$2.$ Our company just won a contract to supply the government with transport vehicles. We have weekly demand to meet over the $3$ months of the contract, given in the table below. We currently do not have the factory or warehouse set up$.$ First we must choose how much capacity to invest in$.$ Factory costs scale linearly, and it costs us $$20,000$ for each unit of weekly capacity we choose to build $($e$.$g$.$ we can pay $$20,000,000$ to be able to make $1,000$ units per week at normal costs, or we can by $$10,000,000$ to have capacity of $500$ units per week$).$ Each unit at normal cost costs $$2000$ to build. We have the capacity to make overtime units at $$3000$ per unit, up to an extra $50\%$ of the capacity $($if our capacity is $1000$ units, we can make up to $500$ OT units each week$).$ We also can store units from month to month. It costs us $$400$ per month to store each unit.

a$)$ How much capacity should we build, and how much should we produce each week in order to minimize our total costs? What will our total cost be$?$