values ρ ( x ) as the This Is Indeed Better . . . desired degrees. Fuzzy Optimization • However, we want values of the membership function Home Page – and these values should be from the interval [0 , 1]. Title Page • However, the pdf can take values larger than 1. ◭◭ ◮◮ • To make all the values ≤ 1, we can normalize these ◭ ◮ values, i.e., divide by the largest of them: Page 7 of 36 ρ ( x ) µ ( x ) = ρ ( y ) . Go Back max y Full Screen • This is a well-known way to get membership functions Close (Coletti, Huynh, Lawry, et al.) Quit
Need for Fuzzy Knowledge Where Fuzzy Degrees . . . 7. Need for Defuzzification Centroid . . . • By using expert knowledge transformed into the nu- Optimization under . . . merical form, we can determine: Probability-Based . . . Let’s Improve . . . – for each possible value u of the control, The Resulting . . . – the degree µ ( u ) to which this value is reasonable. This Is Indeed Better . . . • These degrees can help an expert make better deci- Fuzzy Optimization sions. Home Page • However, if we want to make an automatic system, we Title Page must select a single value u that the system will apply. ◭◭ ◮◮ • Selecting such a value is known as defuzzification . ◭ ◮ Page 8 of 36 Go Back Full Screen Close Quit
Need for Fuzzy Knowledge Where Fuzzy Degrees . . . 8. Centroid Defuzzification: Description, Successes, Centroid . . . and Limitations Optimization under . . . • The most widely used defuzzification procedure is cen- Probability-Based . . . troid defuzzification, in which we select the value Let’s Improve . . . The Resulting . . . � x · µ ( x ) dx x = µ ( x ) dx . This Is Indeed Better . . . � Fuzzy Optimization • It has led to many successful applications of fuzzy con- Home Page trol. Title Page • However, it has two related limitations. ◭◭ ◮◮ • First, it is heuristic, it is not justified by a precise ar- ◭ ◮ gument. Page 9 of 36 • Therefore, we are not sure whether it will always work Go Back well. Full Screen • Second, it sometimes leads to disastrous results. Close Quit
Need for Fuzzy Knowledge Where Fuzzy Degrees . . . 9. Centroid Defuzzification (cont-d) Centroid . . . • For example, when a car encounters an obstacle on an Optimization under . . . empty road, it can go around it: Probability-Based . . . Let’s Improve . . . – by veering to the left or The Resulting . . . – by veering to the right. This Is Indeed Better . . . • The situation is completely symmetric with respect to Fuzzy Optimization the direction to the obstacle. Home Page • As a result, the centroid will lead exactly to the center Title Page – i.e., smack into the obstacle. ◭◭ ◮◮ • The actual fuzzy control algorithms use some tech- ◭ ◮ niques to avoid such as a situation. Page 10 of 36 • However, these techniques are also heuristic – and thus, Go Back not guaranteed to produce good results. Full Screen Close Quit
Need for Fuzzy Knowledge Where Fuzzy Degrees . . . 10. Optimization under Fuzzy Constraints Centroid . . . • Another class of situations in which fuzzy knowledge Optimization under . . . is important is optimization. Probability-Based . . . Let’s Improve . . . • Traditional optimization techniques finds x for which The Resulting . . . the objective function f ( x ) attains its optimal value. This Is Indeed Better . . . • This value can be argest or smallest depending on the Fuzzy Optimization problem. Home Page • These techniques assume – explicitly or implicitly – Title Page that all possible combinations x are possible. ◭◭ ◮◮ • In practice, there are usually constraints restricting ◭ ◮ possible combinations. Page 11 of 36 • In some cases, constraints are formulated in precise terms. Go Back Full Screen • For example, there are regulations limiting noise level and pollution level from a plant. Close Quit
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