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March 19th, 2016, 10:58 AM   #1
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Integer linear programming - auxillary binary range constraint variable

Hello, I'm currently facing this problem in ILP:

I have two matrices of dimension 1*n of variables : $$s_i,x_i$$

I would like to create a matrix of new variables (dimension n*k, where k is an arbitrary natural number). Using constraints, I would like the variables to have the following function:

$$z_{ij} =
\begin{cases}
1 & \text{if } j \geq s_i \text{ and } j \leq x_i\\
0 & \text{otherwise} \\
\end{cases} $$

For example:

$$

s_i = 4 \\
x_i = 8 \\

z_{i5} = 1
$$

My question is, how would I formulate the constraints of $$z_{ij}$$?
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