1. What do you understand by integer programming problem?

2. Discuss the need of integer programming.

**1. Maximize z = 7x _{1} + 9x_{2}**

subject to

-x_{1} + 3x_{2} ≤ 6

7x_{1} + x_{2} ≤ 35

where x_{1}, x_{2} are integers ≥
0

**2. Maximize z = x _{1} + 2x_{2}**

subject to

2x_{2} ≤ 7

x_{1} + x_{2} ≤ 7

2x_{1} ≤ 11

where x_{1}, x_{2} are integers ≥
0

**3. Maximize z = 2x _{1} + 6x_{2}**

subject to

3x_{1} + x_{2} ≤ 5

4x_{1} + 4x_{2} ≤ 9

where x_{1}, x_{2} are integers ≥
0

4. Solve the following by Branch & Bound method

** Maximize z = 5x _{1} + 9x_{2}**

subject to

-x_{1} + 5x_{2} ≤ 3

5x_{1} + 3x_{2} ≤ 27

where x_{1}, x_{2} are non negative integers