1. GOOD MORNING CLASS! In Operation Research Class, WE MEET AGAIN WITH A TOPIC OF :

2. LINEAR PROGRAMMING THE SIMPLEX METHOD :  GAUSSIAN ELIMINATION  SETTING UP THE INITIAL SOLUTION  DEVELOPING THE SECOND SOLUTION  DEVELOPING THE THIRD SOLUTION

3. Gaussian Elimination : IS CHANGING : 54-3 2-43 4-31 X Y Z 230 120 140 =TO 100 010 001 X Y Z 50 40 60 =

4. HOW TO CHANGE ?

5. LP as Three Stage Process : Problem Formulation Problem Solution Solution Interpretation and Implementation Maximize : Profit = 8 T + 6 C Subject to the constraint : 4 T + 2 C = 0 2 T + 4 C = 0

6. Problem Formulation Maximize : Profit = 8 T + 6 C + 0 S1 + 0 S2 Subject to : 4 T + 2 C + 1 S1 + 0 S2 = 60 2 T + 4 C + 0 S1 + 1 S2 = 48 All variables >= 0

7. Parts of the Simplex Tableau Cj column (profits per unit) Product –mix column Constant Column (quantities of product in the mix) Variable columns Real Product Slack Time Cj row Variable row

8. The Initial Simplex Tableau Pivot Point

9. The Replacing Row & Second Simplex

10. Replacing Row of the Third Tableau

11. Summary Of Step In The Simplex Maximization Procedure Set up the inequalities describing the problem constraint Convert the inequalities to equation by adding slack variables Enter the equation in the simplex table Calculate the Zj and Cj – Zj values for this solution Determine the entering variable (optimal column) by choosing the one with highest Cj – Zj value Determine row to be replaced by dividing quantity column values by their corresponding optimal column values and choosing the smallest non negative ratio Compute the values for the replacing rows Compute the values for the remaining rows Calculate Zj and Cj – Zj values for this solution If there is a positive Cj – Zj value return to step 5. If there is no positive Cj – Zj value, the optimal solution has been obtained

12. Exercise for you : The Tekno Fertilizer Company makes two types of fertilizer which are manufactured in two departments. Type A contribution $3 per ton, and type B contributes $4 per ton Department Hours per tonMaximum hours Type AType Bworked per week 12340 23375 Set up a linear programming problem to determine how much of the two fertilizer to make in order to maximize profits. Use simplex algorithm to solve your problem.

13. THANK YOU !!! SEE YOU NEXT WEEK !!

14. SOLUTION Maximize : Profit = $3 A + $4 B Subject to constraint : 2A + 3B <= 40 3A + 3B <=75 A and B >=0