A national conference is planned in a college. The activities are listed down along with their predecessors and time taken. Prepare a network diagram and determine the critical activities.

$\backslash$table$[[$Activity$,$Description,$\backslash$table$[[$Immediate$],[$Predecessor$]],$Duration $($days$)],[$A$,$Confirm lead speaker and topic,$-,5],[$B$,$Prepare brochure,$-,1],[$C$,$Send letters to other speakers,B$,2],[$D$,$Get confirmation from speakers,C$,5],[$E$,$Send letters to participants,C$,$D$,2],[$F$,$Obtain travel plans from speakers,D$,2],[$G$,$Arrange for accommodation for speakers,F$,1],[$H$,$Get handouts from speakers,F$,4],[$I$,$Finalize registrations,G$,$H$,10],[$J$,$Arrange hall and AV$,$I,$1],[$K$,$Conduct of programme,J$,1]]$

Ims.kku.edu.sa $*$

For the PERT problem find the critical path and project duration. What is the probability that the project will be completed in $25$ days?

$\backslash$table$[[$Activity$,$Predecessor,Time$],[$Optimistic$,$Most likely,Pessimistic$],[$A$,-,2,5,14],[$B$,-,1,10,12],[$C$,$A$,0,0,6],[$D$,$A$,1,4,7],[$E$,$C$,3,10,15],[$F$,$D$,3,5,7],[$G$,$B$,1,2,3],[$H$,$EF$,5,10,15],[$I$,$G$,3,6,9]]$

The following table lists the jobs of a network along with their estimates.

$\backslash$table$[[$Activity$,$Time $($Weeks$),$Cost $($Rs$)],[$Normal$,$Crash,Normal,Crash$],[1-2,9,4,1300,2400],[1-3,15,13,1000,1380],[2-3,7,4,7000,1540],[2-4,7,3,1200,1920],[2-5,12,6,1700,2240],[3-6,12,11,600,700],[4-5,6,2,1000,1600],[5-6,9,6,900,1200]]$

a$.$ Draw the project network diagram.

b$.$ Calculate the length and variance of the critical path.

c$.$ What is the probability that the jobs on the critical path can be completed in $41$ days?