The Simplex Algorithm An Algorithm for solving Linear

The Simplex Algorithm An Algorithm for solving Linear Programming Problems

We Start with a Linear Programming Problem • • Maximise P = 4 x +5 y +3 z Subject to the constraints 8 x + 5 y +2 z 4 x +2 y +3 z 1

Setting up the Tableau • First rearrange the equation for P so that it is equal to zero : • P - 4 x - 5 y - 3 z = 0

Introduce Slack variables • • • 8 x + 5 y +2 z 4 becomes 8 x + 5 y + 2 z + s = 4 x +2 y +3 z 1 becomes x +2 y +3 z + t = 1 s and t are called the slack variables

This is the Tableau (Fancy word for table)

Finding a pivot • Chose any negative number in the first row • Consider the positive values in the column below it • Divide the value in the last column by the corresponding value in the chosen column and see which gives you the least • That tells you which is the pivot. . . it goes like this:

4/8 =1/2 1/1=1 1/2 is the least so 8 is the pivot

The next step is to reduce the pivot to 1 by dividing equation by 8

We now reduce the other elements in the column of the pivot to zero:

We have now completed the first iteration of the algorithm • This tells us that • P = 2 when y = z = s = 0 and x = 1/2, t= 1/2 • P= 2 is not the optimal solution as we still have negative numbers in the first row.

We now repeat the process: chose a negative number in the first row and find a new pivot: So is the pivot

We now repeat the process: choose a negative number in the first row and find a new pivot:

Reduce the pivot to 1 by dividing equation ‘ by 13/8 to get equation ”

Reduce the other elements in the column of the pivot to zero

We have now completed the second iteration of the algorithm • This tells us that • P = 210/11 when z = s = t=0 and x = 3/11, y= 4/11 • P= 210/11 is the optimal solution as we have no negative numbers in the first row. • P= 210/11 is the maximum value; we have finished!