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Network Models: Quantitative approaches

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5. Find the shortest route between nodes 1 and 10 in the following network. (Show steps)

(see diagram in the attached file)

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5. Find the shortest route between nodes 1 and 10 in the following network. (Show steps)

 This is the problem regarding identifying shortest path. In this case the source node is 1 and destination node is 10. We have to identify the shortest path to traverse from source to destination node. Nodes 2-9 are called as intermediate nodes. Arcs joining two nodes are called as paths and numbers on the arc indicate cost/time to traverse a particular path. Although not indicated in the figure we will assume that it is directed network. That means any path from a preceding node to succeeding node has got a direction, which is forward. Node 1 is preceding node to 2, 3 and 4. Nodes 2, 3, and 4 are preceding nodes to 5, 6, and 7 and so on. Between any two sets of nodes preceding nodes are denoted by i and succeeding nodes are denoted by j.

Decision variables: Let x12 be the variable to indicate the choice of path 1-2. This variable is a binary/bivalent/Boolean variable. It means it will take value of 1 if a particular path is chosen or otherwise. Since these are integer valued variables they can not take any value between 0 and 1, the upper bound is 1 and lower bound is 0.
Similarly we define one such variable for other paths and they are
x13, x14, x25, x26, x27, x35, x36, x37, x45, x46, x58, x59, x68, x69, x78, x79, ...

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