Linear Programming

Linear Programming

It is a mathematical technique to find the best outcome with limited resources (i.e. to achieve maximum profit with minimum cost.)

This method is possible only with the linear relationships. This technique is also called as linear optimization.

Example of Linear Programming

Consider the function D = 5x + 4y.

Let us find the minimum value of this function in the region defined by the inequalities.

x ≥ 0, y ≥ 0, x + 2y ≤ 4, and x + y ≤ 3.

The feasible region determined by the given constraints is shown.

To find the minimum and maximum value of D, we evaluate the value of D = 5x + 4y at each of the four vertices and we find that the minimum value of the function subject to the constraints is 0.

The process that we adopted in the example above is called Linear Programming.

Solved Example on Linear Programming

Question: 

Find the maximum value of the function C = 6x + y subject to the constrains

x ≥ 0, y ≥ 0 , 5x + 3y ≤ 15.

The Answer: 18

Solution:

Step 1: Objective function is C = 6x + y

Step 2: Constraints are x ≥ 0, y ≥ 0 5x + 3y ≤ 15

Step 3: [Draw the graph.]

The feasible region determined by the given constraints is shown.

Step 4: From the graph, the three vertices are (0, 0), (3, 0), and (0, 5).

Step 5: To evaluate the minimum, maximum values of C, we evaluate C = 2x + y at each of the vertices.

Step 6: [Substitute the values.]

At (0, 0) , C = 6(0) + (0) = 0

Step 7: [Substitute the values.]

At (3, 0) , C = 6(3) + (0) = 18

Step 8: [Substitute the values.]

At (0, 5) , C = 6(0) + (5) = 5

Step 9: 

So, the maximum value of C is 18.



Video of Liner Programming:

Original Resource:

http://www.icoachmath.com/math_dictionary/linear_programming.html