$[20$ points$]$ A youth organization is planning a project to build a jungle gym for kids. The planning team has identified $10$ activities, named A$-$J$,$ and activity times for the project. The tasks, durations $($in days$),$ costs, and predecessor relationships are shown as follows:

$\backslash$table$[[$Activity$,$Immediate Predecessor,Activity Time$],[$A$,$None,$4],[$B$,$None,$3],[$C$,$A$,$ B$,6],[$D$,$A$,7],[$E$,$C$,$ D$,8],[$F$,$D$,7],[$G$,$C$,12],[$H$,$E$,$ F$,9],[$I$,$F$,8],[$J$,$G$,$ I,$2]]$

MGMT $3106$: Management Science and Operations Management

a$.$ Draw the network for this project. If you draw on a paper, scan and insert into the $.$docx or $.$pdf file. If images are uploaded separately, they will not be graded.

b$.$ Create the activity schedule and calculate ES$,$ EF$,$ LS$,$ and LF times. Activity schedule is a table with ES$,$ EF$,$ LS$,$ LF times for each activity.

c$.$ Calculate the slack for each activity. What do they mean?

d$.$ What are the critical path activities? Why are they important?

e$.$ What is the project completion time? Under what conditions is this project completion time valid?