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ip-solve - How to solve Integer Proggramming problems Idea...

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How to solve Integer Proggramming problems? Idea: Ignore the “integer” constraints, solve the LP relaxation, and then round. Problem 1: May need to change some variables that were integer, or change variables by more that 1 unit. x 1 + x 2 3 . 5 x 1 + x 2 16 . 5 x 1 ,x 2 0 integer Problem 2: Rounded soltuion may not be optimal to the IP. max z = x 1 + 5 x 2 s . t . x 1 + 10 x 2 20 x 1 2 x 1 ,x 2 0 integer So what to do???? 1
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Exhaustive enumeration: Check all integer points within the feasible region, and pick the best one. max z = 2 x 1 + x 2 s . t . x 1 + x 2 5 x 1 + x 2 0 6 x 1 + 2 x 2 21 x 1 ,x 2 0 integer Problem: Very slow. 2
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Branch and Bound, Sections 9-3, 9-4. Solve an LP relaxtion of a (sub) problem. If there is a variable x i = f that is required to be integer, but is not, then branch : Create 2 new subproblems, one with x i ≤ ⌊ f , and the other with x i ≥ ⌈ f . Bound: If a subproblem has all required variables integer at its optimal solution, then we get a candidate z . Keep the best candidate.
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