$\backslash$begin$\{$tabular$\}\{|$c$|$c$|$c$|$c$|$c$|$c$|$c$|$c$|$c$|\}$

$\backslash$hline & A & B & C & D & E & F & G & H $\backslash \backslash$

$\backslash$hline $1$ & & & & $\backslash$multicolumn$\{3\}\{|$l$|\}\{$Capital Requirements $(\backslash$$$)\}$ & & $\backslash \backslash$

$\backslash$hline $2$ & Alternative Number & Alternative & Net Present Value $(\backslash$$$)$ & Year $1$ & Year $2$ & Year $3$ & & $\backslash \backslash$

$\backslash$hline $3$ & $1$ & Limited warehouse expansion & $3,500$ & $1,000$ & $1,000$ & $4,000$ & & $\backslash \backslash$

$\backslash$hline $4$ & $2$ & Extensive warehouse expansion & $5,500$ & $2,500$ & $3,500$ & $3,500$ & & $\backslash \backslash$

$\backslash$hline $5$ & $3$ & Test market new product & $10,000$ & $6,000$ & $4,000$ & $5,000$ & & $\backslash \backslash$

$\backslash$hline $6$ & $4$ & Advertising campaign & $3,500$ & $2,000$ & $1,500$ & $1,800$ & & $\backslash \backslash$

$\backslash$hline $7$ & $5$ & Basic Research & $8,500$ & $5,000$ & $1,000$ & $4,000$ & & $\backslash \backslash$

$\backslash$hline $8$ & $6$ & Purchase new equipment & $2,500$ & $1,000$ & $500$ & $900$ & & $\backslash \backslash$

$\backslash$hline $9$ & & Capital funds available & & $9,000$ & $11,000$ & $12,000$ & & $\backslash \backslash$

$\backslash$hline $10$ & & & & & & & & $\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$ DATAfile: one of the warehouse expansion alternatives can be chosen.

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

Max

s$.$t$.$

Year $1$

Year $2$

Year $3$

warehouse

expansions

$\backslash [$

x$\_\{$i$\}=0,1\backslash$text $\{$ for $\}$ 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$$ $\backslash ))?$

$($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?

According to this model, what is the maximum net present value $($in $\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$---\backslash (\backslash$quad $\backslash$checkmark $\backslash )$ with the additional constraint.