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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?

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The options that are available can be enumerated as follows: denote $x_{ni}$ as an additional area contract occurred at $n$th next month which is durable for $i$ periods. Then all of the available options are $$\begin{align}&x_{11},x_{12},x_{13},x_{14},x_{15}\\ &x_{21},x_{22},x_{23},x_{24}\\ &x_{31},x_{32},x_{33}\\ &x_{41},x_{42}\\ &x_{51} \end{align}$$ and the following is the equations that represents the extra space needed for the company. $$\begin{align} &x_{11} + x_{12} + x_{13} + x_{14} + x_{15} = 30000\\ &x_{21} + x_{22} + x_{23} + x_{24} + x_{12} + x_{13} + x_{14} + x_{15} = 20000\\ &x_{31} + x_{32} + x_{33} + x_{22} + x_{23} + x_{24}+ x_{13} + x_{14} + x_{15} = 40000\\ &x_{41} + x_{42} + x_{32} + x_{33} + x_{24} + x_{14} + x_{15} = 10000\\ &x_{51} + x_{42} + x_{33} + x_{24} + x_{15} = 50000 \end{align}$$ Therefore, the cost you are minimizing can be written as $$\begin{align} Z &= 65(x_{11} + x_{21} + x_{31} + x_{41} + x_{51}) \\ &+ 100(x_{12} + x_{22} + x_{32} + x_{42})\\ &+ 135(x_{13} + x_{23} + x_{33})\\ &+ 160(x_{14} + x_{24})\\ &+ 190x_{15} \end{align} $$

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