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