$(15$ points$).$ MPS for product AA is given below. The manufacturing process consists of first injection molding and then assembly. The cycle time information is given below. Assuming that that there are $2$ lines and $1$ shift, develop a mathematical model to minimize the cost incurred and then solve it considering the possibility of

$\backslash (\backslash$rightarrow $\backslash )$ Building in advance and carrying inventory as needed$(\backslash$$$300/$unit$/$period$)$

$\backslash (\backslash$rightarrow $\backslash )$ Doing weekday overtime at additional cost of $(\backslash$$$150/$unit$)$ as needed needed

$($Each line has monthly capacity of $160$ hours$/$shift$,$ do not consider reliability and efficiency$).$

Note: Weekday overtime is limited to $40$ hours$/$line$/$shift$/$month$.$