Problem 2. You are operating a server farm with many GPU servers dedicated to mining bitcoin. When a fraction x, where x [0, 1], of the servers are in use, they generate a revenue of $1000x dollars per hour. Since the GPUs in these servers consume a lot of electricity, your power company charges you a total cost of

For example, you will be charged 5000(0.2)^2 = $200 if you use 20% of your server during the period of 8:00am to 4:00pm; meanwhile, during that 8 hours of time, you will make 8 1000 0.2 = $1600. Note that the more bitcoin you mine, the more electricity you consume, and the more you pay for electricity. You are asked to decide on a schedule for the server farm operation by deciding on the fractions of the server used during different periods of the day. (a) Formulate an optimization problem to maximize the total net revenue per day. For the formulated problem, identify the decision variable, the objective function, the constraint, and the type of optimization problem.

(b) What is the maximum net revenue you can make? and what is the respective schedule?

Hint: The total net revenue is the amount of money made subtracted by the running cost. You should decide on the fraction of servers in use during the three time periods: 12:00am to 8:00am, 8:00am to 4:00pm, 4:00pm to 12:00am.

f1(x) = 100x2 if you use x [0, 1] fraction of your servers from 12:00am to 8:00am. f2(x) = 5000x2 if you use x [0, 1] fraction of your servers from 8:00am to 4:00pm. $3(x) = 10000x2 if you use x [0, 1] fraction of your servers from 4:00pm to 12:00am. f1(x) = 100x2 if you use x [0, 1] fraction of your servers from 12:00am to 8:00am. f2(x) = 5000x2 if you use x [0, 1] fraction of your servers from 8:00am to 4:00pm. $3(x) = 10000x2 if you use x [0, 1] fraction of your servers from 4:00pm to 12:00am