Chapter 7, Problem 1P

Summary Introduction

To determine: The shortest route from node 1 to each of the other six nodes.

Explanation of Solution

Given diagram:

Determine the shortest route:

Step 1:

The initial node from the given diagram is Node 1. In the above table, nodes from Node 1 should be considered. While comparing the distance of each branch, 1-3 has the minimum distance. Hence, it was chosen. The diagram for the above table is as follows:

Step 2:

In the above table, nodes from Node 1 and Node 3 (as Node 3 was chosen in step 1) should be considered. However, the chosen branch should not be considered again. While comparing the distance of each branch, 1-2 has the minimum distance. Hence, it was chosen. The diagram for the above table is as follows:

Step 3:

In the above table, nodes from Node 1, Node 3, and Node 2 (as Node 2 was chosen in step 2) should be considered. However, the chosen branch should not be considered again. While comparing the distance of each branch, 3-4 has the minimum distance. Hence, it was chosen. The diagram for the above table is as follows:

Step 4:

In the above table, nodes from Nodes 1, 3, 2, and 4 (as Node 4 was chosen in step 3) should be considered. However, the chosen branch should not be considered again. While comparing the distance of each branch, 4-7 has the minimum distance. Hence, it was chosen. The diagram for the above table is as follows:

Step 5:

In the above table, nodes from Nodes 1, 3, 2, 4, and 7 (as Node 7 was chosen in step 4) should be considered. However, the chosen branch should not be considered again. While comparing the distance of each branch, 4-6 has the minimum distance. Hence, it was chosen. The diagram for the above table is as follows:

Step 6:

In the above table, nodes from Nodes 1, 3, 2, 4, 7, and 6 (as Node 6 was chosen in step 5) should be considered. However, the chosen branch should not be considered again. While comparing the distance of each branch, 2-5 has the minimum distance. Hence, it was chosen. The final node of the shortest route is Node 5 after Node 1, Node 3, Node 2, Node 4, Node 7, and Node 6 (all the given nodes were chosen for the shortest route). Hence, the steps should be stopped here. The diagram for the above table is as follows:

Formulae to determine each distance from step 1 to step 6:

Final diagram and the shortest route:

Hence, the shortest route should have the following branches:

1-3, 1-2, 3-4, 4-7, 4-6, and 2-5