Question $3$: $(30\backslash \%)$

ABC company is a manufacturing company that has a contract to supply a customer with parts from April through September. However, ABC does not have enough storage space to store the parts during this period, so it needs to lease extra warehouse space during the six$-$month period. Following are $\backslash ($ A B C $\backslash )\text{'}$s space requirements.

The rental agent that $\backslash ($ A B C $\backslash )$ is dealing with has provided it with the following cost schedule for warehouse space. This schedule shows that the longer the space is rented the cheaper it is$.$ For example, if $\backslash ($ A B C $\backslash )$ rents space for all six months, it costs $\backslash (\backslash$$ $1\mathrm{.}00/\backslash$mathrm$\{$ft$\}^\{2\}\backslash )$ per month, whereas if it rents the same space for only month, it costs $\backslash (\backslash$$ $1\mathrm{.}70/\backslash$mathrm$\{$ft$\}^\{2\}\backslash )$ per month.

$\backslash ($ A B C $\backslash )$ can rent amount of warehouse space on a month basis at any time for any number of $($whole$)$ months. $\backslash ($ A B C $\backslash )$ wants to determine the least costly rental agreement that will exactly meet its space needs each month and avoid any unused space.

$($a$)$ Formulate a mathematical model for this problem, the space to rent is exactly the same as required.

$($b$)$ Solve the model using a computer.

$($c$)$ Suppose that $\backslash ($ A B C $\backslash )$ were to relax its restriction that it rent exactly the space it needs every month such that it would rent excess space if it were cheaper. How would this affect the optimal solution?