J. Comp. & Math. Sci. Vol.5 (1), 111-117 (2014)
Development of New Theory of Fuzzy Logic Transformations for Environmental Pollution Problems and Their Solutions R. N. Yadava1, G. P. Chhalotra2, Manisha Chhalotra3, R. K. Tiwari4, Rajendra Kumar Varandani5 and Rajesh Khattri6 1
Former Director, Grade Scientist AMPRI-CSIR, Bhopal, M. P., INDIA. 2 Former Dean, Govt. Engineering College, Jabalpur, M. p., INDIA. 3 Atharawa Institute of Management, Bombay, M. S., INDIA. 4 Director, TIT, Bhopal, M. P., INDIA. 5 Dean Senior Students, Sagar Institute of Reachi Institute 6 Research Scholar, RGPV University, Bhopal, M. P., INDIA. (Received on: February 19, 2014) ABSTRACT Fuzzy logic is well known to all the researchers. It is a set theory of its kind and universal method of transformation of uncertain problems Here we have developed a new method to solve the pollution problems. The pollutants are described by their physical electrical mechanical, thermal and electro thermal parameters. The parameters are Fuzzified and then defuzzified to obtain the Fuzzy logic cardinality and relive Fuzzy cardinality. This can find the highly hazardous pollutant. The effects of pollution on the society can be found. Keywords: Fuzzy Logic, Fuzzy Set, Fuzzification.
1. INTRODUCTION L.A. Zadeh developed the Fuzzy logic in 1965 to solve the vague and uncertain problems. Most of our decisions
are taken in the dark environment. One cannot assume the good accuracy and precision in the decision. Bart Kosko developed a joint model of Fuzzy systems and neural networks for dynamic systems.
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The pollution is a dynamic system, consisting of several gases flow and drift in the air. There are several methods to calculate the Fuzzy grades of truth for known system. For unknown systems one has to assign the Fuzzy grades of truth by own heuristics and decision. The pollution materials are well known and can be analyzed using Fuzzy membership functions. One can assume the exponential distribution function and we bull function for the failure rate of the pollutants. 2. FUZZY LOGIC Fuzzy logic is a mathematical tool to study a graph of any event from 0 to 1 and rising with a variable x. This is necessary for the rockets aero planes, satellites and many other vehicles rising to a height of 1 from zero. The path assigned to the rockets may be Fuzzy path. Similarly the rockets may be landing according to a given curve. The Fuzzy paths are random but can be designed according to well known mathematical tools or newly developed mathematical toos to find a very good path. Fig. 1 represents several Fuzzy distribution curves to rise from 0 to 1 with a time t one can predict easily which graph is dangerous and hazardous for a rocket to rise and coming down.
The graphs I and II are more risky than graphs II and IV. The rate of rise is extremely high for the air vehicles and they will burn down. If the vehicles come down with the paths I and II they will burn down. Similarly any thing which rises in a Fuzzy way carries greater risk. 3. FUZZY SET A system is defined when input and output are specified with its elements X; The elements may be Fuzzy in nature but can be specified by its element hood or Fuzzy grade of truth or Fuzzy membership function (x) where x is the Fuzzy element Fig. 2 represents a Fuzzy system with element Xi
Fig 2 A Fuzzy System
This Fuzzy system can be simulated with a Fuzzy set. F( z ) ( ( x1 )x1 , ( x2 )x2 , ( x3 )x3 ,........ ) Fuzzy logic is a transformation of complicated systems which have no reference no context and no origin. 4. FUZZY SET OF AIR POLLUTION The pollution of the environment is highly uncertain but can be Fuzzified with density (D), melting point (MP), Boiling point (BP), thermal conductivity (TC), specific heat capacity (SP), latent heat of evaporation (Le) enthalpy
Fig. 1 Fuzzy graphs for a rocket to rise in the steady stead path from 0 to 1
F( Z) ( D, MP , BP , TC , SP , Le, H , P , Er , r , , tan , Q, n, CT , CP , CV , , Bm E, ST , Se, Ce ) F( P )
Journal of Computer and Mathematical Sciences Vol. 5, Issue 1, 28 February, 2014 Pages (1-122)
R.N. Yadava, et al., J. Comp. & Math. Sci. Vol.5 (1), 111-117 (2014) ( H ), electrical resistivity (p), dielectric
constant (Er), magnetic permeability ( ) , loss tangent ( t a n ) , Q-factor (Q), refractive index(n), critical temperature (cT), critical pressure (cP) critical volume (CV), viscosity ( ) , Bulk modulus ( B m ) Young’s modulus (E), surface tension (ST) specific latent heat of vaporization (Se), cubic expensivity (Ce). r
F ( Z ) ( D , MP , BP , TC , SP , Le , H , P , Er , r , , tan , Q , n, CT , C P , CV , , Bm E , ST , Se, Ce ) F ( P )
All the 22 properly elements follow the mass law of mixing and logarithmic law of the heterogeneous mixtures. The Air pollution is a mixture of several gases. The Fuzzy set of the pollution would be.
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3.214 kg/m3) and its used to Fuzzify all other elements as follows: F(D ) = (0.365, 0.4023, 0.2398, 0.555, 0.615, 0.3889, 1.00, 0.727, 0.547, 0.0557, 0.28, 0.5102, 0.478, 0.223, 0.417, 0.388, 0.615, 0.445, 0.9107, 0.2489, 0.445, 0.423) = (x) This is a symmetrical Fuzzy set having an elementhood 1.00. One can defuzzify the Fuzzy set F(Dm) by getting a Fuzzy cardinality as follows A
z
22
1
( x ) dx 10 .0275, ( x 10 and x 11 cardinal events )
And relative Fuzzy cardinality will be: A
1 n
z
n
1
(x)dx 0.4557954
F ( Z ) ( C2 H 2 Air , NH 4 , Ar , CO2 , CO , Cl2 , C 2 N 2 , C 2 H 4 , He He , H 2 , HCl , H 2 S , CH 4 , NO , N 2 , N 2 O ,
O2 , SO2 , H 2 O vapours NO , NO2 , SO3 ) F ( poll )
F(D) = Fuzzy set of density of the pollutant will be F(D) = (1.173, 1.293, 0.771, 1.784, 1.977, 1.250, 3.214, 2.337, 1.76, 0.179, 0.09, 1.64, 1.538, 0.717, 1.34, 1.25, 1.978, 1.429, 2.927, 0.80, 1.43, 1.36, 1927 (Solid)) In this manner one can form 22 Fuzzy sets of the above property parameters to make a Fuzzy decision of the air pollution. The Air is itself a Fuzzy element is the pollution. O2 and N2 are also the elements in the Fuzzy set F(Z) = F(poll). Attempts are made to analyze the air pollution effects and properties of the pollution mixture. One can drop the SO3 as it does not belong to this Fuzzy set as the density is extremely higher to oxyven and other gaseous pollutants. 5. FUZZIFICATION OF F(D) FUZZY SET The Fuzzy set F(D) has an element Cl2 that has a maximum density (Dmax =
6. THE MASS LAW OF MIXING AND FUZZY LOGIC The density (D), melting point (MP) etc follow the mass law of maxing of the components as follows. One can mark this large Fuzzy matrix to a small Fuzzy matrix as follows: ( D, BP , SP , C P , CP / CV , TC , , (n 1), CT , CP , CV )
[P] [M] [V] and [V] [M]1 [P]
The [M]-1 is very difficult because mass law of maxing is only in one directional. The Fuzzy logic can provide the [M]-1 by probability of distribution function and frequency of occurrence of the Vi the volumetric contents of the pollutants. This problem is solved here using the Fuzzy cardinality method. The MOM and COA methods are also used in the study. All the 22 parameters of the 22 pollutants can be obtained from the tables and some of them may be calculated using mass law of maxing. For Air one can use the parameters as follows:
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LM D OP MM M p PP MM TS pc PP MM L e PP MM H PP MM PE r PP MM r PP MM t a n PP MM Q PP MM nC PP MM C p PP MM C PP MM PP MM BE PP MM S T PP MM S e PP NCe Q T
v
m
LM D MM MB PP MM T C MM S P MM L e MM HP MM E r MM r MM t a Qn MM n MM C MM C p MM C MM B MM E MM SS Te MN C e
D2
. . . D 22
M P2 B P2
. . . M P2 2 . . . B P2 2
TC2 S P2
. . . T C 22 . . . S P2 2
1
1
1 1
1
1
1
L e2 H
2
. . . L e 22 . . . H 22
P2 E r2
1
1
1
1
r2 tan
. . . P2 2 . . . E r2 2 2
. . . r2 2 . . . ta n 22
Q2 n2
. . . Q 22 . . . n 22
1
CT2 Cp2
. . . C T 22 . . . C p 22
v1
Cv2
. . . C v 22 . . . 22
1
2
T1
1
2
Bm2 E2
... Bm1 . . . E 22
1
S T2 S e2
. . . S T2 2 . . . S e1
1
C e2
. . . C e1
m1 1
1
LM V MM VV MM V MM V MM V MM V MM VV MM V MM V MM V MM VV MM V MM V MM V MM VV MM V MM V MN V
OP PP PP PP PP PP PP PP PP PP PP PP PP PP PP PP PQ
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
OP PP PP PP PP PP PP PP PP PP PP PP PP PP PP PP PP PQ
Table – 1 (Fuzzy Set) Xi 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
(X) 0.365 0.4023 0.2398 0.555 0.615 0.3889 1.00 0.727 0.547 0.0557 0.028 0.5102 0.478 0.223 0.417 0.388 0.615 0.445 0.9107 0.2489 0.445 0.423
1.0078 0.9105 1.428 0.588 0.4861 0.944 0 0.318 0.6032 2.887 3.575 0.673 0.738 1.50 0.874 0.946 0.486 0.8096 0.0935 1.3906 0.809 0.8603
R 0.3678 0.366 0.342 0.326 0.298 0.3672 0 0.2317 0.345 0.1608 0.1001 0.343 0.3528 0.334 0.364 0.3673 0.2989 0.3603 0.085 0.346 0.3603 0.3639
Sec 0.632 0.6337 0.657 0.6732 0.7010 0.6327 -1 0.7682 0.654 0.839 0.899 0.656 0.647 0.665 0.635 0.632 0.7010 0.6396 0.914 0.6538 0.6397 0.636
T 0.992 1.098 0.7002 1.7006 2.057 1.0593
3.144 1.582 0.3463 0.279 1.48 1.355 0.666 1.144 1.057 2.057 1.235 10.69 0.7191 1.236 1.162
A 0.332 0.386 0.211 0.511 0.588 0.332 0.886 0.668 0.489 0.042 0.011 0.423 0.422 0.186 0.376 0.311 0.552 0.411 0.886 0.182 0.411 0.386
M 0.346 0.384 0.206 0.498 0.562 0.346 0.912 0.682 0.496 0.046 0.012 0.446 0.426 0.192 0.366 0.322 0.546 0.421 0.866 0.186 0.408 0.356
Journal of Computer and Mathematical Sciences Vol. 5, Issue 1, 28 February, 2014 Pages (1-122)
R.N. Yadava, et al., J. Comp. & Math. Sci. Vol.5 (1), 111-117 (2014)
( D, BP , SP , CP , CP / CV , TC , , (n 1), CT , CP , CV ) 3
4
(1.293 kg/m , 83K, 21.4 x 10 J/kg 993 J/kgk, 1.102, 241 x 10-4 W/mk, 18.325 x 10-6 NS/m2, 292 x 10-6, 132K, 3.77 MPa, 98m3/mol) The eleven parameters are given in table of Clarke and eleven are calculated. The electrical parameters can be provided for the air ( P, Er , r , tan ) (100m, 10006, 1.00006, 0.002)
The failure rate will be Xi 1. 2. 3. 4. 5. 6. 7. 8.
Fuzzy Grade of Truth Reliability (R) Failure Rate ( ) MTBF (T) Risk ( R) Security (1 – R) Availability (A) Maintainability (M) Distribution
log 10 ( x ) 0. 7857 0.4343
MTBF = 1/ = 1.2727 time units The Fuzzy grade of truth (x) = e-xt = e- t= elementhood one can form a table of the Fuzzy set F(D) to study the pollution F(Poll) below. The Fuzzy set has 22 elements and form the Fuzzy parameters as follows in Table – 2.
Table – 2 (Fuzzy Set) 0.455 A ( )
0.486
0.493
0.486 0.78 1.5 0.336 0.664 0.466 0.476
0.466 0.762 1.89 0.362 0.672 0.483 0.486
e (( 1 2 )( 0 ) e t
e (( 1 2 )( 0 )
0.472 0.766 0.179 0.342 0.668 0.496 0.52 e.rfex
One can form all the Fuzzy sets of property parameters in Table 3 Xi
F(Z)
A ( )
MTBF
1. 2. 3. 4. 5. 6. 7.
F(D) F(Mp) F(Bp) F(Tc) F(Sp) F(Le)
0.46 0.52 0.612 0.551 0.662 0.712 0.776
0.776 0.654 0.491 0.596 0.412 0.339 0.2536
1.28 1.52 2.036 1.677 2.424 2.944 3.943
8. 9. 10. 11. 12. 13.
F(p) F(Er) F( r) F(tan ) F(Q)
0.456 0.667 0.812 0.766 0.812 0.766
0.785 0.404 0.2082 0.2665 0.2082 0.266
1.273 2.46 4.801 3.75 4.801 3.751
14. 15. 16. 17.
F(CT) F(CP) F(CV)
0.812 0.462 0.511 0.612
0.2082 0.772 0.671 0.491
4.801 1.295 1.489 2.036
18. 19. 20. 21. 22.
F(Bm) F(E) F(ST) F(Se) F(Ce)
0.717 0.812 0.776 0.712 0.462
0.3326 0.2082 0.253 0.3396 0.772
3.005 4.801 3.94 2.944 1.295
F(DH )
F( )
F( )
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The air itself is a Fuzzy system and its components change every moment and follow mass law of mixing in table – 4 Table – 4 Vi
Xi 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
N2 O2 Ar CO2 Ne He CH4 Kr H2 N2 O Xe Rn
78.09 20.95 0.93 0.03 1.8x10-3 5.2x10-4 2x10-4 10 -4 5x10-5 5x10-5 9x10-6 6x10-18
A ( ) 1.00 0.886 0.662 0.512 0.442 0.331 0.226 0.116 0.106 0.120 0.216 0.117
R
Sec
T
0 0.121036 0.412 0.669 0.816 1.1056 1.487 2.154 2.244 2.12 1.532 2.145
0 0.1072 0.273 0.342 0.3608 0.366 0.336 0.2498 0.237 0.254 0.331 0.251
-1 0.892 0.727 0.657 0.369 0.634 0.663 0.75 0.762 0.745 0.668 0.748
8.262 2.427 1.49 1.225 0.904 0.672 0.464 0.445 0.4716 0.652 0.466
The composition of the air (dry) is remarkably constant all over the globe and throughout entire troposphere. The proportions by volume of the various components are brought to this study using Fuzzy logic set theory and the results are under the assumed new Fuzzy model. 7. EFFECTS OF AIR POLLUTION SOCIETY Effects of air pollution on human being are difficult using physical methods. This depends of psychological and philosophical effects which are quite different. The Fuzzy logic may predict the physical effects and chemical effects using mass law of mixing of human body parameters. The electrical parameters change rapidly with a senility. The biotic systems have constant parameters and Fuzzy logic is effective to predict such adverse effect of air pollution. 8. DISCUSSION A new theory of Fuzzy logic is developed to find out Fuzzy cardinality of
A
M
0.78 0.811 0.612 0.483 0.411 0.306 0.196 0.098 0.88 0.091 1.196 0.112
0.662 0.823 0.622 0.433 0.422 0.308 0.184 0.096 0.86 0.092 0.184 0.113
the pollutions in the air. Mass law of mixing which is pending for a long time for the determination of inverse of the matrix is now possible by the New theory of Fuzzy logic. A new method is used to find Fuzzy grades of truth and their failure rate. All systems in the environment fail due to failure rates, FOR, NOR and POR. Several effects of air pollution are observed on green leaves in the open environment using new theory of Fuzzy logic transformation. 9. REFERENCES 1. L.A. Zadeh “Fuzzy Sets” Control 8, (1965). 2. Bart Kosko “Neural Networks and Fuzzy systems” Prentice Hall of India, New Delhi (Book), (2000). 3. Riza, C. Berkan and S.L. Trubatch “Fuzzy system design Principles” IEEE., press 1997, (Book) (2001). 4. B. Yenarayan “Artificial Neural Networks” Prentice Hall of India (2001). 5. G.P. Chhalotra, Neena Chhalotra, Rishi Chhalotra and Manisha Chhalotra, “A study of Fuzzy logic systems in the well
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known spaces and its reliability attributes Paper No. BSR/T-178/CP/NRNE-33, 58, IE(I) Calcutta (2003). 6. G.P. Chhalotra, Ravindra Sharma, R.S. Parihar and Neena Chhalotra “Study of fault and transmissing line surges using Fuzzy logic transformation of of electrical power system”, Paper No. 02369 (2D) AMSE France June (2003).
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7. G.P. Chhalora “Reliability Engineering and its applications” Security and Adequacy (Book), Khanna Publishers Delhi, PP 102 – 108 (1986). 8. G.P. Chhalotra, et al., “A study of dielectric Air pollution using Fuzzy logic” University of Rorkee, International Conf. Proceedings Sep. 4-6, (1998).
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