Part $1(25$ pts$)$

You and a few friends are backcountry camping in Western NC$.$ You plan to trek $8$ miles and then set up camp. When setting up camp you and your friends will break in to teams to do tasks in unison to set up camp faster. However, some tasks are dependent on others and cannot be completed before the prior tasks have been performed. The itinerary for camping is as follows:

You and your friends hike a total of $8$ miles to the nearest campsite $($Activity A$).$ Upon arriving you break off into teams. One team will set up tents and hammocks for everyone to sleep in $($Activity B$).$ At the same time, another team will get wood for a fire $($Activity C$).$ Once they have the wood, that same team then proceeds to start the fire $($Activity D$).$ The third team will walk to find water $($Activity E$)$ and once they find it they will to collect water and bring it back to camp $($Activity F$).$ Once the fire is ready and the water is brought back to camp the two groups$-$ the one collecting firewood and the one collecting water$-$ will then purify the water $($Activity G$).$ Once the camp is set up and the water is purified everyone will come together and cook dinner around the campfire Activity $H.$ Once dinner is prepared and eaten half of the group stays behind to put out the fire $($Activity I$)$ and half of the group walks away from the campsite to dispose of the food waste $($Activity J$).$ After disposing of the food waste the second half of the group returns to the camp $($Activity K$).$ Once everyone is at the campsite the day of camping ends.

$\backslash$table$[[$Activity$,\backslash$table$[[$Optimistic$],[$Time to$],[$Complete$]],\backslash$table$[[$Most Likely$],[$Time to$],[$Complete$]],\backslash$table$[[$Pessimistic$],[$Time to$],[$Complete$]]],[$A Hike to the campsite,$2,4,7],[$B Set up campsite,$4,6,7],[$C Collect wood for fire,$1,2,4],[$D Start a fire,$0\mathrm{.}5,1,1\mathrm{.}5],[$E Find water,$0\mathrm{.}5,1,3],[$F Collect water and bring it back to camp,$1,1\mathrm{.}5,3\mathrm{.}5],[$G Purify the water,$1,2,4],[$H Cook and eat dinner,$0\mathrm{.}5,2,3],[$I Put out fire,$1,2,4],[$J Dispose of food waste,$0\mathrm{.}5,2,3],[$K Return from food waste disposal,$0\mathrm{.}25,0\mathrm{.}5,1]]$

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ISE$352$

a$.(10$ points$)$ Assuming that the PERT chart is deterministic $($i$.$e$.,$ based on most likely event times$),$ complete a PERT chart model for the camping trip by determining and placing activity $($label with the letter$),$ activity times, the earliest start times, and the latest start times on the model.