Given the Following All-Integer Linear Program

question 16

Essay

Given the following all-integer linear program:
MAX 3x₁ + 2x₂
s.t.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 of 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.

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