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Tutorial_9

Tutorial_9 - 0 1 2 14 5 4 3 2 7 4 1 4 x 1 1 x 2 1...

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IELM202 Tutorial 9 1/6/11 1. Use the branch-and bound method to solve the following integer program: Maximize Z=4X 1 -2X 2 +7X 3 -X 4 Subject to X 1 +5X 3 <= 10 X 1 +X 2 -X 3 <=1 6X 1 -5X 2 <=0 -X 1 +2X 3 -2X 4 <=3 X j >= 0, for j = 1, 2, 3, 4 X j is integer, for j = 1, 2, 3 Subproblem 1: Subproblem 2: maximize Z = 4x 1 - 2x 2 + 7x 3 - x 4 maximize Z = 4x 1 - 2x 2 + 7x 3 - x 4 subject to x 1 + 5x 3 10 subject to x 1 + 5x 3 10 x 1 + x 2 - x 3 1 x 1 + x 2 - x 3 1 6x 1 - 5x 2 0 6x 1 - 5x 2 0 -x 1 + 2x 3 - 2x 4 3 -x 1 + 2x 3 - 2x 4 3 x 1 1 x 1 2 x j 0, for j = 1, 2, 3, 4 x j 0, for j = 1, 2, 3, 4 x j is integer, for j = 1, 2, 3. x j is integer, for j = 1, 2, 3. The following diagram shows this step of branching. 0 ( , , , 0) 1 2 14 5 4 3 2 7 4 1 4 x 1 1 x 2 1

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IELM202 Tutorial 9 1/6/11 b. bounding : Solve the LP relaxation of these two new subproblems and the branching tree now becomes:
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Unformatted text preview: ( , , , 0) 1 2 14 5 4 3 2 7 4 1 4 x 1 1 x 2 1 infeasible ( 1 , , , 0) 6 5 9 5 14 1 5 fathomed ( , , , 0) 1 2 14 5 4 3 2 7 4 1 4 x 1 1 x 2 1 infeasible ( 1 , , , 0) 6 5 9 5 14 1 5 fathomed 3 4 x 1 2 x 2 2 ( , 1 , , 0) 5 6 11 6 14 1 6 ( , 2 , , 0) 5 6 11 6 12 1 6 IELM202 Tutorial 9 1/6/11 ( , , , 0) 1 2 14 5 4 3 2 7 4 1 4 x 1 1 x 2 1 infeasible ( 1 , , , 0) 6 5 9 5 14 1 5 fathomed 3 4 x 1 2 x 2 2 ( , 1 , , 0) 5 6 11 6 14 1 6 ( , 2 , , 0) 5 6 11 6 12 1 6 5 6 x 1 x 1 1 infeasible ( 0 , 0 , 2 , ) 1 2 Z*=13 1 2 fathomed fathomed fathomed The optimal solution is (0, 0, 2, ½) and Z*=27/2...
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