$[25$ points$]$ Project Management. Completing a project $x$ requires tasks $A,B,C,D,E$ to be completed, and

some of these tasks have predecessors that must be completed before the task can be started. For these

tasks, the normal durations are given, and in some cases cost per week for the resources required to complete

a task earlier than its regular duration is also provided. Refer to Table $1$ for the data. $1.$ Formulate an

IP to find out the earliest completion date of project $x$ if all tasks are completed according to their regular

duration. $2.$ Assume the minimum duration of the project from part $1$ is $D^{**}.$ If the project manager receives

$25K$ bonus for every week before $D^{**}$ the project is completed, formulate an IP that maximizes the project

manager's profit.