Question $3(20$ points$)$

Consider a mixed integer program with three variables $x_1,x_2,x_3.x_1$ and $x_2$ are required to take

integer values. $x_3$ can take any value. The picture below shows the branch$-$and$-$bound tree in the

middle of solving this problem.

$($a$)(15$ points$)$ For subproblems SP$5,$ SP$6,$ SP$7,$ SP$8$ and SP$9,$ indicate whether you need to

continue branching. If the answer is no$,$ briefly explain why not.

$($b$)(5$ points$)$ According to this branch$-$and$-$bound tree, what is the range $($lower and upper

bounds$)$ in which the optimal objective function value of the mixed integer program lies?

Explain your answer.