I have a binary decision variable \(x_{kn}\) and I would like to model a situation like the following:

If \(x_{kn}=1\), then

  • \(A_n\geqslant b_k\) or
  • \(A_n+C_{n+1}\geqslant b_k\) if \(C_{n+1}>0\) or
  • \(A_n+C_{n+1}+C_{n+2}\geqslant b_k\) if \(C_{n+1}>0\) and \(C_{n+2}>0\) or
  • ... or
  • \(A_n+C_{n+1}+\ldots+C_{n+p}\geqslant b_k\) if \(C_{n+1}>0\) and \(C_{n+2}>0\) and \(\ldots\) \(C_{n+p}>0\)

Where, \(p\geqslant2\) is given, \(C_n\) and \(b_k\) are non-negative integers given as inputs and \(A_n\) is a non-negative integer decision variable.

At least one constraint of the above \(p+1\) constraints has to be true. Can I model this in linear integer programming?

I thought of adding binary variables \(y_i\) such that \(x_{kn}=\sum_{i=1}^{p+1}y_{ikn}\) but I cannot continue (if I am even right).

asked 10 Jan '18, 11:41

zbir's gravatar image

zbir
1714
accept rate: 0%

edited 10 Jan '18, 14:31


I suspect you have some indexing errors in your question. If all \(C_n\) are nonnegative, each constraint implies the next, so the OR operator reduces to the last constraint.

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answered 10 Jan '18, 12:58

Rob%20Pratt's gravatar image

Rob Pratt
1.2k26
accept rate: 28%

So you mean I just have to write the constraint like \(A_n+C_{n+1}+\ldots+C_{n+p}\geqslant b_k x_{kn}\) ?

(10 Jan '18, 13:42) zbir

Yes, if your question is stated correctly.

(10 Jan '18, 14:01) Rob Pratt

Let me give an example to illustrate my problem. Assume \(p=3\) and \(b_k=3\) and \(C_{n+1}=0\), \(C_{n+2}=0\) and \(C_{n+3}=1\). Also, say, the variable \(A_{n}=2\). Here, \(A_n+C_{n+1}+C_{n+2}+C_{n+3}\geqslant b_k\) but I cannot set \(x_{kn}=1\) because \(A_n+C_{n+1}+C_{n+2}< b_k\).

(10 Jan '18, 14:29) zbir

I have updated the question.

(10 Jan '18, 14:31) zbir
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Asked: 10 Jan '18, 11:41

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Last updated: 10 Jan '18, 14:31

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