Q$6.$ Schedule tasks

Your friend Ananth has a set of tasks to do$,$ labeled set $,$ that he would like to accomplish

as fast as possible. Each task $$ in $$ will take $$ amount of time to complete, and he can

work on more than one task at a time. There is a catch, however: certain tasks need to

completed before other ones can begin. You are given a second set $$

that indicates

precedence. For each task $$ in $,$ the tasks in set $$ need to be completed before $$ can be

started. Note that $,$ in $.$

a$)$ Formulate this as an LP$.$ formulation should include definitions of sets, parameters,

decision variables, constraints, and the objective. Provide the formulation in your

homework write up$.$

Hints: Let $$ be a decision variable that represents the start time of task $$ in $.$ For the

objective function, it may be helpful to define a new dummy task $+1$ whose precedence

is all the other tasks $(+1=)$ and has no duration $+1=0$