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Mike
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I would like to seek some advice on modeling the following logical condition:

Given two groups of binary decision variables $$A_{i}, i=1...n,$$ and $$B_{j}, j=1...m$$.

$$A_{i}=1- B_{j}, \forall i, \forall j$$

i.e., if one of $$A_{i}=1$$, all $$B_{j}$$ must be zero, and vice-versa.

Besides, the above equality constraint, I would like to include tighter cuts, but have only managed to come up with the following, which seems loose:

$$\left\lvert B\right\rvert *A_{i}\le \left\lvert B\right\rvert-\sum_{j=1}^{j=m}B_{j}, \forall i$$

$$A_{i}\ge 1-\sum_{j=1}^{j=m}B_{j}, \forall i$$

Thank you!

Mike
• 663
• 3
• 10

# Assistance in formulating binary constraint(s)

I would like to seek some advice on modeling the following logical condition:

Given two groups of binary decision variables $$A_{i}, i=1...n,$$ and $$B_{j}, j=1...m$$.

$$A_{i}=1- B_{j}, \forall i, \forall j$$

i.e., if one of $$A_{i}=1$$, all $$B_{j}$$ must be zero, and vice-versa.

I would like to include tighter cuts, but have only managed to come up with the following, which seems loose:

$$\left\lvert B\right\rvert *A_{i}\le \left\lvert B\right\rvert-\sum_{j=1}^{j=m}B_{j}, \forall i$$

$$A_{i}\ge 1-\sum_{j=1}^{j=m}B_{j}, \forall i$$

Thank you!