Network Optimization Models

1. Network Optimization Models

2. Network Terminology • A network consists of a set of nodes and arcs. • The arcs may have some flow through them. • If flow is allowed in only one direction, the arc is a directed arc. • If flow is allowed in both directions, the arc is an undirected arc.

3. Network Terminology (Cont’d) • A path between two nodes is a sequence of distinct arcs connecting these two nodes. • A directed path consists of directed arcs in same direction. • An undirected path allows arc of any direction. A B

4. Network Terminology (Cont’d) • A path that begins and ends at the same node is a cycle. • In a directed network, a cycle is either a directed cycle or an undirected cycle depending on the directions of the arcs in the path.

5. Network Terminology (Cont’d) • Two nodes are connected if there is a path between them. • A connected network is a network where every pair of nodes is connected.

6. Network Terminology (Cont’d) • A tree is a connected network that contains no cycles. • A spanning tree is a tree that connects all nodes in the network. • If a connected network has n nodes, every spanning tree has exactly n-1 arcs .

7. Network Terminology (Cont’d) • In a directed network with flow, the arc capacity is the maximum flow allowed on an arc. • A supply node (or source node) has net flow out of the node. • A demand node (or sink node) has net flow into the node. • A transshipment node (or intermediate node) satisfies conservation of flow; flow in equals flow out. A 3 Sink (Demand) 9 1 5 D 4 Source (Supply) 7 B 6 5 1 4 2 4 E C

8. Shortest Route Problem • Find the shortest route between Source and Sink. • Idea is to start with the source, or origin, and find the shortest route to each of the nodes in the network sequentially until the destination is reached. A 7 5 2 D Sink 2 4 5 B Source 7 3 1 4 1 4 C E

9. Algorithm for Shortest Route Problem • Start with the source, or origin. Label it with zero distance from origin. It is now called a solved node. • Find the next nearest node: do this by • Consider all unsolved (unlabelled) nodes that are directly connected to a solved (labelled) node – these are the candidate nodes • For each candidate, calculate the distance to origin by adding the branch distance to the labelled distance • Find the minimum distance, and label that node with the distance and the preceding node. This node is now called a solved node. (if there is a tie, label both nodes.) • Repeat until the destination becomes a solved node. The shortest route can then be traced backwards.

10. Minimal Spanning Tree Problem • Find the spanning tree of the shortest length A 7 2 2 5 D Sink 4 5 B Source 7 4 1 1 3 4 C E

11. Algorithm for Minimal Spanning Tree Problem • Start with any node (source node is OK). Label it with 0. This is now a labelled node. • Find the closest unlabelled node to any labelled node. Label it with the preceding node. • Repeat until all nodes have been labelled. The minimal spanning tree can then be traced. Ties may be broken arbitrarily, but multiple optimal solutions can be identified by pursuing all ties.

12. Maximum Flow Problem • Find the max flow from Source to Sink. A 3 9 1 Sink 5 D 4 7 B Source 6 5 1 4 2 4 E C

13. The Augmenting Path Algorithm for the Maximum Flow Problem • Identify a flow augmenting path in the residual network (initially the original network), by checking all nodes connected to the source node with residual capacity along the arcs. If no augmenting path exists, the current flow is optimal. • Identify the residual capacity c* of this augmenting path by finding the minimum residual capacities along all arcs in the path. Increase the flow in this path by c*. • Decrease the residual capacities on each arc in the path by c* (if arc is in the opposite direction, increase its residual capacity by c*). Go to step 1.

14. Minimum Cost Flow Problem Demand nodes Supply nodes Transshipment nodes Let xij be flow though arc i→j.