I have to represent this problem using general linear programming, which is not a problem in itself, but I can't figure out what would the $x_1, x_2,...,x_n$ be. A company is trying to minimize their cost for storing merchandise, and they know exactly how much extra space they'll need over the next 5 months.
$$\begin{array}{|c|c|} \hline \text{Month} & \text{Additional space required (sqr. ft)} \\ \hline \ 1 & \ 30,000 \\ \hline \text{2} & \ 20,000\\ \hline \text{3} & \ 40,000\\ \hline \text{4} & \ 10,000\\ \hline \text{5} & \ 50,000\\ \hline \end{array}$$
The company can either rent space monthly, or for have a special rate for periods of 2+ months. The costs go as follows:
$$\begin{array}{|c|c|} \hline \text{Months} & \text{Cost (\$/sqr. ft)} \\ \hline \ 1 & \ 65 \\ \hline \text{2} & \ 100\\ \hline \text{3} & \ 135\\ \hline \text{4} & \ 160\\ \hline \text{5} & \ 190\\ \hline \end{array}$$
The company has to minimize their cost while ensuring they have the space required.
I originally thought that I would have 5 $x_n$ variables ($x_1$ to $x_5$) and that each $x_n$ would be the amount of sqr. ft. for each rate period and that my mimizing function would look like:
$$ min \ Z = 65x_1 + 100x_2 + 135x_3 + 160x_4 + 190x_5$$
but that wouldn't make sense, because if space is rented using a 2 months rate, that space can't be rented the next month. So I'm lost. What am I missing? What would be my $x_n$ and my constraints?
