Linear ProgrammingApplication Using Matrices

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.

To maximize:

f = c1x+c2y+c3z ...

Subjected to constraints :0<= ax + by + cz + ... <= P10<= dx + ey + fz + ... <= P2...

Application

STANDARD FORM(x >= 0 y >= 0 ...)

To minimize:

f = c1x+c2y+c3z ...

We maximize:

g = -f = -(c1x+c2y+c3z ...)

1. L acres of land2. Two crops to be planted : potato and ladyfinger3. Budget :

a. F for fertilisersb. P for pesticides

4. Crops has the following requirements/ returns per acre per season:

Crop Plantation Problem

Crop Water Manure Pesticide Profit

Potato W1 M1 P1 R1

Ladyfinger W2 M2 P2 R2

AimDistribute land to Maximize profit.

Simplex Algorithmx = Potato area y = Ladyfinger area

Constraints :1. x , y >= 0 (non negative)2. x + y <= L (land)3. 0<= xP1 + yP2 <= P (Pesticide)4. 0<= xM1 + yM2 <= M (Manure)

Aim : To Maximize Profit (f)f = xR1 + yR2

Simplex MethodIntroduce slack variables & remove inequalities

Constraints

1. x + y <= L x + y + u = L2. xP1 + yP2 <= P xP1 + yP2 + v = P3. xM1 + yM2 <= M xM1 + yM2 + w = M

-xR1 - yR2 + f = 0

For solution purpose, let :P1 = 10, P2 = 12, P = 18 | L = 6M1 = 5, M2 = 7, M = 10 | R1 = 3 ; R2 = 6

Constraints Slacks Values

Algorithm1) In constraints, select the column with min. negative value at bottom

Constraints

-6 < -

3

Algorithm2) Pivot element in the selected row is min(value/respected value)

= 7

Algorithm3) Apply row operations to make pivot element = 1 and all other elements in that column = 0

1. R3 = R3 + R42. R1 = R1 - R33. R2 = R2 - 2R4

Algorithm4) Repeat until all elements in the last row of constraints become >=0

SolutionThe last element of last row is the optimal solution.

Determining x,yFrom final matrix we get the following equations :

1. 0.28x + 1u -0.14w = 4.572. 10x + 1v = 183. 0.7x + 1y + 0.14w = 1.424. 1.28x + 0.85w + 1f = 8.57

Therefore f is 8.57 (max) when x = 0, w = 0y = 1.42 (using x,w,(3))

Graphical Interpretationhttp://fooplot.com/plot/ipyhavtwvc

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

Reference1. Wikipedia

2. Logic of how simplex method works by Mathnik http://explain-that.blogspot.in/2011/06/logic-of-how-simplex-method-works.html

3. Youtube : http://www.youtube.com/watch?v=qxls3cYg8to

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

Credits

Thank You