2. The Simplex Algorithm An Algorithm for solving Linear Programming Problems

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

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

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

6. This is the Tableau (Fancy word for table)

7. 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:

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

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

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

11. 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.

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

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

14. Reduce the pivot to 1 by dividing equation ‘ by 1 3/8 to get equation ”

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

16. We have now completed the second iteration of the algorithm This tells us that P = 2 10/11 when z = s = t=0 and x = 3/11, y= 4/11 P= 2 10/11 is the optimal solution as we have no negative numbers in the first row. P= 2 10/11 is the maximum value; we have finished!