Introduction to
Management Science
with Spreadsheets
Stevenson and Ozgur
First Edition
Part 2 Deterministic Decision Models
Chapter 7 Supplement
Branch and Bound Method of
Solving Integer Programming
Problems
McGraw-Hill/Irwin
Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved.
Learning Objectives
After completing this chapter, you should be able to:
1. Use the branch and bound method to solve integer
programming problems.
2. Use the enumeration method to solve 0–1 integer
programming problems.
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McGraw-Hill/Irwin 7S–2
Integer Programming Problems
• Bound
–An upper or lower limit on the value of the objective
function at a given stage of the analysis of an integer
programming problem.
• Branch
–Selection of an integer value of a decision variable to
examine for a possible integer solution to a problem.
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Solving Integer Programming
• Branch and Bound method
–Creates and solves a sequence of subproblems to the
original problem that are increasingly more restrictive
until an optimal solution is found.
• Enumeration Method
–A method used with 0–1 problems that involves listing
every possible outcome in order to identify the optimal
solution.
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Integer Programming Problems
• Mixed-Integer Problem
–Requires that some, but not all, decision variables
have integer values.
• Pure-Integer Problem
–requires that all decision variables have integer
values.
• Mixed-Integer Problem
–requires that some, but not all, decision variables have
integer values.
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Figure 7S–1
Graph of an Integer Programming Problem
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Example 7S-1 All-Integer Problem
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Figure 7S–2
The LP Relaxation Solution
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Figure 7S–3
Node I Summarizes the LP Relaxation Solution
Figure 7S–4
Branching from the First Node
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Figure 7S–5
Subproblems and Solution Points for Nodes 2 and 3
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Figure 7S–6
Solutions for Nodes 2 and 3
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Figure 7S–7
Subproblem and Solution Point for Node 4
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Figure 7S–8
Solutions for Nodes 4 and 5
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Figure 7S–9
Subproblems and Solutions for the Next Two Nodes
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Figure 7S–10
Solutions for Nodes 6 and 7
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Figure 7S–11
The LP Relaxation Solution Point
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Figure 7S–12
Solution at Node 1 for Example 2 and Branches to Nodes 2 and 3
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Figure 7S–13
Subproblems and Solution Points for Nodes 2 and 3
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Figure 7S–14
Solutions for Nodes 2 and 3
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Figure 7S–15
The LP Relaxation Solution Point
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Figure 7S–16
Node 1 Solution and Branches
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Figure 7S–17
Subproblems and Solution Points for Nodes 2 and 3
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Figure 7S–18
Solutions for Nodes 2 and 3 (Node 3 Contains the Optimal)
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Figure 7S–19
Tree Diagram for Example 4
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Figure 7S–20
Tree for Example 5
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Figure 7S–21
Node Diagram for Example 7S-6
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