Branch and Bound Algorithm for Solving Integer Linear Programming

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Branch and Bound Algorithm for Solving Integer
Linear Programming

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Introduction

• Branch and Bound is a general optimization
  method.
• Starting by considering the original problem, the
  lower-bounding and upper-bounding procedures are
  applied to the root problem.
• Recursively divide the feasible region into two
  or more regions and solve the subproblems.
• If the lower bound for a node exceeds the best
  known feasible solution for minimization problem,
  no globally optimal solution can exist in the
  subspace of the feasible region represented by
  the node. Therefore, the node can be removed from
  consideration.
• The search proceeds until all nodes have been
  solved or pruned.

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Example

• Max Z 21x111x2
• s.t. 7x14x2 13
• x1 0, x2 0
• x1 ,x2 are integers

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Example (cont.)

• Step 0 Create Problem 1.
• Step 1 Remove Problem 1
• from the master list.
• Step 2 Solve Problem 1.
• Step 3 Branch on X1, since
• X1 not integer-valued.
• Step 4 Create Problem 2 3.
• Place on master list.

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Example (cont.)

• Step 1 Remove Problem 2
• from the master list.
• Step 2 Solve Problem 2.
• Step 3 No feasible solution.
• Stop.

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Example (cont.)

• Step 1 Remove Problem 3
• from the master list
• Step 2 Solve Problem 3
• Step 3 Branch on X2, since
• X2 not integer-valued
• Step 4 Create Problem 4 5
• place on master list

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Example (cont.)

• Step 1 Remove Problem 4
• from the master list.
• Step 2 Solve Problem 4.
• Step 3 Solution satisfies integer
• constraint. Record the
• solution and stop!

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Example (cont.)

• Following the same steps,
• terminate computations
• until master list is empty.