Hint$($s$)$

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three years are summarized as follows:

$\backslash$table$[[,$Capital Requirements $($$$)],[$Alternative$,$Net Present Value $($$$),$Year $1,$Year $2,$Year $3],[$Limited warehouse expansion,$4,000,3,000,1,000,4,000],[$Extensive warehouse expansion,$6,000,2,500,3,500,3,500],[$Test market new product,$10,500,6,000,4,000,5,000],[$Advertising campaign,$4,000,2,000,1,500,1,800],[$Basic research,$8,000,5,000,1,000,4,000],[$Purchase new equipment,$3,000,1,000,500,900],[$Capital funds available,,$10,500,7,000,8,750]]$

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

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

Select the correct answers below.

Optimal Solution:

Net Present Value: $