Integer linear programming - auxillary binary range constraint variable

[Akcope]

My question as an image with TeX for better readability:

< link to objectionable page removed >

 

[stapel]

The text in the graphic is too small for me to read. Sorry.

 

[Akcope]

The text in the graphic is too small for me to read. Sorry.

Looks like the forum downsized it. Here is a direct link: < link to objectionable page removed>

 

[stapel]

I'd put the image here, but it's way too wide. Whoever typed it up didn't wrap the text in a helpful manner. So, for other viewers, here's the text at the link:

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I have two matrices of 1*n of variables:

. . . . .$\displaystyle s_i,\, x_i$

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:

. . . . .\(\displaystyle z_{ij}\, =\, \begin{cases}1&\mbox{ if }\, j\, \geq\, s_i\, \mbox{ and }\, j\, \leq\, x_i \\ 0&\mbox{ otherwise} \end{cases}\)

For example:

. . . . .$\displaystyle s_i\, =\, 4$

. . . . .$\displaystyle x_i\, =\, 8$

. . . . .$\displaystyle z_{i5}\, =\, 1$

My question is, how would I formulate the constraints of

. . . . .$\displaystyle z_{ij}$

?

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If I have typoed something in the above, please reply with corrections. Thank you!