Duff Avenue Storage Management Company $($DASMC$)$ has two items that are subject to a

possible space constraint with the following data:

$D_1=80$ units per year for item $1$;

$D_2=100$ units per year for item $2.$

$A_1=$$$50$ for each setup of item $1$ ;

$A_2=$$$40$ for each setup of item $2.$

$f_1=45$ cubic feet per unit for item $1$ ;

$f_2=75$ cubic feet per unit for item $2.$

$c_1=$$$60$ per unit for item $1$

$c_2=$$$100$ per unit for item $2.$

$i=10\%$ per year

$F=3000$ cubic feet

$($a$)$ What are the optimal order quantity for Item $1$ and the optimal order quantity for Item $2?$

$($b$)$ What is the corresponding optimal total cost per year?

$($c$)$ Suppose a sales person offers to rent out $1$ cubic foot of space to DASMC in return of $$0\mathrm{.}05$

per year. Should DASMC accept the offer? Explain the reason for your answer.

$($d$)$ Suppose now, the upper management decides that DASMC will add Item $3$ for its business.

The relevant data on Item $3$ are as follows: $D_3=40$ units per year for item $3$;$A_3=$$$90$ for each

setup of item $3$;$f_3=90$ cubic feet per unit for item $3$;$c_3=$$$120$ per unit for item $3$ ;

What are the corresponding $Q_1^{*},Q_2^{*}{Q}_3^{*},$ and ${}^{*}?$