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1. Consider the following linear programming problem.

Maximize ?
EMGT 6912: Homework #1

 

1. Consider the following linear programming problem.

 

Maximize

 


 

?1 + 32

 


 

subject to

 


 

21 + 32 ? 6

 

1 ? 32 ? ?3

 

1 , 2 ? 0

 


 

a. Solve the problem graphically.

 

b. Formulate the dual problem.

 

c. Solve the dual problem graphically.

 


 

2. Consider the following linear programming problem.

 

Maximize

 


 

?1 + 32 + 21 + 2

 


 

subject to

 


 

21 + 32 + 1 + 22 = 10

 

(1 , 2 ) ? = {(1 , 2 ) ? 2 : 1 ? 1 ? 2, 1 ? 2 ? 3}

 

1 , 2 ? 0

 


 

a. Enumerate all extreme points in .

 

b. Formulate the Dantzig-Wolfe Master Problem using all extreme points you enumerated in

 

(a).

 

c. Formulate a Restricted Dantzig-Wolfe Master Problem (RMP) using the first extreme point

 

you enumerated in (a).

 

d. Formulate the dual problem of your RMP in (c).

 

e. Solve your dual problem in (d) graphically.

 

f.

 


 

Write the subproblem using your answer in (e) to compute ? .

 


 

g. Is your solution to RMP optimal? Explain based on your answer of (f).

 


 

3. Solve the following binary integer knapsack problem by the LP-based Branch-and-Bound

 

method. Provide the tree, where each node displays necessary information (as in the lecture

 

video). Use the Best-first rule for the node selection strategy.

 


 

Maximize

 


 

161 + 192 + 233 + 284

 


 

subject to

 


 

21 + 32 + 43 + 54 ? 7

 

1 , 2 , 3 , 4 ? 0 and binary

 


 

4. Consider the following mixed-integer program (MIP).

 

Maximize

 


 

21 + 32 + 1 + 52

 


 

subject to

 


 

31 + 22 + 1 + 42 ? 5

 

1 + 22 ? 1 + 22 ? 2

 

1 , 2 ? 0 and integer

 

1 , 2 ? 0

 


 

a. Formulate the master program (MP) and the subproblem (SP) for applying Benders?

 

partitioning to this problem.

 

b. Solve the LP relaxation of MIP using Benders? algorithm. Start with (1 , 2 ) = (0,0) to derive

 

the first Benders? cut.

 

c. Using the starting set of cuts from (b), solve MIP by Benders? algorithm. Summarize your

 

work of (b) and (c) in a table of the following type.

 

Iteration No.

 


 

MP solution (? , ?)

 


 

SP solution: (? ), ?, and ? values

 


 


 


 

Note: You may solve MPs and SPs in any way you like (graphically, logically, or by computer).

 


 

 


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