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$.$ Eirst 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 $(e_{.}$ 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$?$

b$)$ Show a graph and describe what happens to the amount of capacity that we build as the

Overtime Cost goes from $$\frac{2000}{?}$ unit to $$\frac{4000}{?}$ unit? Why does this happen?

c$)$ What happens to the amount of capacity we build if the storage cost is $$0?$ What if its

$$1000$ per unit? What if it's $$2000?$ Why does this happen?