Chapter 9 Integer Programming

Part 2 Assoc. Prof. Dr. Arslan M. ÖRNEK

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9.3. Branch-and-Bound Method (Pure IP)

• Branch-and-Bound methods find the optimal solution to an IP by efficiently enumerating the points in a subproblem’s feasible region.

• IMPORTANT OBSERVATION: If you solve the LP relaxation of a pure IP and obtain a solution in which all variables are integers, then the optimal solution to the LP relaxation is also the optimal solution to the IP.

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9.3. Branch-and-Bound Method (Pure IP)

• The Telfa Corporation manufactures tables and chairs.

• A table requires 1 hour of labor and 9 square board feet of wood, and a chair requires 1 hour of labor and 5 square board feet of wood.

• Currently, 6 hours of labor and 45 square board feet of wood are available.

• Each table contributes $8 to profit, and each chair contributes $5 to profit.

• Formulate and solve an IP to maximize Telfa’s profit.

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9.3. Branch-and-Bound Method (Pure IP)

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9.3. Branch-and-Bound Method (Pure IP)

Upper bound

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9.3. Branch-and-Bound Method (Pure IP)

• Chose any variable that is fractional at the moment.

• Let’s chose x1:

• At the optimal solution it can be either <=3 or >=4.

• We partition the solution space by branching on this variable.

• Two new subproblems:

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9.3. Branch-and-Bound Method (Pure IP)

• Chose a subproblem to solve (arbitrarily):

• Subproblem 2 LP relaxation:

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9.3. Branch-and-Bound Method (Pure IP)

A node of the

branch and

bound tree

An arc of the

branch and

bound tree

Solution not

integer, branch

on x2

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9.3. Branch-and-Bound Method (Pure IP)

Chose this node to

solve (LIFO):

Infeasible –

Fathomed.

Then,

chose this

node

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9.3. Branch-and-Bound Method (Pure IP)

Upper bound

Subproblem 5:

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9.3. Branch-and-Bound Method (Pure IP)

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9.3. Branch-and-Bound Method (Pure IP)

Lower bound on

the original IP:

Incumbent

Solution

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9.3. Branch-and-Bound Method (Pure IP)

NEW Lower bound

on the original IP

New Incumbent

Solution

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9.3. Branch-and-Bound Method (Pure IP)

• Only remaining subproblem:

• Subproblem 3: Solve LP relaxation:

This is smaller than the current LB , so fathom.

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9.3. Branch-and-Bound Method (Pure IP)

Incumbent

Solution =

Best LB =

Optimal

Solution

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9.3. Branch-and-Bound Method (Pure IP)

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9.4. Branch-and-Bound Method (Mixed IP)

• In a mixed IP, some variables are required to be integers and others are allowed to be either integers or nonintegers.

• Branch only on variables that are required to be integers.

Optimal solution of the LP-relaxation:

Branch-and-Bound Method (Binary Prog.)

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Prototype Example: Bounding

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Prototype Example: Bounding

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Prototype Example: Fathoming • A subproblem can be fathomed in three ways:

1. LP relaxation of the subproblem gives an integer solution, so we do not need to branch further, the subproblem is solved optimally.

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Prototype Example: Fathoming

• A subproblem can be fathomed in three ways:

2. A subproblem is fathomed if its bound ≤ incumbent soln.

Ex. Since Z*=9, there is no need to consider any solution whose bound is below 9.

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Prototype Example: Fathoming

• A subproblem can be fathomed in three ways:

3. If a subproblem’s LP relaxation has no feasible solution, then the subproblem itself must have no feasible solutions, so it can be dismissed (fathomed).

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Prototype Example continued – Iteration 2

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Prototype Example continued – Iteration 2

Cannot fathom.

Cannot fathom. 25

Prototype Example continued – Iteration 2

Resulting solution tree after iteration 2:

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Prototype Example continued- Iteration 3

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Prototype Example continued – Iteration 3

Resulting solution tree after iteration 3:

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Prototype Example continued- Iteration 4

New incumbent!

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Prototype Example continued – Iteration 3

Resulting solution tree after iteration 4 (final):

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B&B for Minimization Problems

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