1. Linear Programming The Table Method

2. Objectives and goals Solve linear programming problems using the Table Method

3. Example Solve the linear programming problem. Maximize subject to

4. Step 1 Change the inequalities to equations by adding slack variables, one for each equation. We will call the x and y in the initial inequalities, the initial variables. s t

5. Step 2 Create a table having the initial variables, slack variables, and the equation for P. xystP = 50x + 80y

6. Step 3 Place zeros strategically in each row. The number of zeros is the same as the number of initial variables. xystP = 50x + 80y 00 00 00 00 00 00

7. Step 4 For each row, substitute in zeros into the equations from step 1 and solve for the other variables. xystP = 50x + 80y 00 00 00 00 00 00

8. Step 4 (Continued) For each row, substitute in zeros into the equations from step 1 and solve for the other variables. xystP = 50x + 80y 00 3284 00 00 00 00 00

9. Step 4 (Continued) For each row, substitute in zeros into the equations from step 1 and solve for the other variables. xystP = 50x + 80y 00 3284 0 16 0 20 00 00 00 00

10. Step 4 (Continued) For each row, substitute in zeros into the equations from step 1 and solve for the other variables. xystP = 50x + 80y 00 3284 0 16 0 20 0 21-10 0 32 00 -12 28 0 4 0 206 00

11. Step 5 If both the initial and slack variables are not negative, then substitute in the values of the initial variables into the profit model P. xystP = 50x + 80y 0032840 016020 021-100 3200-12 28040 20600

12. Step 5 (continued) If both the initial and slack variables are not negative, then substitute in the values of the initial variables into the profit model P. xystP = 50x + 80y 0032840 0160201280 021-100 3200-12 28040 20600

13. Step 5 (continued) If both the initial and slack variables are not negative, then substitute in the values of the initial variables into the profit model P. xystP = 50x + 80y 0032840 0160201280 021-100--- 3200-12--- 28040 20600

14. Step 5 (continued) If both the initial and slack variables are not negative, then substitute in the values of the initial variables into the profit model P. xystP = 50x + 80y 0032840 0160201280 021-100--- 3200-12--- 280401400 206001480

15. Step 6 The highest value of P is the maximum and the values of the initial variables is when it happens xystP = 50x + 80y 0032840 0160201280 021-100- 3200-12- 280401400 206001480

16. The Idea Behind the Table Method Graph created using Desmos.com