Detailed Explanation to Integer Programming

1. Solve the following problem:

Max X + 2Y
s.t. 6X + 8Y < 48
7X + 5Y > 35
X, Y > 0
Y integer

a. Find the optimal solution to the LP Relaxation. Round down to find a feasible integer solution. Is this solution optimal?
b. Find the optimal solution.

2. The Westfall Company has a contract to produce 10,000 garden hoses for a large discount chain. Westfall has four different machines that can produce this kind of hose. Because these machines are from different manufacturers and use differing technologies, their specifications are not the same.

Machine.....Fixed Cost to Set Up Production Run....Variable Cost Per Hose ...Capacity
1 .............................750 .......................................1.25 ............................6000
2 ...............................500 .......................................1.50 ...........................7500
3 ............................1000 .......................................1.00 ............................4000
4 ...............................300 .......................................2.00 ............................5000

a. The company wants to minimize total cost. Give the objective function.
b. Give the constraints for the problem.
c. Write a constraint to ensure that if machine 4 is used, machine 1 cannot be