$($a$)$ Develop an integer programming model for maximizing the net present value $($in $\backslash (\backslash$$ $\backslash )).($Let $\backslash ($ x$\_\{$l$\}=1\backslash )$ if investment alternative $\backslash ($ i $\backslash )$ is selected, $0$ otherwise, for $\backslash ($ i$=1,2,3,4,5,6\backslash ).)$

According to this model, what is the maximum net present value $($in $\backslash$$$)?$

$\backslash (\backslash$$ $\backslash )$

$($b$)$ Suppose that if the Advertising campaign is chosen, then the Test market new product alternative must be chosen. Modify your model from part $($a$)$ to take this into account.

In addition to the constraints from part $($a$),$ what additional constraint should be added to the integer programming model?

According to this model, what is the maximum net present value $($in $\backslash (\backslash$$ $\backslash ))?$

$\backslash (\backslash$$ $\backslash )$

$($c$)$ Suppose the Extensive warehouse expansion must be chosen. Modify your formulation from part $($b$)$ to reflect this new situation.

In addition to the constraints from part $($a$)$ and part $($b$),$ what additional constraint should be added to the integer programming model?

$\backslash (\backslash$xrightarrow$\{\backslash$text $\{$ In adaition to the $\}\}\backslash )$

According to this model, what is the maximum net present value $($in $\backslash (\backslash$$ $\backslash ))?$

$\backslash (\backslash$$ $\backslash )$

What is the impact of this constraint on the optimal NPV in comparison to the optimal NPV from part $($b$)?$

Compared to the optimal NPV from part $($b$),$ the optimal NPV $-$ Select$--\backslash ($ V $\backslash )$ with the additional constraint.