CSC 445/545 Notes 7.2: Solving Integer Programs by Cutting Plane Techniques

*************** 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 7.2, 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  <=  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 7.2, 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 
     3.00 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 =   4.00 - 3.00 X1 - 1.00 X2 
-------------------------------------------------
z  =  -0.00 +28.00 X1 +11.00 X2 


The final dictionary:
X3 =   1.00 + 4.00 X1 + 6.00 X5 
X4 =   1.00 - 1.00 X1 + 0.00 X5 
X2 =   4.00 - 3.00 X1 - 1.00 X5 
-------------------------------------------------
z  =  44.00 - 5.00 X1 -11.00 X5 

The optimal solution: 44.00
X1  =  0.00  X2  =  4.00  X3  =  1.00  X4  =  1.00  
X5  =  0.00