2. LP History
LP first developed by Leonid
Kontorovich in 1939 to plan
expenditures and returns
during WW 2.
It was kept secret until 1947. Revealed after
publication of Dantzig's Simplex Algorithm.
3. Application
To maximize:
f = c1x+c2y+c3z ...
Subjected to constraints :
0<= ax + by + cz + ... <= P1
0<= dx + ey + fz + ... <= P2
...
STANDARD FORM
(x >= 0 y >= 0 ...)
4. To minimize:
f = c1x+c2y+c3z ...
We maximize:
g = -f = -(c1x+c2y+c3z ...)
5. Crop Plantation Problem
1. L acres of land
2. Two crops to be planted : potato and ladyfinger
3. Budget :
a. F for fertilisers
b. P for pesticides
4. Crops has the following requirements/ returns
per acre per season:
Crop
Water
Manure
Pesticide
Profit
Potato
W1
M1
P1
R1
Ladyfinger
W2
M2
P2
R2
7. Simplex Algorithm
x = Potato area
y = Ladyfinger area
Constraints :
1.
2.
3.
4.
x , y >= 0
x + y <= L
0<= xP1 + yP2 <= P
0<= xM1 + yM2 <= M
(non negative)
(land)
(Pesticide)
(Manure)
Aim : To Maximize Profit (f)
f = xR1 + yR2
8. Simplex Method
Introduce slack variables & remove inequalities
Constraints
1. x + y <= L
2. xP1 + yP2 <= P
3. xM1 + yM2 <= M
x+y
xP1 + yP2
xM1 + yM2
-xR1 - yR2
+ u
+
+
+
v
w
f
=L
=P
=M
=0
9. For solution purpose, let :
P1 = 10, P2 = 12, P = 18
|L=6
M1 = 5, M2 = 7, M = 10
| R1 = 3 ; R2 = 6
Constraints
Slacks
Values
12. Algorithm
3) Apply row operations to make pivot element = 1
and all other elements in that column = 0
1. R3 = R3 + R4
2. R1 = R1 - R3
3. R2 = R2 - 2R4
15. Determining x,y
From final matrix we get the following equations :
1.
2.
3.
4.
0.28x + 1u -0.14w = 4.57
10x + 1v
= 18
0.7x + 1y + 0.14w = 1.42
1.28x + 0.85w + 1f = 8.57
Therefore f is 8.57 (max) when x = 0, w = 0
y = 1.42 (using x,w,(3))
19. Reference
1. Wikipedia
2. Logic of how simplex method works by Mathnik
http://explain-that.blogspot.in/2011/06/logicof-how-simplex-method-works.html
3. Youtube : http://www.youtube.com/watch?
v=qxls3cYg8to
20. Credits
1. Matrix images : Roger's Online Equation Editor
http://rogercortesi.com/eqn/
2. Title font : Amatic Sc by Vernon Adams https://plus.
google.com/107807505287232434305/posts