L30_Integer Linear Programming - Branch and Bound Algorithm

L30_Integer Linear Programming - Branch and Bound Algorithm...

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    INTEGER LINEAR PROGRAMMING There are many LP problems in which the decision variables will take only integer values. If all the decision variables will only take integer values it is called a pure integer LPP; otherwise the problem is called a mixed integer LPP. We discuss Land-Doig Branch and Bound algorithm to solve an integer LPP.
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    Branch-and-Bound Algorithm (Problem 2a problem set 9.2A page 378) Maximize 2 1 2 3 x x z + = 9 5 2 2 1 + x x 9 2 4 2 1 + x x Subject to 0 , 2 1 x x and integers
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    The set of feasible region consists of the Lattice points (0,0),(1,0),(2,0),(0,1),(1,1). The associated LPP, LP0 is defined by removing the integer restrictions. Solving we get x 1 =1.69, x 2 =1.13, z=7.31. Because the optimum LP0 solution does not satisfy the integer requirements, the B&B algorithm modifies the solution space in a manner that eventually identifies the integer optimum solution. First we select one of the variables whose optimum value at LP0 is not an integer. We select x 1 .
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    We replace the original LP0 with two new LPPs, LP1 and LP2 defined as LP1 space = LP0 space + LP2 space = LP0 space + ) 1 ( 1 x ) 2 ( 1 x We solve the LP1 problem (which is given by adding to LP0, the constraint
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L30_Integer Linear Programming - Branch and Bound Algorithm...

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