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- 1. On Fuzzy Concepts in Engineering & Technology Surender Singh Asstt. Prof. , School of Mathematics Shri Mata Vaishno Devi University , Katra –182320 National Seminar on Engineering Applications of Mathematics (NSEAM) N.C College of Engineering, Israna 17th March, 2012
- 2. Outline • Introduction •crisp set •Fuzzy set •Fuzzy logic •Fuzzy logic System •Example ( To build a fuzzy controller) •Fuzzy concepts in Engineering •Some Probabilistic Divergence measures and their fuzzy analogue •A Model for strategic decision making •Illustrative Example •Conclusion
- 3. Introduction The present communication is intended to serve as introductory material on fuzzy sets and fuzzy logic . Some contextual usage of fuzziness in Engineering are presented. Three divergence measures between fuzzy sets are introduced and these measures are used to propose a Model for strategic decision making environment.
- 4. Crisp set Recall that a crisp set A in a universe of discourse U (which provides the set of allowable values for a variable) can be defined by listing all of its members or by identifying the elements x A. One way to do the latter is to specify a condition by which x A; thus A can be defined as A = { x / x meets some condition}. Alternatively, we can introduce a zero-one membership function (also called a characteristic function, discrimination function, or indicator function) for A, denoted µA(x) such that A µA(x) = 1 if x and µA(x) = 0 if x A. Subset A is mathematically equivalent to its membership function µA(x) in the sense that knowing µA(x) the same as knowing A itself.
- 5. Fuzzy Set Definition: Let a universe of discourse X = {x1, x2, x3… an} then a fuzzy subset of universe X is defined as A = {(x; µA(x)) / x ε X; µA(x): X [0; 1]} Where µA(x): X [0; 1] is a membership function defined as follow 0 if x does not belong to A and there is no ambiguity µA(x) = 1 if x belong to A and there is no ambiguity 0.5 if there is maximum ambiguity whether x belongs to A or not
- 6. Fuzzy set (cont…)In fact µA(x) associates with each x ε X a grade of membership of the set A. Some notions related to fuzzy sets [5].
- 7. Example 1 • A car can be viewed as “domestic” or “foreign” from different perspectives.
- 8. Fuzzy Logic Fuzzy logic is superset of the Boolean logic and it adds degrees between absolute true and absolute false in the sense that some propositions may to more true than others. Like the extension of the crisp set theory to fuzzy set theory, fuzzy logic is an extension of the crisp logic, in which the bivalent membership function is replaced by the fuzzy membership functions. In crisp logic the truth values acquired by the proposition are two valued, namely true as ‘1’ and false as ‘0’ while in the fuzzy logic the truth values acquired by the proposition are multi-valued, as absolutely true , partially true, absolutely false etc. represented numerically as real value between ‘0’ and ‘1’.
- 9. Fuzzy Logic System Fig.2 Fuzzy Logic System
- 10. Fuzzy Logic System (Cont…) • Rules may be provided by experts (you may be such a person) or can be extracted from numerical data. A collection of prepositions containing linguistic variables ; the rules are expressed in the form: IF x is A and y is B … THEN z is C. where x , z are variables ( e.g. distance , time etc.) and A,B,C are linguistic variables ( e.g. small ,far ,near etc.) • The fuzzifier maps crisp numbers into fuzzy sets. It is needed in order to activate rules which are in terms of linguistic variables, which have fuzzy sets associated with them.
- 11. Fuzzy Logic System (Cont…) • The inference engine of the FLS maps fuzzy sets into fuzzy sets. It handles the way in which rules are combined. • The defuzzifier maps output sets into crisp numbers. In a controls application, for example, such a number corresponds to a control action.
- 12. Example 2[1] (To build a fuzzy controller) • The temperature of a room equipped with an fan/air conditioner should be controlled by adjusting the motor speed of fan/ air conditioner. Fig3 describes the control of room temperature. In this example the goal is to Design a motor speed controller for fan.
- 13. (To build a fuzzy controller) Fig. 3
- 14. (To build a fuzzy controller) • Step 1: Assign input and output variables Let X be the temperature in Fahrenheit and Y be the motor speed of the fan. • Step 2: Pick fuzzy sets (Fuzzification) Define linguistic terms of the linguistic variables temperature (X) and motor speed (Y) and associate them with fuzzy sets .For example, 5 linguistic terms / fuzzy sets on X may be Cold, Cool, Just Right, Warm, and Hot. Let 5 linguistic terms / fuzzy sets on Y be Stop, Slow, Medium, Fast, and Blast.
- 15. (To build a fuzzy controller)
- 16. (To build a fuzzy controller)
- 17. (To build a fuzzy controller) Step 3: Assign a motor speed set to each temperature set (Rule or Fuzzy controller) • If temperature is cold then motor speed is stop • If temperature is cool then motor speed is slow • If temperature is just right then motor speed is medium • If temperature is warm then motor speed is fast • If temperature is hot then motor speed is blast
- 18. (To build a fuzzy controller)
- 19. (To build a fuzzy controller)
- 20. (To build a fuzzy controller)
- 21. (To build a fuzzy controller)
- 22. (To build a fuzzy controller)
- 23. (To build a fuzzy controller) Step 4: Defuzzification In this example crisp input is X= 63 Fo and crisp output is Y= 42%.
- 24. Fuzzy concepts in Engineering • A list of fuzzy terms (see table 1) that are widely used in control, signal processing and communications. However we always strive for their crisp values still these are used in fuzzy control, where they convey more useful information than would a crisp values.
- 25. Table1. Engineering Terms whose Contextual usages is usually quite fuzzy Terms Contextual Usage Alias None , a bit , high Bandwidth Narrowband, broadband Blur Somewhat ,quite , very Correlation Low, medium, high, perfect Errors Large ,medium, small, a lot of, so great, very large, very small, almost zero Frequency High , low , ultra-high Resolution Low , high Sampling Low-rate, medium-rate, high-rate Stability Lightly damped, highly damped, over damped, critically damped ,unstable
- 26. Fuzzy concepts in Engineering (cont…) • Correlation is an interesting example, because it can be defined mathematically so that, for a given set of data, we can compute a crisp number for it. Let’s assume that correlation has been normalized so that it can range between zero and unity, and that for a given set of data we compute the correlation value as 0.15. When explaining the amount of data correlation to someone else, it is usually more meaningful to explain it as “this data has low correlation.”When we do this, we are actually fuzzifying the crisp value of 0.15 into the fuzzy set “low correlation.” Other fuzzy terms appearing in Table 1 can also be interpreted accordingly.
- 27. Applications of Fuzzy logic in Engineering and interdisciplinary sciences • A short list of applications of FL includes: Controls Applications-aircraft control (Rockwell Corp.), Sendai subway operation (Hitachi), cruise control (Nissan), automatic transmission (Nissan, Subaru), self-parking model car (Tokyo Tech. Univ.), and space shuttle docking (NASA): Scheduling and Optimization-elevator scheduling (Hitachi, Fujitech, Mitsubishi) and stock market analysis (Yamaichi Securities); and Signal Analysis for Tuning and Interpretation - TV picture adjustment (Sony), handwriting recognition (Sony Palm Top), video camera autofocus (Sanyol Fisher, Canon) and video image stabilizer (Matshushita Panasonic). For many additional applications, see [1], [2], [3], [7] and [8].
- 28. Some probabilistic divergence measures be the set of all complete finite discrete probability distributions. Then for all P,Q ε Гn. Bhatia, Singh and Kumar [6] proposed three probabilistic divergence measures to discriminate between two probability distributions as follow:
- 29. Some probabilistic divergence measures (cont…)
- 30. Fuzzy Analogue of Prob. Div. Measures Where A and B are fuzzy sets and µA(x), µB(x) are their respective membership functions.
- 31. . . Model for Strategic Decision making Let the organization X want to apply m strategies S1, S2,… Sm to meet a target. Let each strategy has varied degree of effectiveness if cost associated with it is varied, let {C1,C2,…Cn} be cost set. Let the fuzzy set X denotes the effectiveness of a particular strategy with uniform cost. Therefore Further, let Cj be a fuzzy set denotes the degree of effectiveness of a strategy when a it implemented with cost Cj . . where j= 1,2...,n.
- 32. Model for Strategic Decision making (cont…) Taking A=X and B = Cj in the fuzzy divergence measures and calculate the value of Then . Let the minimum value is attained at Ct , With this Ct find , let it corresponds to Sp , Thus if the strategy Sp is Implemented with cost Cp then organization will meet its target in most cost effective manner. Determines the suitability of Cj
- 33. Illustrative Example Let m = n = 5 in the above model. The table below shows the effectiveness of strategies at uniform cost. Table:2
- 34. Illustrative Example (cont…) The table below shows the effectiveness of strategies at particular cost. Table:3
- 35. Illustrative Example (cont…) The table below shows the divergence between X and Cj , j = 1,2,3 ,4 ,5. Table:4
- 36. Illustrative Example (cont…) According to the divergence measures presented in the table 4 budget C2 is more suitable and after examining the table 3 , it is observed that strategy S1 is most effective. Therefore the organization will achieve its target in most cost effective manner if the strategy S1 is implemented with a budget C2 .
- 37. Scope for further research In this communication the basics of fuzzy set and fuzzy logic are discussed. There are some advanced concepts , like fuzzy c-means , Intustic fuzzy valued sets etc. These concepts can also be applied in certain areas.The concept of fuzziness can be used in the research related to digital image registration, image processing , pattern recognition , genome analysis for effective gene selection, network and queuing theory.
- 38. References [1]B. Kosko, Fuzzy Thinking: The New Science of Fuzzy Logic. New York Hyperion, 1993 [2] C. C. Lee, “Fuzzy logic in control systems: Fuzzy logic controller, part I,” IEEE Trans. Syst., Man, and Cybern., vol.SMC-20, no. 2, pp. 404-418, 1990. [3] D. Schwartz, G. J. Klir, H. W. Lewis 111, and Y. Ezawa, “Applications of fuzzy sets and approximate reasoning,” IEEE Proc., vol. 82, pp. 482-498, 1994. [4] G.J Klir And T.A Folger, Fuzzy sets ,Uncertainty and Information ,Prentice Hall International 1998.
- 39. References (cont…) [5] L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, pp.338-353 ,1965. [6] P.K Bhatia, Surender Singh And Vinod Kumar. Some New Divergence Measures and Their Properties. Int. J. of Mathematical Sciences and Applications,1(3), 2011,1349-1356 [7] T. Terano, K. Asai, and M. Sugeno, Fuzzy Systems Theory and Its Applications. New York Academic, 1992. [8] J. Yen, R. Langari, and L. Zadeh, Eds., Industrial Applications of Fuzy Logic and Intelligent Systems.
- 40. THANKS FOR YOUR ATTENTION

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