The Minimal Spanning Tree Problem

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The Minimal Spanning Tree Problem

In the shortest route problem presented in the previous section, the objective was to determine the shortest routes between the origin and the destination nodes in the network. In our example, we determined the best route from Los Angeles to each of the six destination cities. The minimal spanning tree problem is similar to the shortest route problem, except that the objective is to connect all the nodes in the network so that the total branch lengths are minimized. The resulting network spans (connects) all the points in the network at a minimum total distance (or length).

The minimal spanning tree problem is to connect all nodes in a network so that the total branch lengths are minimized .

To demonstrate the minimal spanning tree problem, we will consider the following example. The Metro Cable Television Company is to install a television cable system in a community consisting of seven suburbs. Each of the suburbs must be connected to the main cable system. The cable television company wants to lay out the main cable network in a way that will minimize the total length of cable that must be installed. The possible paths available to the cable television company (by consent of the town council) and the feet of cable (in thousands of feet) required for each path are shown in Figure 7.11.

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Figure 7.11. Network of possible cable TV paths

Management Science Application: Reducing Travel Costs at the Defense Contract Management Agency

The Defense Contract Management Agency (DCMA), headquartered at Fort Belvoir, Virginia, has 15,000 people stationed in offices around the United States. These federal employees have recurring professional development training requirements, and the associated costs for travel to training sites is a substantial portion of the DCMA's annual budget. In 1998 the DCMA faced a 25% budget cut, without any reduction in training requirements. At that time training sites were selected arbitrarily at the discretion of training planners. Because travel costs vary with the location, the selection of training sites can have a dramatic impact on the total temporary duty (i.e., travel) costs for employees . With the help of consultants , DCMA developed a decision support system (DSS) called OffSite that would select the minimum-cost training sites from among many cities. The DSS employs a shortest route model to determine the lowest -cost location for program participants arriving from geographically dispersed locations.

A shortest route network was created that connected 261 U.S. cities with government-contracted airfares. The minimum-cost training location is based on the federal per-diem travel rate (including meals and lodging) for a specific city, the duration of the training event, the number of participants at the city, and the cost of travel to the city from other cities. The shortest route method determines the minimum-cost route from each of the 261 cities to a specific training site, enabling training planners to assess the overall costs of bringing attendees to various alternate locations. OffSite was tested using data from a 2-day training event in Monterey, California, attended by representatives from every DCMA office. The total cost of this meeting was $37,582, which was over $10,000 (i.e., 39%) more than the least-cost alternative identified by OffSite, which was Tulsa, Oklahoma. Since OffSite's development in 1998, it has been used throughout the federal government. At the Defense Logistics Agency alone, savings have been over $1 million.

Source: J. L. Huisingh, H. M. Yamauchi, and R. Zimmerman, "Saving Federal Travel Dollars," Interfaces 31, no. 5 (SeptemberOctober 2001): 1323.

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In Figure 7.11 the branch from node 1 to node 2 represents the available cable path between suburbs 1 and 2. The branch requires 16,000 feet of cable. Notice that the network shown in Figure 7.11 is identical to the network in Figure 7.2 that we used to demonstrate the shortest route problem. The networks were intentionally made identical to demonstrate the difference between the results of the two types of network models.

The Minimal Spanning Tree Solution Approach

The solution approach to the minimal spanning tree problem is actually easier than the shortest route solution method. In the minimal spanning tree solution approach, we can start at any node in the network. However, the conventional approach is to start with node 1. Beginning at node 1, we select the closest node (i.e., the shortest branch) to join our spanning tree. The shortest branch from node 1 is to node 3, with a length of 9 (thousand feet). This branch is indicated with a heavy line in Figure 7.12. Now we have a spanning tree consisting of two nodes: 1 and 3. The next step is to select the closest node not now in the spanning tree. The node closest to either node 1 or node 3 (the nodes in our present spanning tree) is node 4, with a branch length of 15,000 feet. The addition of node 4 to our spanning tree is shown in Figure 7.13.

Figure 7.12. Spanning tree with nodes 1 and 3

Figure 7.13. Spanning tree with nodes 1, 3, and 4

Start with any node in the network and select the closest node to join the spanning tree. Select the closest node to any node in the spanning area .

Next, we repeat the process of selecting the closest node to our present spanning tree (nodes 1, 3, and 4). The closest node not now connected to the nodes in our spanning tree is node 2. The length of the branch from node 4 to node 2 is 12,000 feet. The addition of node 2 to the spanning tree is shown in Figure 7.14.

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Figure 7.14. Spanning tree with nodes 1, 2, 3, and 4

Our spanning tree now consists of nodes 1, 2, 3, and 4. The node closest to this spanning tree is node 5, with a branch length of 14,000 feet to node 4. Thus, node 5 joins our spanning tree, as shown in Figure 7.15.

Figure 7.15. Spanning tree with nodes 1, 2, 3, 4, and 5

The spanning tree now contains nodes 1, 2, 3, 4, and 5. The closest node not currently connected to the spanning tree is node 7. The branch connecting node 7 to node 5 has a length of 8,000 feet. Figure 7.16 shows the addition of node 7 to the spanning tree.

Figure 7.16. Spanning tree with nodes 1, 2, 3, 4, 5, and 7

Now our spanning tree includes nodes 1, 2, 3, 4, 5, and 7. The only remaining node not connected to the spanning tree is node 6. The node in the spanning tree closest to node 6 is node 7, with a branch length of 14,000 feet. The complete spanning tree, which now includes all seven nodes, is shown in Figure 7.17.

Figure 7.17. Minimal spanning tree for cable TV network
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The spanning tree shown in Figure 7.17 requires the minimum amount of television cable to connect the seven suburbs72,000 feet. This same minimal spanning tree could have been obtained by starting at any of the six nodes other than node 1.

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Notice the difference between the minimal spanning tree network shown in Figure 7.17 and the shortest route network shown in Figure 7.10. The shortest route network represents the shortest paths between the origin and each of the destination nodes (i.e., six different routes). In contrast, the minimal spanning tree network shows how to connect all seven nodes so that the total distance (length) is minimized.

In summary, the steps of the minimal spanning tree solution method are as follows :

  1. Select any starting node (conventionally, node 1 is selected).

  2. Select the node closest to the starting node to join the spanning tree.

  3. Select the closest node not presently in the spanning tree.

  4. Repeat step 3 until all nodes have joined the spanning tree.

Steps of the minimal spanning tree solution method .

Computer Solution of the Minimal Spanning Tree Problem with QM for Windows

The minimal spanning tree solution for our Metro Cable Television Company example using QM for Windows is shown in Exhibit 7.6.

Exhibit 7.6.