$($lecture $8)$ Please consider section $8\mathrm{.}2$ Project Selection of lecture note.

$($a$)$ If one of projects $1$ or $5$ must be done at most, what is the new constraint?

$($b$)$ If project $3$ is chosen only if project $2$ is chosen, what is the new constraint?

$($c$)$ If at least two of projects $1,2$ or $4$ must be done, what is the new constraint?

$($d$)$ If projects $1$ and $4$ are mutually exclusive ${?}^1,$ what is the new constraint ${?}^2?$

$($e$)$ If projects $2,3,$ and $5$ are mutually exclusive, what is the new constraint?

$($lecture $8)$ Please consider section $8\mathrm{.}3$ Selection of Sites of lecture note. If the response time is within $8$ minutes, what are the new constraints for each tract?

$($lecture $8)($problem $12\mathrm{.}5-2$ for the ${11}^{th}$ edition, problem $12\mathrm{.}5-3$ for the ${10}^{th}$ edition$)$ Maximize $Z=200x_1+80x_2$

$($c$)$ Please solve the problem by brute force, that is$,$ compute the objective function value for each feasible solution and compare with the result in problem $($b$).$

$($lecture $9)$ Please consider the example in Section $9\mathrm{.}1$ Branch$-$and$-$Bound Method of lecture note.

$($a$)$ After the first iteration, why do we consider the branch with $x_1=1?$

$($b$)$ For the third iteration, why do we dismiss the branch with $x_3=1?$

$($c$)$ For the forth iteration, path of the problem $($b$),$ which constraints are violated with $x_4=1$ for the original problem?

Please help me with my operations research homework please make sure calculations are clear less of explanations if not needed and if it involves drawing show the drawing too