In each case we need bounds on how much the

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Unformatted text preview: a constraint is satisfied, and 0 otherwise. In each case, we need bounds on how much the constraint could be violated in a solution that satisfies every other constraint of the problem. Example 1. ⎧1 ⎪ w= ⎨ ⎪0 ⎩ if x ≥ 1 if x = 0. In this example, x is an integer variable. And suppose that we know x is bounded above by 100. (We may know the bound on x because it is one of the constraints of the problem. We may also know the bound of 100 on x because it is implied by one or more of the other constraints. For example, suppose that one of the constraints was "3x + 4y + w : 300." We could infer from this constraint that 3x : 300 and thus x : 100. Equivalent constraint: w : x : 100w. w E {0,1}. In any feasible solution, the definition of w is correct. If x � 1, then the first constraint is satisfied whether w = 0 or w = 1, and the second constraint forces w to be 1. If x = 0, then the first constraint forces w to be 0, and the second constraint is satisfied. ⎧1 ⎪ if x ≥ 7 Example 2. w = ⎨ otherwise. ⎪ 0 ⎩ Agai...
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