Misconceptions about Fuzzy concepts

1. Misconceptions about Fuzzy concepts

2. Fuzziness is not Vague • we shall have a look at some propositions. • Dimitris is six feet tall • The first proposition (traditional) has acrisp truth valueof either TRUE or FALSE. • He is tall • Thesecond proposition is vague. • It does not provide sufficient information for us to make a decision, either fuzzy or crisp. • We do not know the value of the pronoun. • Is it Dimitris, John or someone else?

3. Fuzziness is not Vague • Andrei is tall • This proposition is a fuzzy proposition. • It is true to some degree depending in the context, i.e., the universe of discourse. • It might be SomeWhat True if we are referring to basketball players or it might be Very True if we are referring to horse-jockeys.

4. Fuzziness is not Multi-valued logic • The limitations of two-valued logic were recognised very early. • A number of different logic theories based on multiple values of truth have been formulated through the years. • For example, in three-valued logic three truth values have been employed. • These are TRUTH, FALSE, and UNKNOWN represented by 1, 0 and 0.5 respectively. • In 1921 the first N-valued logic was introduced. • The set of truth values Tn were assumed to be evenly divided over the closed interval [0,1]. • Fuzzy logic may be considered as an extension of multi-valued logic but they are somewhat different. • Multi-valued logic is still based on exact reasoning whereas fuzzy logic is approximate reasoning.

5. Fuzziness is not Probability!!!

6. Fuzziness is not Probability • Let X be the set of all liquids (i.e., the universe of discourse) . • Let L be a subset of X which includes all suitable for drinking liquids. A B The label of bottle B is marked as probability of L is 0.9. Bottle A label is marked as membership of L is 0.9. Which one would you drink?

7. Fuzziness is not Probability • This is better explained using an example. • Let X be the set of all liquids (i.e., the universe of discourse) . • Let L be a subset of X which includes all suitable for drinking liquids. • Suppose now that you find two bottles, A and B. • The labels do not provide any clues about the contents. • Bottle A label is marked as membership of L is 0.9. • The label of bottle B is marked as probability of L is 0.9. • Given that you have to drink from the one you choose, the problem is of how to interpret the labels.

8. Fuzziness is not Probability • Well, membership of 0.9 means that the contents of A are fairly similar to perfectly potable liquids. • If, for example, a perfectly liquid is pure water then bottle A might contain, say, tonic water. • Probability of 0.9 means something completely different. • You have a 90% chance that the contents are potable and 10% chance that the contents will be unsavoury, some kind of acid maybe. • Hence, with bottle A you might drink something that is not purebut with bottle B you might drink something deadly. So choose bottle A.

9. Fuzziness is not Probability • Opening both bottles you observe beer (bottle A) and hydrochloric acid (bottle B). • The outcome of this observation is that the membership stays the same whereas the probability drops to zero. • All in all: • probability measures the likelihood that a future event will occur, • fuzzy logic measures the ambiguity of events that have already occurred. • In fact, fuzzy sets and probability exist as parts of a greater Generalized Information Theory. • This theory also includes: • Dempster-Shafer evidence theory, • possibility theory, • and so on.

10. Applications of Fuzzy concepts

11. Fuzzy inference for practical control by Mamdani The most commonly used fuzzy inference technique is the so-called Mamdani method. In 1975, Professor EbrahimMamdaniof London University built one of the first fuzzy systems to control a steam engine and boiler combination. He applied a set of fuzzy rules supplied by experienced human operators.

12. Mamdani versus Sugeno Models • Most of our examples were for Mamdani Model. • Another famous model comes from Sugeno. • We will discuss and compare both models.Mo

13. Intelligent Systems and Soft Computing Sugeno fuzzy inference nMamdani-style inference, as we have just seen, requires us to find the centroid of a two-dimensional shape by integrating across a continuously varying function. In general, this process is not computationally efficient. nMichioSugenosuggested to use a single spike, a singleton, as the membership function of the rule consequent. A singleton, or more precisely a fuzzy singleton, is a fuzzy set with a membership function that is unity at a single particular point on the universe of discourse and zero everywhere else.

14. Intelligent Systems and Soft Computing Sugeno-style fuzzy inference is very similar to the Mamdani method. Sugeno changed only a rule consequent. Instead of a fuzzy set, he used a mathematical function of the input variable. The format of theSugeno-style fuzzy rule is IF x is A AND y is B THEN z is f (x, y) where x, y and z are linguistic variables; A and B are fuzzy sets on universe of discourses X and Y, respectively; and f (x, y) is a mathematical function.

15. Intelligent Systems and Soft Computing The most commonly used zero-order Sugeno fuzzy model applies fuzzy rules in the following form: IF x is A AND y is B THEN z is k wherek is a constant. In this case, the output of each fuzzy rule is constant. All consequent membership functions are represented by singleton spikes.

16. Sugeno-style rule evaluation

17. Sugeno-style aggregation of the rule outputs Intelligent Systems and Soft Computing

18. Intelligent Systems and Soft Computing Weighted average (WA): Sugeno-style defuzzification

19. Intelligent Systems and Soft Computing How to make a decision on which method to apply – Mamdani or Sugeno? nMamdani method is widely accepted for capturing expert knowledge. It allows us to describe the expertise in more intuitive, more human-like manner. However, Mamdani-type fuzzy inference entails a substantial computational burden. nOn the other hand, Sugeno method is computationally effective and works well with optimisation and adaptive techniques, which makes it very attractive in control problems, particularly for dynamic nonlinear systems.

20. Example 4 (Mamdani Fuzzy Model) Single input single output Mamdani fuzzy model with 3 rules: If X is small then Y is small  R1 If X is medium then Y is medium  R2 Is X is large then Y is large  R3 X = input [-10, 10] Y = output [0,10]

21. Single input single output antecedent & consequent MFs Using centroid defuzzification, we obtain the following overall input-output curve Overall input-output curve

22. Intelligent Systems and Soft Computing Example 5 (Mamdani Fuzzy model) Two input single-output Mamdani fuzzy model with 4 rules: If X is small & Y is small then Z is negative large If X is small & Y is large then Z is negative small If X is large & Y is small then Z is positive small If X is large & Y is large then Z is positive large

23. Two-input single output antecedent & consequent MFs X = [-5, 5]; Y = [-5, 5]; Z = [-5, 5] with max-min composition & centroid defuzzification, we can determine the overall input output surface

24. Overall input-output surface Intelligent Systems and Soft Computing X = [-5, 5]; Y = [-5, 5]; Z = [-5, 5] with max-min composition & centroid defuzzification, we can determine the overall input output surface Z Y X

25. Overall input-output surface Intelligent Systems and Soft Computing X = [-5, 5]; Y = [-5, 5]; Z = [-5, 5] with max-min composition & centroid defuzzification, we can determine the overall input output surface

26. Example of Mamdani:Cement Kiln Example EXAMPLE 6

30. Examples for Sugeno Fuzzy Models Example 7: Single output-input Sugeno fuzzy model with three rules If X is small then Y = 0.1X + 6.4 If X is medium then Y = -0.5X + 4 If X is large then Y = X – 2 If “small”, “medium” & “large” are nonfuzzy sets then the overall input-output curve is a piece wise linear

31. If X is small then Y = 0.1X + 6.4 If X is medium then Y = -0.5X + 4 If X is large then Y = X – 2

32. However, if we have smooth membership functions (fuzzy rules) the overall input-output curve becomes a smoother one

33. Examples for Sugeno Fuzzy Models Example 8: Two-input single output fuzzy model with 4 rules R1: if X is small & Y is small then z = -x +y +1 R2: if X is small & Y is large then z = -y +3 R3: if X is large & Y is small then z = -x +3 R4: if X is large & Y is large then z = x + y + 2 Overall input-output surface

34. EXAMPLE 8 Building a fuzzy expert system: case study

35. Intelligent Systems and Soft Computing Building a fuzzy expert system: case study nA service centre keeps spare parts and repairs failed ones. nA customer brings a failed item and receives a spare of the same type. nFailed parts are repaired, placed on the shelf, and thus become spares. nThe objective here is to advise a manager of the service centre on certain decision policies to keep the customers satisfied.

36. Intelligent Systems and Soft Computing Process of developing a fuzzy expert system 1. Specify the problem and define linguistic variables. 2. Determine fuzzy sets. 3. Elicit and construct fuzzy rules. 4. Encode the fuzzy sets, fuzzy rules and procedures to perform fuzzy inference into the expert system. 5. Evaluate and tune the system.

37. Intelligent Systems and Soft Computing Step 1: Specify the problem and define linguistic variables • There are four main linguistic variables: • average waiting time (mean delay) m, • repair utilization factor of the service centre r (is the ratio of the customer arrival day to the customer departure rate) , • number of servers s, • initial number of spare parts n.

38. Linguistic variables and their ranges

39. Intelligent Systems and Soft Computing • Step 2: Determine fuzzy sets • Fuzzy sets can have a variety of shapes. • However, • a triangle or a trapezoid can often provide an adequate representation of the expert knowledge, • and at the same time, significantly simplifies theprocess of computation.

40. Intelligent Systems and Soft Computing Fuzzy sets of Mean Delay m

41. Fuzzy sets of Number of Servers s

42. Fuzzy sets of Repair Utilisation Factor r

43. Fuzzy sets of Number of Spares n

44. Step 3: Elicit and construct fuzzy rules To accomplish this task, we might ask the expert to describe how the problem can be solved using the fuzzy linguistic variables defined previously. Required knowledge also can be collected from other sources such as books, computer databases, flow diagrams and observed human behavior. The matrix form of representing fuzzy rules is called fuzzy associative memory (FAM).

45. The square FAM representation

46. The rule table

47. Rule Base 1

48. Cube FAM of Rule Base 2

49. Step 4: Encode the fuzzy sets, fuzzy rules and procedures to perform fuzzy inference into the expert system To accomplish this task, we may choose one of two options: to build our system using a programming language such as C/C++ or Pascal, or to apply a fuzzy logic development tool such as MATLAB Fuzzy Logic Toolbox, Fuzzy Clips, or Fuzzy Knowledge Builder.

50. Step 5: Evaluate and tune the system The last, and the most laborious, task is to evaluate and tune the system. We want to see whether our fuzzy system meets the requirements specified at the beginning. Several test situations depend on the mean delay, number of servers and repair utilization factor. The Fuzzy Logic Toolbox can generate surface to help us analyze the system’s performance.