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

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