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Unformatted text preview: • Solve the LP relaxation of problem P (relaxing the integrality constraint for nonﬁxed integer variables). • Let x * be the optimal solution obtained, and z P the optimal objective function value. • (bounding) If z P ≥ U we delete the problem (as z P is a lower bound on all integer solutions in this branch of the tree). • (branching) Pick some x j whose optimal solution is fractional, and construct two problems by adding the constraints x j ≤ b x * j c and x j ≥ d x * j e in each problem. 1 Example Apply the branch and bound method to ﬁnd a solution for min x 12 x 2 s . t .4 x 1 + 6 x 2 ≤ 9 x 1 + x 2 ≤ 4 x 1 , x 2 ≥ x 1 , x 2 integer Apply the branch and bound method to ﬁnd a solution for max 12 x 1 + 8 x 2 + 7 x 3 + 6 x 4 s . t . 8 x 1 + 6 x 2 + 5 x 3 + 4 x 4 ≤ 15 x i ∈ { , 1 } , i = 1 , . . . , 4 2...
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 Spring '05
 YY
 Linear Programming, Optimization, integer variables, mixed integer programs, nonfixed integer variables, integer feasible solution

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