(Linear Optimization)

Youaretheoperator ofthreepowerplants.Youneedtodeterminetheoptimalamount oftimetokeep each power plant running in order to meet the energy demand while minimizing your operational cost. Eachplanthasanhourlycostofoperation_!=6,_"=15,_#=2andeachonecansupply_!=

10,_"=20,_#= 3unitsofenergyeachhour.Thereisatotaldemandof= 400unitsofenergyforthenext day. Todeterminetheoptimaloperationtimeforeachpowerplant, youdecidedtosetupandsolve alinear optimization problem. The decision variables are_!,_",_#. These decision variables describe the amount of time each power plant needs to be operational. Due tolocalregulationsonnoisepollution,thetotalnumberofhourspowerplants1and2canoperate together in total is at most 24 hours.

Duetoefficiencyconcerns, eachpowerplantcanrunatmost20hoursa day.

In this question, you need to formulate thelinearoptimizationproblem.Foreachoftheconstraints,specify ifitshouldbeaninequality (or ) or equality (=).

1. Write theobjectivefunction intermsofthedecision variables.
2. Write the3constraintslimitingthenumberofhourseachpowerplant can be operational.
3. Write alinear(constraint)toensurethatthedemand forenergyisfulfilledby your plant (the demand constraint).
4. Write alinear(constraint)toensurepowerplants1and2mustoperateat most a total of 24 hours (the noise constraint).
5. Doweneednon-negativityconstraints?Ifso,please writetheseaswell.