2. Fuzzy Rules 1965 paper: “Fuzzy Sets” (Lotfi Zadeh) Apply natural language terms to a formal system of mathematical logic http://www.cs.berkeley.edu/~zadeh 1973 paper outlined a new approach to capturing human knowledge and designing expert systems using fuzzy rules

3. Fuzzy Rules A fuzzy rule is a conditional statement in the familiar form: IF x is A THEN y is B x and y are linguistic variables A and B are linguistic values determined by fuzzy sets on the universe of discourses X and Y, respectively

4. Linguistic Variables A linguistic variable is a fuzzy variable e.g. the fact “ John is tall ” implies linguistic variable “ John ” takes the linguistic value “ tall ” Use linguistic variables to form fuzzy rules : IF‘project duration’ is long THEN‘risk’ is high IFrisk is very high THEN‘project funding’ is very low

5. Fuzzy Expert Systems A fuzzy expert system is an expert system that uses fuzzy rules, fuzzy logic, and fuzzy sets Many rules in a fuzzy logic system will fire to some extent If the antecedent is true to some degree of membership, then the consequent is true to the same degree

6. Fuzzy Expert Systems Two distinct fuzzy sets describing tall and heavy :

7. Fuzzy Expert Systems IF height is tall THEN weight is heavy

8. Fuzzy Expert Systems Other examples (multiple antecedents): e.g. IF‘ project duration ’ is long AND‘ project staffing ’ is large AND‘ project funding ’ is inadequate THEN risk is high e.g. IF service is excellent OR food is delicious THEN tip is generous

9. Fuzzy Expert Systems Other examples (multiple consequents): e.g. IF temperature is hot THEN‘ hot water ’ is reduced ; ‘ cold water ’ is increased

10. Fuzzy Inference Named after Ebrahim Mamdani, the Mamdani method for fuzzy inference is: 1.Fuzzify the input variables 2.Evaluate the rules 3. Aggregate the rule outputs 4. Defuzzify

11. Fuzzy Inference – Example Rule 1: IF x is A3 OR y is B1 THEN z is C1 Rule 2: IF x is A2 AND y is B2 THEN z is C2 Rule 3: IF x is A1 THEN z is C3 Rule 1: IF‘ project funding ’ is adequate OR‘ project staffing ’ is small THEN risk is low Rule 2: IF‘ project funding ’ is marginal AND‘ project staffing ’ is large THEN risk is normal Rule 3: IF‘ project funding ’ is inadequate THEN risk is high x, y, and z are linguistic variables A1, A2, and A3 are linguistic values on X B1 and B2 are linguistic values on Y C1, C2, and C3 are linguistic values on Z

12. Fuzzy Inference – Example 1.Fuzzification project funding project staffing inadequate marginallarge small

13. Fuzzy Inference – Example 2.Rule 1 evaluation project funding project staffing adequatesmall risk low

14. Fuzzy Inference – Example 2.Rule 2 evaluation project fundingproject staffing marginallarge risk normal

15. Fuzzy Inference – Example 2.Rule 3 evaluation project funding inadequate risk high

16. Fuzzy Inference – Example 3.Aggregation of the rule outputs risk highnormallow

17. Fuzzy Inference – Example 4.Defuzzification e.g.use the centroid method in which a vertical line slices the aggregate set into two equal halves How can we calculate this?

18. Fuzzy Inference – Example 4.Defuzzification Calculate the centre of gravity ( cog ): xxdx x

19. Fuzzy Inference – Example 4.Defuzzification Use a reasonable sampling of points

20. Applications of Fuzzy Logic Why use fuzzy expert systems or fuzzy control systems ? Apply fuzziness ( and therefore accuracy ) to linguistically defined terms and rules Lack of crisp or concrete mathematical models exist When do you avoid fuzzy expert systems ? Traditional approaches produce acceptable results Crisp or concrete mathematical models exist and are easily implemented

21. Applications of Fuzzy Logic Real-world applications include: Control of robots, engines, automobiles, elevators, etc. Sendai Subway system in Sendai, Japan Cruise-control in automobiles Temperature control Handwriting recognition, OCR Predictive and diagnostic systems (e.g. cancer)