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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)
An arc of the
branch and
bound tree
A node 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)
Subproblem 5:
Upper bound
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9.3. Branch-andBound Method
(Pure IP)
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9.3. Branch-andBound Method
(Pure IP)
Lower bound on
the original IP:
Incumbent
Solution
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9.3. Branch-andBound 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-andBound 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:
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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.
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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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