*************** Problem 1 ***************
Maximize
28.00 X1 +11.00 X2
subject to
14.00 X1 + 6.00 X2 <= 25.00
X1 , X2 >= 0
Phase 1: Input dictionary.
X3 = 25.00 -14.00 X1 - 6.00 X2
-------------------------------------------------
z = -0.00 +28.00 X1 +11.00 X2
The final dictionary:
X1 = 1.79 - 0.43 X2 - 0.07 X3
-------------------------------------------------
z = 50.00 - 1.00 X2 - 2.00 X3
The optimal solution: 50.00
X1 = 1.79 X2 = 0.00 X3 = 0.00
Notes 7.1, Page 2
*************** Problem 2 ***************
Maximize
28.00 X1 +11.00 X2
subject to
14.00 X1 + 6.00 X2 <= 25.00
- 1.00 X1 + 0.00 X2 <= -2.00
X1 , X2 >= 0
Phase 1: Input dictionary.
X3 = 25.00 -14.00 X1 - 6.00 X2
X4 = -2.00 + 1.00 X1 + 0.00 X2
-------------------------------------------------
z = -0.00 +28.00 X1 +11.00 X2
Original problem is INFEASIBLE.
Notes 7.1, Page 3
*************** Problem 3 ***************
Maximize
28.00 X1 +11.00 X2
subject to
14.00 X1 + 6.00 X2 <= 25.00
1.00 X1 + 0.00 X2 <= 1.00
X1 , X2 >= 0
Phase 1: Input dictionary.
X3 = 25.00 -14.00 X1 - 6.00 X2
X4 = 1.00 - 1.00 X1 + 0.00 X2
-------------------------------------------------
z = -0.00 +28.00 X1 +11.00 X2
The final dictionary:
X2 = 1.83 - 0.17 X3 + 2.33 X4
X1 = 1.00 + 0.00 X3 - 1.00 X4
-------------------------------------------------
z = 48.17 - 1.83 X3 - 2.33 X4
The optimal solution: 48.17
X1 = 1.00 X2 = 1.83 X3 = 0.00 X4 = 0.00
Notes 7.1, Page 4
*************** Problem 4 ***************
Maximize
28.00 X1 +11.00 X2
subject to
14.00 X1 + 6.00 X2 <= 25.00
1.00 X1 + 0.00 X2 <= 1.00
0.00 X1 - 1.00 X2 <= -2.00
X1 , X2 >= 0
Phase 1: Input dictionary.
X3 = 25.00 -14.00 X1 - 6.00 X2
X4 = 1.00 - 1.00 X1 + 0.00 X2
X5 = -2.00 + 0.00 X1 + 1.00 X2
-------------------------------------------------
z = -0.00 +28.00 X1 +11.00 X2
The final dictionary:
X1 = 0.93 - 0.07 X3 - 0.43 X5
X4 = 0.07 + 0.07 X3 + 0.43 X5
X2 = 2.00 + 0.00 X3 + 1.00 X5
-------------------------------------------------
z = 48.00 - 2.00 X3 - 1.00 X5
The optimal solution: 48.00
X1 = 0.93 X2 = 2.00 X3 = 0.00 X4 = 0.07
X5 = 0.00
Notes 7.1, Page 5
*************** Problem 5 ***************
Maximize
28.00 X1 +11.00 X2
subject to
14.00 X1 + 6.00 X2 <= 25.00
1.00 X1 + 0.00 X2 <= 1.00
0.00 X1 - 1.00 X2 <= -2.00
- 1.00 X1 + 0.00 X2 <= -1.00
X1 , X2 >= 0
Phase 1: Input dictionary.
X3 = 25.00 -14.00 X1 - 6.00 X2
X4 = 1.00 - 1.00 X1 + 0.00 X2
X5 = -2.00 + 0.00 X1 + 1.00 X2
X6 = -1.00 + 1.00 X1 + 0.00 X2
-------------------------------------------------
z = -0.00 +28.00 X1 +11.00 X2
Original problem is INFEASIBLE.
Notes 7.1, Page 6
*************** Problem 6 ***************
Maximize
28.00 X1 +11.00 X2
subject to
14.00 X1 + 6.00 X2 <= 25.00
1.00 X1 + 0.00 X2 <= 1.00
0.00 X1 - 1.00 X2 <= -2.00
1.00 X1 + 0.00 X2 <= 0.00
X1 , X2 >= 0
Phase 1: Input dictionary.
X3 = 25.00 -14.00 X1 - 6.00 X2
X4 = 1.00 - 1.00 X1 + 0.00 X2
X5 = -2.00 + 0.00 X1 + 1.00 X2
X6 = 0.00 - 1.00 X1 + 0.00 X2
-------------------------------------------------
z = -0.00 +28.00 X1 +11.00 X2
The final dictionary:
X5 = 2.17 - 0.17 X3 + 2.33 X6
X4 = 1.00 + 0.00 X3 + 1.00 X6
X2 = 4.17 - 0.17 X3 + 2.33 X6
X1 = 0.00 + 0.00 X3 - 1.00 X6
-------------------------------------------------
z = 45.83 - 1.83 X3 - 2.33 X6
The optimal solution: 45.83
X1 = 0.00 X2 = 4.17 X3 = 0.00 X4 = 1.00
X5 = 2.17 X6 = 0.00
Notes 7.1, Page 7
*************** Problem 7 ***************
Maximize
28.00 X1 +11.00 X2
subject to
14.00 X1 + 6.00 X2 <= 25.00
1.00 X1 + 0.00 X2 <= 1.00
0.00 X1 - 1.00 X2 <= -2.00
1.00 X1 + 0.00 X2 <= 0.00
0.00 X1 - 1.00 X2 <= -5.00
X1 , X2 >= 0
Phase 1: Input dictionary.
X3 = 25.00 -14.00 X1 - 6.00 X2
X4 = 1.00 - 1.00 X1 + 0.00 X2
X5 = -2.00 + 0.00 X1 + 1.00 X2
X6 = 0.00 - 1.00 X1 + 0.00 X2
X7 = -5.00 + 0.00 X1 + 1.00 X2
-------------------------------------------------
z = -0.00 +28.00 X1 +11.00 X2
Original problem is INFEASIBLE.
Notes 7.1, Page 8
*************** Problem 8 ***************
Maximize
28.00 X1 +11.00 X2
subject to
14.00 X1 + 6.00 X2 <= 25.00
1.00 X1 + 0.00 X2 <= 1.00
0.00 X1 - 1.00 X2 <= -2.00
1.00 X1 + 0.00 X2 <= 0.00
0.00 X1 + 1.00 X2 <= 4.00
X1 , X2 >= 0
Phase 1: Input dictionary.
X3 = 25.00 -14.00 X1 - 6.00 X2
X4 = 1.00 - 1.00 X1 + 0.00 X2
X5 = -2.00 + 0.00 X1 + 1.00 X2
X6 = 0.00 - 1.00 X1 + 0.00 X2
X7 = 4.00 + 0.00 X1 - 1.00 X2
-------------------------------------------------
z = -0.00 +28.00 X1 +11.00 X2
The final dictionary:
X3 = 1.00 +14.00 X6 + 6.00 X7
X4 = 1.00 + 1.00 X6 + 0.00 X7
X2 = 4.00 + 0.00 X6 - 1.00 X7
X1 = 0.00 - 1.00 X6 + 0.00 X7
X5 = 2.00 + 0.00 X6 - 1.00 X7
-------------------------------------------------
z = 44.00 -28.00 X6 -11.00 X7
The optimal solution: 44.00
X1 = 0.00 X2 = 4.00 X3 = 1.00 X4 = 1.00
X5 = 2.00 X6 = 0.00 X7 = 0.00
Notes 7.1, Page 9
*************** Problem 9 ***************
Maximize
28.00 X1 +11.00 X2
subject to
14.00 X1 + 6.00 X2 <= 25.00
1.00 X1 + 0.00 X2 <= 1.00
0.00 X1 + 1.00 X2 <= 1.00
X1 , X2 >= 0
Phase 1: Input dictionary.
X3 = 25.00 -14.00 X1 - 6.00 X2
X4 = 1.00 - 1.00 X1 + 0.00 X2
X5 = 1.00 + 0.00 X1 - 1.00 X2
-------------------------------------------------
z = -0.00 +28.00 X1 +11.00 X2
The final dictionary:
X3 = 5.00 +14.00 X4 + 6.00 X5
X1 = 1.00 - 1.00 X4 + 0.00 X5
X2 = 1.00 + 0.00 X4 - 1.00 X5
-------------------------------------------------
z = 39.00 -28.00 X4 -11.00 X5
The optimal solution: 39.00
X1 = 1.00 X2 = 1.00 X3 = 5.00 X4 = 0.00
X5 = 0.00