Business Statistics with Quantitative Analysis

1. Business Statistics with Quantitative Analysis Network Models Unit 7 Seminar Janet Kaplan JKaplan3@kaplan.edu

2. Network Models • A network is simply a collection of points connected by lines. • The points are called nodes or vertices and are usually represented by circles or squares. • The lines are called edges or links or arcs. • Each edge has frequently an associated “length”. Network models are used to represent a variety of situations in the world of business, and other fields.

3. Network Problems–Solving Techniques • This unit covers three techniques used to solve a variety of network-related problems. • The minimal spanning tree technique finds a path that connects all the nodes while minimizing the total distance. • The maximal flow technique finds the maximum flow of substance that can flow between two given nodes. • The shortest route technique finds the shortest path between two given points. • Note: Large scale problems may require hundreds or thousands of iterations, making efficient computer programs a necessity.

4. Networks–Solving Techniques Using Technology • I recommend to watch the video Networks, available in the Live Binder for Unit 7, for step-by-step explanations of how to perform the calculations associated with the three techniques using QM for Windows . • Note: You could use Excel QM for Shortest-Path and Maximal Flow, but to solve Minimal-Spanning Tree problems you must use QM for Windows or do the problems “by hand”.

5. Minimal Spanning Tree Technique • The minimal spanning tree problem involves connecting all the points of a network together while minimizing the total distance. • For example, an electrical company may want to connect power distribution centers while minimizing the amount of electric cable used. • The minimal spanning tree techniqueis used to determine a path that connects all the points of a network while minimizing the total distance.

6. Minimal Spanning Tree Technique • Steps of the minimal spanning tree technique: • 1. Select any node in the network. • 2. Connect this node to the nearest node that minimizes the total distance. • 3. Considering all the nodes that are now connected, find and connect the nearest unconnected node. If there is a tie, select one arbitrarily. • Note: A tie suggests that there are more than one optimal solution. • 4. Repeat the step 3 until all nodes are connected.

7. Minimal Spanning Tree Technique–Example • Lauderdale Construction • The Lauderdale Construction Company is developing a housing project. • It wants to determine the least expensive way to provide water and power to each house. • There are eight houses in the project. • The distance between houses is shown in the next slide.

8. 3 2 5 4 3 5 3 7 7 1 2 2 3 8 3 5 1 6 2 6 4 Gulf Minimal Spanning Tree Technique–Example • Lauderdale Construction–Network Figure 11.1

9. Minimal Spanning Tree Technique–Example 1 • Lauderdale Construction–First Iteration • Start by (arbitrarily) selecting node 1. The nearest node is node 3, at a distance of 2 (200 feet); connect those nodes. Figure 11.2

10. Minimal Spanning Tree Technique–Example 1 • Lauderdale Construction–Second Iteration • Consider nodes 1 and 3, look for the next nearest node, which is node 4, the closest to node 3. Connect nodes 3 and 4. Figure 11.3

11. Minimal Spanning Tree Technique–Example 1 • Lauderdale Construction–Third Iteration • Look for the unconnected node nearest to nodes 1, 3, and 4. This is either node 2 or node 6. Pick node 2 and connect it to node 3. Figure 11.3

12. Minimal Spanning Tree Technique–Example 1 • Lauderdale Construction–Fourth Iteration • Look for the nearest unconnected node to nodes 1, 2, 3, and 4. This is either node 5 or node 6. Pick node 5 and connect it to node 2. Figure 11.4

13. Minimal Spanning Tree Technique–Example 1 • Lauderdale Construction–Fifth Iteration • Look for the nearest unconnected node to nodes 1, 2, 3, 4, and 5. This is node 6, connect it to node 3. Figure 11.4

14. Minimal Spanning Tree Technique–Example 1 • Lauderdale Construction–Sixth Iteration • Look for the nearest unconnected node to nodes 1, 2, 3, 4, 5, and 6. This is node 8, connect it to node 6. Figure 11.5

15. Minimal Spanning Tree Technique–Example 1 • Lauderdale Construction–Seventh (and Final) Iteration • Look for the nearest unconnected node to nodes 1, 2, 3, 4, 5, 6, and 8. This is node 7, connect it to node 8. Figure 11.5

16. Minimal Spanning Tree Technique–Example 1 Lauderdale Construction–Total Distance To find the total distance add up the distances in the arcs used in the spanning tree: 2 + 2 + 3 + 3 + 3 + 1 + 2 = 16, or 1,600 feet Figure 11.5

17. Minimal Spanning Tree Technique–Example 1 • Lauderdale Construction–Summary Table 11.1

18. Minimal Spanning Tree Technique–Example 1 • Lauderdale Construction–QM for Windows Input Distance

19. Minimal Spanning Tree Technique–Example 1 • Lauderdale Construction–QM for Windows Output Total Distance Program 11.1

20. Minimal Spanning Tree Technique–Example 2 • Video Cameras • Problem 7-14 (Page 301) • The director of security wants to connect security video cameras to the main control site from five potential trouble locations. Ordinarily, cable would simply be run from each location to the main control site. However, because the environment is potentially explosive, the cable must be run in a special conduit that is continually air purged. This conduit is very expensive but large enough to handle five cables (the maximum that might be needed). Use the minimal-spanning tree technique to find a minimum distance route for the conduit between the locations noted in Figure 7.24. (Note that it makes no difference which one is the main control site.)

21. Minimal Spanning Tree Technique–Example 2 • Video Cameras–QM for Windows Input Distance Problem 7-14

22. Minimal Spanning Tree Technique–Example 2 • Video Cameras–QM for Windows Output Total Distance

23. Maximal Flow Problem • The maximal flow problem is to maximize the amount of material or items that can flow through a given network from one node, the source or origin, to another node, the sink or destination, when each arc has limited capacity. • For example, an oil company may want to find the maximum amount of oil that can flow from an oil field (source) to a refinery (sink) through a network of limited capacity pipelines. • The maximal flow technique determines the maximum amount of material that can flow through a network from source to destination.

24. Maximal Flow Technique • Steps of the maximal-flow technique: • 1. Pick any path from the source to the sink, with some flow. If none exists, the optimal solution has been found. • 2. Find the arc on this path with the smallest flow capacity available. Call this capacity C. This represents the maximum additional capacity that can be allocated to this route. • 3. For each node on this path, decrease the flow capacity in the direction of flow by the amount C, and increase the flow capacity in the reverse direction by the amount C. • 4. Repeat until an increase in flow is no longer possible.

25. Maximal Flow Technique–Example 1 • The town of Waukesha, Wisconsin is developing a road system for the downtown area. • Town leaders want to determine the maximum number of cars that can flow through the town from west to east. • The road network is shown in Figure 11.6 (next slide). • The numbers by the nodes indicate the number of cars that can flow from the node.

26. 2 2 1 2 East Point 6 1 1 3 0 West Point 1 2 1 0 1 10 4 1 6 1 5 3 2 0 3 Maximal Flow Technique–Example 1 • Waukesha–Road Network Capacity in Hundreds of Cars per Hour Figure 11.6

27. 2 2 1 6 2 3 1 Maximal Flow Technique–Example 1 • Waukesha Road System–Capacity Adjustment for • Path 1–2–6 Iteration 1 • 1. Pick up (arbitrarily) the path 1–2–6. • 2. The maximum flow is 2 units from node 2 to node 6. • 3. Adjust the path capacity by adding 2 to the westbound flows and subtracting 2 from the eastbound flows. Add 2 Subtract 2 Old Path Figure 11.7

28. 0 3 4 1 2 6 1 Maximal Flow Technique–Example 1 • Waukesha Road System–Capacity Adjustment for • Path 1–2–6 Iteration 1 • The result is this new path: Figure 11.7 New Path

29. 2 6 1 3 1 1 1 4 1 1 Maximal Flow Technique–Example 1 • Waukesha Road System–Second Iteration • 1. Pick path 1–2–4–6. • 2. The maximum capacity along this path is 1. • 3. Adjust the path capacity by adding 1 to the westbound flows and subtracting 1 from the eastbound flows. Add 1 Subtract 1 Old Path Figure 11.8

30. 0 2 4 4 6 0 2 0 0 1 2 2 0 0 10 4 1 6 1 5 3 2 0 3 Maximal Flow Technique–Example 1 • Waukesha Road System–Second Iteration • The result is this new path: Figure 11.8 New Network

31. 0 2 4 4 6 0 2 0 2 1 2 2 0 0 8 4 1 4 3 5 3 0 2 3 Maximal Flow Technique–Example 1 • Waukesha Road System–Third (and Final) Iteration • Pick path 1–3–5–6. • The maximum capacity along this path is 2. • The result is this new path: Figure 11.9

32. Maximal Flow Technique–Example 1 • Since there are no more paths from nodes 1 to 6 with unused capacity, the third iteration 3 is final. • The maximum flow through this network is 500 cars.

33. Maximal Flow Technique–Example 2 • PetroChem • Problem 7-2 (Page 298) • PetroChem, an oil refinery located on the Mississippi River south of Baton Rouge, Louisiana, is designing a new plant to produce diesel fuel. Figure 7.17 shows the network of the main processing centers along with the existing rate of flow (in thousands of gallons of fuel). The management at PetroChem would like to determine the maximum amount of fuel that can flow through the plant, from node 1 to node 7.

34. Maximal Flow Technique–Example 2 • PetroChem–QM for Windows Input Inbound Outbound Arcs

35. Maximal Flow Technique–Example 2 • PetroChem–QM for Windows Output Total Flow

36. Shortest Route Technique • The shortest route probleminvolves finding the shortest “distance” between two given nodes through the network. • For example, a delivery company may want to determine the shortest truck route from their warehouse to a delivery point. • The shortest-route technique finds how a person or item can travel from one location to another while minimizing the total “distance” traveled.

37. Shortest Route Technique • Steps of the shortest-route technique: • 1. Find the nearest node to the origin. Put the distance in a box by the node. • 2. Find the next-nearest node to the origin and put the distance in a box by the node. Several paths may have to be checked to find the nearest node. • 3. Repeat this process until you have gone through the entire network. The last distance at the ending node will be the distance of the shortest route.

38. Shortest Route Technique–Example • Ray Design, Inc. transports beds, chairs, and other furniture from the factory to the warehouse. • The company would like to find the route with the shortest distance. • The road network is shown in the next slide.

39. 200 2 4 Plant 100 100 100 50 150 1 6 200 100 40 3 5 Warehouse Shortest Route Technique–Example • Ray DesignNetwork Figure 11.10

40. 200 2 4 Plant 100 100 100 50 150 1 6 200 100 40 3 5 Warehouse Shortest Route Technique–Example • Ray Design–First Iteration • 1. The nearest node to the plant (node 1), is node 2. • 2. Connect these two nodes. 100 Figure 11.11

41. 100 200 2 4 Plant 100 100 100 50 150 1 6 200 100 40 3 5 Warehouse Shortest Route Technique–Example • Ray Design–Second Iteration • 1. Node 3 is the next nearest node but there are two possible paths. • 2. The shortest path is 1–2–3 with a distance of 150. Figure 11.12 150

42. 100 200 2 4 Plant 100 100 100 50 150 1 6 200 100 40 3 5 Warehouse 150 Shortest Route Technique–Example • Ray Design–Third Iteration • The next nearest node is node 5 by going through node 3. Figure 11.13 190

43. 100 200 2 4 Plant 100 100 100 50 150 1 6 200 100 40 3 5 Warehouse 190 150 Shortest Route Technique–Example • Ray Design–Fourth (and Final) Iteration • The next nearest node is either 4 or 6; turns out to be closer. • The shortest path is 1–2–3–5–6 with a distance of 290 miles. 290 Figure 11.14

44. Shortest Route Technique–Example 2 • Transworld Moving • Problem 7-11 (Page 301) • Transworld Moving has been hired to move the office furniture and equipment of Cohen Properties to their new headquarters. What route do you recommend? • The network of roads is shown in Figure 7.23 of the textbook.

45. Shortest Route Technique–Example 2 • Transworld Moving • QM for Windows Input

46. Shortest Route Technique–Example 2 • Transworld Moving–QM for Windows Output Shortest Route Total Distance

47. Unit 7 Graded Assignments • By Tuesday at midnight you should have completed the following graded assignments: • Discussion • Seminar • Project • Quiz