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 8 Network Optimization Models Part 2 Deterministic Decision Models

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–2 Learning Objectives 1.State why network models are important tools for problem solving. 2.Describe the kinds of problems that can be solved using the shortest-route algorithm and use the algorithm to solve typical shortest-route problems. 3.Formulate the shortest-route problem as a linear programming problem. 4.Solve the shortest-route problem using Excel. After completing this chapter, you should be able to:

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–3 Learning Objectives (cont’d) 5.Describe the kinds of problems that can be solved using the minimal-spanning-tree algorithm and use the algorithm to solve typical minimal-spanning-tree problems. 6.Describe the kinds of problems that can be solved using the maximal-flow algorithm and use the algorithm to solve typical maximal-flow problems. 7.Formulate the maximal-flow problem as a linear programming problem. 8.Solve the maximal-flow problem using Excel. After completing this chapter, you should be able to:

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–4 NetworksNetworks Network –A set of nodes and connecting arcs or branches. Can be useful in representing various systems, such as distribution systems, production systems, and transportation systems. Network models are an important approach for problem solving because: –They can be used to model a wide range of problems. –They are relatively simple to work with –They provide a visual portrayal of a problem.

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–5 Frequently Used Algorithms Shortest-Route Algorithm –Used for determining the shortest time, distance, or cost from an origin to a destination through a network. Minimum Spanning Tree Algorithm –Used in determining the minimum distance (cost, time) needed to connect a set of locations into a single system. Maximal Flow Algorithm –Used for determining the greatest amount of flow that can be transmitted through a system in which various branches, or connections, have specified flow capacity limitations.

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–6 Figure 8–1A Simple Network Diagram

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–7 Example 8-1 Labeling Procedure A label is developed for each node. The labels consist of two numbers separated by a comma: The first number refers to the distance from Node 1 to the labeled node along a certain path, while the second number refers to the node that immediately precedes this node along that path.

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–8 Example 8-1 (cont’d)

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–9 Example 8-1 (cont’d)

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–10 Figure 8–2Network for Tri-State Shipping Company Shortest-Route Problem

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–11 Exhibit 8-1Input Worksheet for the Tri-State Shipping Company Shortest- Route Problem (Beginning Node 1, Ending Node 7)

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–12 Exhibit 8-2Parameter Specification Screen for the Tri-State Shipping Company Shortest-Route Problem (Beginning Node 1, Ending Node 7)

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–13 Exhibit 8–3Output Worksheet for the Tri-State Shipping Company Shortest-Route Problem (Beginning Node 1, Ending Node 7)X

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–14 Exhibit 8–4Output Worksheet for the Tri-State Shipping Company Shortest-Route Problem (Beginning Node 1, Ending Node 6)

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–15 Exhibit 8–5Parameter Specification Screen for the Tri-State Shipping Company Shortest-Route Problem (Beginning Node 1, Ending Node 6)

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–16 Figure 8–3Network for Oil Pipeline Problem Note: Arc lengths are not precisely proportional to the distances shown on the arcs.

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–17 Example 8-3

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–18 Example 8-3 (cont’d)

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–19 Example 8-3 (cont’d)

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–20 Figure 8–4A Flow Network

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–21 Figure 8–5Updated Flow Network

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–22 Figure 8–6Second Revision of the Network

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–23 Figure 8–7Third Revision of Flow Network

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–24 Figure 8–8Final Solution for Flow Network Example

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–25 Figure 8–9Flow Network for Demonstration of Reverse Flow

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–26 Figure 8–10Flow Network for Demonstration of Reverse Flow

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–27 Figure 8–11Revised Flow Network

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–28 Table 8–1Branches of the Network and Flow Capacities of the Branches for the Turkmen Oil Ltd. Maximal Flow Problem

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–29 Figure 8–12Network Diagram for the Maximal Flow Problem for Turkmen Oil Ltd.

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–30 Exhibit 8–6Input Worksheet for the Turkmen Oil Ltd. Maximal Flow Problem

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–31 Exhibit 8–7Parameter Specification Screen for the Turkmen Oil Ltd. Maximal Flow Problem

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–32 Exhibit 8–8Output Worksheet for the Turkmen Oil Ltd. Maximal Flow Problem

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–33 Table 8–2 Summary of the Optimal Flow Quantities for Turkmen Old Ltd.

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–34 Exhibit 8–9Worksheet for Solved Problem 1, the Shortest-Route Problem (Beginning Node 1, Ending Node 5)

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–35 Exhibit 8–10Parameter Specification Screen for Solved Problem 1, the Shortest- Route Problem (Beginning Node 1, Ending Node 5)

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–36 Exhibit 8–11Worksheet for Solved Problem 3, the Maximal Flow Problem

Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 8–37 Exhibit 8–12Parameter Specification Screen for Solved Problem 3, the Maximal Flow Problem