1. The Simplex Method Standard Form

2. Furniture Manufacturing Problem 63100096 11010018 26001072 -80-7000010 x y u v w M Carp. Finish. Uph.

3. Selecting the Pivot Column: 63100096 11010018 26001072 -80-7000010 x y u v w M Carp. Finish. Uph. Objective Function: 80x + 70y = M Which would we want to make first chairs or sofas?

4. Selecting the Pivot Column: 63100096 11010018 26001072 -80-7000010 x y u v w M Carp. Finish. Uph. PIVOT COLUMN Largest Negative in the bottom row

5. Selecting the Pivot Row: 63100096 11010018 26001072 -80-7000010 x y u v w M Carp. Finish. Uph. First a bit of Trial and Error…

6. 6 3100096 11010018 26001072 -80-7000010 x y u v w M CFUCFU CFUCFU 63100096 1 1010018 26001072 -80-7000010 CFUCFU 63100096 11010018 2 6001072 -80-7000010 1½ 1/6 00016 0½ -1/6 1002 05 -1/3 01040 0-30 40/3 001 1280 CFUCFU CFUCFU 0-31-600-12 11010018 040-21036 01008001 1440 CFUCFU 0-1510-30-120 0-201-½-½0-18 1300½036 0 170 00401 2880 x y u v w M

7. 1½ 1/6 00016 0½ -1/6 1002 05 -1/3 01040 0-30 40/3 001 1280 CFUCFU CFUCFU 0-31-600-12 11010018 040-21036 01008001 1440 CFUCFU 0-1510-30-120 0-201-½-½0-18 1300½036 0 170 00401 2880 x y u v w M x = 16 y = 0 u = 0 v = 2 w = 40 M = 1280 x = 18 y = 0 u = -12 v = 0 w = 36 M = 1440 x = 36 y = 0 u = -120 v = -18 w = 0 M = 2880

8. Selecting the Pivot Row: 6310009696/6 = 16 1101001818/1 = 18 2600107272/2 = 36 -80-7000010 x y u v w M RATIOS Carp. Finish. Uph.

9. Selecting the Pivot Row: 6310009696/6 = 16 1101001818/1 = 18 2600107272/2 = 36 -80-7000010 x y u v w M RATIOS Carp. Finish. Uph. PIVOT ROW Smallest Non-Negative Ratio

10. Putting it all together: 6 3100096 11010018 26001072 -80-7000010 x y u v w M Carp. Finish. Uph.

11. After Completing the Pivot: Are We Done Yet??? 1½1/600016 0½-1/61002 05-1/301040 0-3040/30011280 x y u v w M Carp. Finish. Uph.

12. Pivot Column: 1½1/600016 0½-1/61002 05-1/301040 0-3040/30011280 x y u v w M Carp. Finish. Uph. PIVOT COLUMN Largest Negative in the bottom row

13. Pivot Row: 1½1/60001616/(1/2) = 32 0½-1/610022/(1/2) = 4 05-1/30104040/5 = 8 0-3040/30011280 x y u v w M Carp. Finish. Uph. PIVOT ROW Smallest Non-Negative Ratio

14. After Completing the Pivot: Are We Done Yet??? 101/30014 01-1/32004 004/3-101020 0010/360011400 x y u v w M Carp. Finish. Uph.

15. FINAL ANSWER: 101/30014 01-1/32004 004/3-101020 0010/360011400 x y u v w M Carp. Finish. Uph. 14 chairs, 4 sofas Profit of $1400

16. Maximize 10x + 12y + 10z subject to:

17. x - 2y + u = 6 3x + z +v = 9 y + 3z +w = 12 -10x - 12y - 10z +M = 0 System of Equations:

18. Simplex Tableau: 1-2010006 30101009 013001012 -10-12-1000010 x y z u v w M

19. Simplex Tableau: 1-20100066/(-2) = -3 301010099/(0) undefined 01300101212/(1) = 12 -10-12-1000010 x y z u v w M Ratios PIVOT COLUMN PIVOT ROW

20. New Simplex Tableau: 106102030 30101009 013001012 -1002600121144 x y z u v w M x = 0, y = 12, z = 0 M = 144

21. New Simplex Tableau: 106102030 30/1 = 30 301010099/3 = 3 01300101212/0 -1002600121144 x y z u v w M PIVOT COLUMN PIVOT ROW

22. New Simplex Tableau: 0017/31- 1/32027 101/30 003 013001012 0088/3010/3121174 x y z u v w M x = 3, y = 12, z = 0 M = 174