Suppose there is a start-up cost of operatingapowerplantfortheday.Ifplant1isoperated,thenthereisafixedstart-upcosto _!.Similarly, for plants 2 and 3 there are fixed start-up costs _",_#, respectively. To model this, weintroducebinaryvariables_!,_",_#thatshouldrepresentwhetherplant1,2,and3respectively,arebeingoperated.

• Writelinear constraintsthatenforcethatifthestart-upcosts arenotpaid, then theplantcannotoperate. Thatis,if_!is0,then_!isalso0.Otherwise,if_!is1,then_!can take any value.
• Writeanewobjectivefunction thatincorporatesthestart-upcosts.
• To simplify the noise constraint, we want to enforce a constraint so that at mostoneofplants1and2canoperate duringtheentire day.Writealinearconstraintto enforce this. Write a linear constraint using only variables _!,_",_#enforcing that plant 1 cannotbetheonlyplantoperated(inotherwords ifplant1isused, thenatleastoneof plants 2 and 3 must also be operated).