Given the Following All-Integer Linear Program:
Max $15 x _ { 1 } + 2 x _ { 2 }$

Given the following all-integer linear program:
Max $15 x _ { 1 } + 2 x _ { 2 }$

s. t. $7 x _ { 1 } + x _ { 2 } \leq 23$
$3 x _ { 1 } - x _ { 2 } \leq 5$
$x _ { 1 } , x _ { 2 } \geq 0$ and integer
a. Solve the problem as an LP, ignoring the integer constraints.
b. What solution is obtained by rounding up fractions greater than or equal to 1/2? Is this the optimal integer solution?
c. What solution is obtained by rounding down all fractions? Is this the optimal integer solution? Explain.
d. Show that the optimal objective function value for the ILP is lower than that for the optimal LP.
e. Why is the optimal objective function value for the ILP problem always less than or equal to the corresponding LP's optimal objective function value? When would they be equal? Comment on the MILP's optimal objective function compared to the corresponding LP & ILP.