Problem $4$: Fundraising for Academic Agricultural Projects $(30$ points total$)$

A gentleman farmer works wisely on academic projects to raise funds for his true passion, farming.

He has identified seven tasks and developed three$-$time estimates $($all in days$)$ of the next big

project on the farm, replacing the fence at the comedy pasture. Tasks, precedence requirements,

and time estimates including optimistic, most likely, and pessimistic completion times in days, are

shown in the table below. For some, the expected duration and variance have already been

computed.

$($a$)(2$ points$)$ What is the value of the missing item labeled $($a$)$ in the table above? $($NOTE:

Show your calculation below$)$

$($b$)(2$ points$)$ What is the value of the missing item labeled $($b$)$ in the table above?

$($c$)(10$ points$)$ Draw the network diagram $($AON$)$ for this project. Clearly label earliest start,

earliest finish, latest start, and latest finish for each activity.

$($d$)(2$ points$)$ Specify the critical path.

$($e$)(2$ points$)$ State the project's expected duration.

$($f$)(3$ points$)$ What is the slack of each path? Show $2$ methods.

$($g$)(3$ points$)$ What is the standard deviation of the critical path? $($NOTE: Round to $2$ decimals$)$

$($h$)(3$ points$)$ What is the probability that they complete this project between $31$ and $35$ days?

$($NOTE: You only need to consider the critical path$)$

$($i$)(3$ points$)$ The project manager wants to know the duration within which he accepts no

more than $1\%$ risk of not finishing the project. $($NOTE: Only consider the critical path$)$