Problems:
All-integer linear program
Given the following all-integer linear program:
MAX 3x1 + 2x2
such that
3x1 + x2 <= 9
x1 + 3x2 <= 7
-x1 + x2 <= 1
x1, x2 >= 0 and integer
a. Solve the problem as a linear program ignoring the integer constraints. Show that the optimal solution to the linear program gives fractional values for both x1 and x2.
b. What is the solution obtained by rounding fractions greater than or equal to 1/2 to the next larger number? Show that this solution is not a feasible solution.
c. What is the solution obtained by rounding down all fractions? Is it feasible?
d. Enumerate all points in the linear programming feasible region in which both x1 and x2 are integers, and show that the feasible solution obtained in (c) is not optimal and that in fact the optimal integer is not obtained by any form of rounding.