Skip to main content

GENETIC WAY TO SOLVE A PROBLEM

 
 We grow as an adult; complete our studies; find a good carrier; then select a suitable spouse; get married.  Offsprings get portions of our genes.  They grow as adults….
The story continues generation after generation.  The point here is; the fittest get married and produce off-springs.
This principle can be used to solve a maths problem.
 
f(x) = 3.x-x^2/6.62
The above function yields different results for different values of X.  I want to find a particular value of X for which the function yields maximum.  That is, I want to maximize the function.  Let us assume that X can only take values from 0 to 31.  In a binary system, the range is 00000 to 11111(for the explanation of binary number see note 1)
Let us randomly select six x values in the range 0 to 31 and evaluate the function.


X- decimal
X- binary
f(x) = 3.x-x^2/6.62
selected?
2
00010
5.395
yes
8
01000
14.33
yes
12
01100
14.24
yes
18
10010
5.057
no
22
10110
-7.111
no
28
11100
-34.42
no
Here x values 2, 8, and 12 which yields maximum results are selected.  The corresponding 3 binary values are taken as genes of X.  We are going to mate them and produce six off-springs.  The function will again be evaluated with the off-springs for maximum yield.  
Here mating means, portions of the binary digits are swapped to produce next-generation X as given below.    
I  generation x      II generation x
8    01000             01010   10         
2    00010             00000   0
       Binary or genes, here, colored portions are swapped
I gen. x
Binary or genes
offsprings
II gen. x
f(x)
8
01000
01100
12
14.24
12
01100
01000
8
14.33
8
01000
01010
10
14.82, best
2
00010
00000
0
0
12
01100
01010
10
14.82, best
2
00010
00100
4
9.583
 From the table, we understand X = 10 gives the maximum value of the function 14.82.  Few more runs of this procedure will confirm the result X = 10.  
    
    This method is known as a genetic algorithm in computer parlance.  For complicated problems, the computer will run the algorithm for hundreds of generations and the best result will be taken.  
Note 1: In a binary system, all numbers are represented using only 0 s and 1 s as follows.  (If required, leading zeros can always be added).


2. The following graph of the function proves the result.

Comments

Post a Comment

Popular posts from this blog

LISSAJOUS FIGURES

  Definition:  "When a particle is subjected to two sine wave motion or two oscillatory motion at right angles, the particle describes lissajous figures".      We know sine wave motion and circular motion is basically same.  Hence we draw two circles A and B perpendicular to each other.  The circle B rotates twice faster than circle A.  That is, frequency of circle B is two times than that of A.        A particle at the intersection of two circles is subjected to two sine wave motion   A and B at 90 degree simultaneously.  The particle will describe figures depending on the frequency and phase of A and B .  In our case, the ratio of frequency is  1:2 and the two waves are in phase.        To draw lissajous figures :  A moving point in both the circles are chosen.   Here we should remember; during the time taken by the circle A to complete one rotation, circle B completes two.  Hence the points are marked on the circles according to their speed.  Then straight lines
Post a Comment
Read more

THE PARABOLA

          A jet of water shooting from a hose pipe will follow a parabolic path.  What is the so special about parabola.    Y= x^2 Draw a graph for the above equation.  It will result in a parabola.  This parabola is also called unit parabola.  Any equation involving square will yield a parabola. Example:  Y = 2x^2 +3x+3 (also called quadratic equation)    X= 2 and -2, both  satisfies the equation 4 = X^2.  Parabolic equations always have two solutions.     Any motion taking place freely under gravity follows parabolic path. Examples:   An object dropped from a moving train,   A bomb dropped from flying plane,  A ball kicked upwards.      If a beam of light rays fall on the parabolic shaped mirror, they will be reflected and brought to focus on a point.  This fact is made use of in Dish Antenna, Telescope mirrors, etc.      Inverted parabola shape is used in the construction of buildings and bridges.  Because the shape is able to bear more weight.      A plane
Post a Comment
Read more

CASINO'S GAME

           Let us find out how the casino survives with mathematics.      Say, your friend invite you for a game of dice.  You must bet (wager) 2 dollars.  If you roll 'six' you will get back 8 dollars.  The game will go on for 30 rounds.  All sounds good.      The probability of rolling 'six' is 1/6.  Since the game will be played for 30 times, the 'expected win' is 30*1/6 = 5.  That is, you are expected to win 5 rounds out of 30.  Hence your gain will be 5 * 8 =40 dollars.  ok.  This also implies that you will loose 25 rounds.  Hence your loss will be 25*2 =50 dollars.  Your net gain will be gain-less = 40-50 = -10 dollars. For 30 rounds, the loss is -10 dollars, Hence, for one round =-10/30 = -1/3 dollars.  There will be a loss of -1/3 or 0.33 dollars per round.  It is not a fair game.     Let us make a simple formula to calculate  'Pay out per round\. The probability for a win = p The pay-out in case of win = V No. of rounds = n The expect
4 comments
Read more