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I am working on a branch and bound problem and I was wondering if it is correct to prune the node with z = 21.667 and if so, how do we know that we will not get a better solution if we branch on this node? enter image description here

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In branch and bound, you prune a branch if you have an upper bound on the best solution that this branch can provide, and this upper bound is smaller than your current best solution. In your case, you have no current best solution (since you haven't considered any integer solutions), and in particular, you cannot prune any branch.

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