1. Linear Programming
Application Using Matrices

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

6. Aim
Distribute land to Maximize profit.

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

10. Algorithm
1) In constraints, select the column with min.
negative value at bottom
-6
<
-3
Constraints

11. Algorithm
2) Pivot element in the selected row is min
(value/respected value)
=7

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

13. Algorithm
4) Repeat until all elements in the last row of
constraints become >=0

14. Solution
The last element of last row is the optimal solution.

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))

16. Graphical Interpretation
http://fooplot.com/plot/ipyhavtwvc

17. Simplex method
mechanically
traverses every
corner point
starting with (0,0)

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

21. Thank You