The activities for building an prototype and related data are as follows.

Activity Normal Time Crash Time Normal Cost Crash Cost Immediate Predecessor

$\backslash$table$[[$Activity$,\backslash$table$[[$Normal$],[$Time$]],$Crash Time,$\backslash$table$[[$Normal$],[$Cost$]],\backslash$table$[[$Crash$],[$Cost$]],\backslash$table$[[$Immediate$],[$Predecessor$]]],[$A$,3,2,1000,1500,-],[$B$,4,3,2000,2600,$A$],[$C$,7,3,1200,1800,$A$],[$D$,4,2,900,1200,$B$],[$E$,3,1,3500,4000,$C$],[$F$,6,4,1500,2000,$DEE$]]$

$($a$)$ What is the project completion date?

$($b$)$ If you want to crash this project to $18$ weeks, which activities need to be crashed?

$($c$)$ Use your answer to part $($a$)$ as the expected completion time of this project. Suppose the variance of the critical path is $4.$

$($d$)$ What is the probability that the project will be finished in $18$ weeks or less?

$($e$)$ What is the probability that the project will take longer than $20$ weeks?

$($f$)$ The project manager wishes to set the due date for the completion of the project so that there is a $90\%$ chance of finishing on schedule. What should this due date be$?$