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### Chapter 7

Network Flow Models

Shortest Route Problem

- Given distances between nodes, find the shortest route between any pair of nodes.

Example: p.282 (291)

Solution Methods

- Dijkstra algorithm:
- Introduced in book.
- Not required for this course

- Using QM:
- Required for this course
- Data input format -

Discussion

- What if the ‘cost’, instead of ‘distance’, between two nodes are given, and we want to find the ‘lowest-cost route’ from a starting node to a destination node?
- What if the cost from a to b is different from the cost from b to a? (QM does not handle this situation.)

Minimal Spanning Tree Problem

- Given costs (distances) between nodes, find a network (actually a “tree”) that covers all the nodes with minimum total cost.
- Applications:

Example: p.290 (299)

Solution Method: Using QM.

Shortest Route vs. Minimal Spanning

- The minimal spanning tree problem is to identify a set of connected arcs that cover all nodes.
- The shortest route problem is to identify a route from a particular node to another, which typically does not pass through every node.

Maximal Flow Problem

- Given flow-capacities between nodes, find the maximum amount of flows that can go from the origin node to the destination node through the network.
- Applications:

Example: p.294 (303)

Solution Method: Using QM.

Network Flow Problem Solving

- Given a problem, we need to tell what ‘problem’ it is (shortest route, minimal spanning tree, or maximal flow); then use the corresponding module in QM to solve it.

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