1. Integer Programming
Set up the mathematical programming problem to choose the set of capital investments to make that will maximize the Net Present Value within the annual yearly expenditure of $150 million required. All amounts are in millions. Project NPV Expenditure Soln: Max Z = 25x1 + 40x2 + 70x3 + 10x4 + 20x5 s.t. 30x1 + 40x2 + 50x3 + 60x4 + 80x5< 150 xi=1if invest in project i, 0 else

2. Integer Programming
Consider the problem of assigning 4 workers to 4 jobs so as to minimize the total time required to complete all jobs. The time (in days) for worker i to do job j is given below. Job Worker Formulate an IP model for this assignment problem. Soln: Min Z = 1x11 + 1x12 + 2x13 + 2x14 + 2x21 + 2x22 + 4x23 + 2x24 + 1x31 + 5x32 + 1x33 + 1x34 + 4x41 + 3x42 + 2x43 + 3x44 s.t. x11 + x12 + x13 + x14 = 1 x21 + x22 + x23 + x24 = 1 x31 + x32 + x33 + x34 = 1 x41 + x42 + x43 + x44 = 1 x11 + x21 + x31 + x41 = 1 x12 + x22 + x32 + x42 = 1 x13 + x23 + x33 + x43 = 1 x14 + x24 + x34 + x44 = 1 Note this can be solved using IP branch and bound, LP/Transportation, or Hungarian.

3. Integer Programming
Solve using the branch and bound technique Max Z = 6x1+4x2 s.t. 2x1+ x2 < 10 x1+ x2 < 8 0 < xi < 8 ; xi = integer
Initial Solution feasible and integer valued (2,6) Z=36
36 is initial set.
Try with 0 < x1 < 4 and 5 < x1 < 8

4. Integer Programming
Solve using the branch and bound technique
Initial Solution feasible and integer valued (2,6) Z=36
36 is initial set.
Try with 0 < x1 < 4
We knew this before trying to solve. Why?
Try with 5 < x1 < 8