Q$5\mathrm{.}3**($Graduation Party$)$ Thierry, Ute, and Vishal are getting ready for their last period in the MBA program. Following the final exams, they intend to throw a big party with many of their friends from back home. Presently, they have identified the following set of activities that need to be completed. They decide to not spend any work on preparing the party until all final exams are over. Moreover, they aim to spend a $3-$day beach vacation as early as possible, but not before all party planning activities are completed.

On June $10,$ they will enter the final exam week, which will take $5$ days. They then want to arrange for live music $($which will take $5$ days$),$ evaluate a number of potential party sites $(6$ days$),$ and prepare a guest list, which includes inviting their friends and receiving the RSVPs $(7$ days$).$ They want to visit their two most promising party sites, which they expect to take $4$ days. However, this can only be done once they have completed the list of party sites. Once they have finished the guest list and received the RSVPs$,$ they want to book hotel rooms for their friends and create a customized T$-$shirt with their names on it as well as the name of the guest. Hotel room reservation $(3$ days$)$ and T$-$shirt creation $(6$ days$)$ are independent from each other, but both of them require the guest list to be complete. Once they have picked the party site, they want to have a meeting on site with an event planner, which they expect to take $4$ days. And then, once all work is completed, they plan to take off to the beach.

a$.$ Create a dependency matrix for the activities described.

b$.$ Build a project graph, visually depicting the evolution of this project.

c$.$ Find the critical path. What is the earliest time that the three can go to the beach?

d$.$ For each activity, compute the late start, the late completion, and the slack time.