$(a)$ Develop and solve $an$ integer programming model for maximizing the net present value.

Let $x_i=\{\begin{array}{ccc}1& if\text{i}\text{n}\text{v}\text{e}\text{s}\text{t}\text{m}\text{e}\text{n}\text{t}\text{a}\text{l}\text{t}\text{e}\text{r}\text{n}\text{a}\text{t}\text{i}\text{v}\text{e}iis\text{s}\text{e}\text{l}\text{e}\text{c}\text{t}\text{e}\text{d}& \\ 0& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$

Select the correct answers below.

Optimal Solution:

$x_1,-$ Select your answer $-vv$

$x_2,-$ Select your answer $-vv$

$x_3,-$ Select your answer $-vv$

$x_4,-$ Select your answer $-vv$

$x_5,-$ Select your answer $-vv$

$x_6,-$ Select your answer $-vv$

Net Present Value: $$

$(b)$ Assume that only one $of$ the warehouse expansion projects can $be$ implemented. Modify your model $of$ part $(a).$ Which $of$ the following constraint must $be$ added $to$ the model?

$(i)x_1=x_2$

$(ii)x_1+x_{2}1$

$(iii)x_1+x_{2}1$

$(iv)$ None $of$ the above.

$(i)x_3{x}_4$

$(ii)x_4{x}_3$

$(iii)x_3+x_{4}1$

$(iv)x_3+x_{4}1$