1. Stevenson and Ozgur First Edition Introduction to Management Science with Spreadsheets McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 7 Supplement Branch and Bound Method of Solving Integer Programming Problems Part 2 Deterministic Decision Models

2. Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 7S–2 Learning Objectives 1.Use the branch and bound method to solve integer programming problems. 2.Use the enumeration method to solve 0–1 integer programming problems. After completing this chapter, you should be able to:

3. Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 7S–3 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.

4. Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 7S–4 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.

5. Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 7S–5 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.

6. Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 7S–6 Figure 7S–1Graph of an Integer Programming Problem

7. Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 7S–7 Example 7S-1 All-Integer Problem

8. Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 7S–8 Figure 7S–2The LP Relaxation Solution

9. Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 7S–9 Figure 7S–3Node I Summarizes the LP Relaxation Solution Figure 7S–4Branching from the First Node

10. Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 7S–10 Figure 7S–5Subproblems and Solution Points for Nodes 2 and 3

11. Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 7S–11 Figure 7S–6Solutions for Nodes 2 and 3

12. Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 7S–12 Figure 7S–7Subproblem and Solution Point for Node 4

13. Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 7S–13 Figure 7S–8Solutions for Nodes 4 and 5

14. Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 7S–14 Figure 7S–9Subproblems and Solutions for the Next Two Nodes

15. Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 7S–15 Figure 7S–10Solutions for Nodes 6 and 7

16. Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 7S–16 Figure 7S–11The LP Relaxation Solution Point

17. Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 7S–17 Figure 7S–12Solution at Node 1 for Example 2 and Branches to Nodes 2 and 3

18. Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 7S–18 Figure 7S–13Subproblems and Solution Points for Nodes 2 and 3

19. Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 7S–19 Figure 7S–14Solutions for Nodes 2 and 3

20. Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 7S–20 Figure 7S–15The LP Relaxation Solution Point

21. Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 7S–21 Figure 7S–16Node 1 Solution and Branches

22. Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 7S–22 Figure 7S–17Subproblems and Solution Points for Nodes 2 and 3

23. Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 7S–23 Figure 7S–18Solutions for Nodes 2 and 3 (Node 3 Contains the Optimal)

24. Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 7S–24 Figure 7S–19Tree Diagram for Example 4

25. Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 7S–25 Figure 7S–20Tree for Example 5

26. Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 7S–26 Figure 7S–21Node Diagram for Example 7S-6