1. Network Flow Problems – Shortest Path Problem
Consider the following flow network:
d26 2 6
d12
d36
d23
d13 1 3
d56
d35
d34
d14
d45 4 5
The objective is to find the shortest path from node 1 to node n. The distance from node i to j, dij does not have to equal the distance from node j to i, dji.

2. Network Flow Problems – Shortest Path Problem
Dijkstra’s labeling algorithm: 9 2 6 3 6 2 7 1 3 3 3 1 4 3
Summary: Nodes are
permanently labeled or
temporarily labeled. All nodes that can be immediately reached from permanently labeled nodes are temporarily labeled. The label is the minimum of the sum of the distance from node 1 to the permanent node plus the distance to the temporary node. Start with node 1 as the only permanently labeled node. Find the temporarily labeled node with minimum distance. Make this node permanent. Repeat this process until all nodes are permanently labeled. 4 5

3. Network Flow Problems – Shortest Path Problem
Dijkstra’s labeling algorithm: 9 2
(3,1) 6 3 6 2 7 1
[0] 3
(7,1) 3 3 1 4 3 4
(4,1) 5
[ ] – permanent label
(d,f) – temporary label; d – distance, f - from node
L(0) = [0, 3, 7, 4, inf., inf.] – choose node 2 *

4. Network Flow Problems – Shortest Path Problem
Dijkstra’s labeling algorithm: 9 2
[3,1] 6
(12,2) 3 6 2 7 1
[0] 3
(5,2) 3 3 1 4 3 4
(4,1) 5
L(1) = [0, 3, 5, 4, inf., 12] – choose node 4
* *

5. Network Flow Problems – Shortest Path Problem
Dijkstra’s labeling algorithm: 9 2
[3,1] 6
(12,2) 3 6 2 7 1
[0] 3
(5,2) 3 3 1 4 3 4
[4,1] 5
(7,4)
L(2) = [0, 3, 5, 4, 7, 12] – choose node 3
* * *

6. Network Flow Problems – Shortest Path Problem
Dijkstra’s labeling algorithm: 9 2
[3,1] 6
(11,3) 3 6 2 7 1
[0] 3
[5,2] 3 3 1 4 3 4
[4,1] 5
(7,4)
L(3) = [0, 3, 5, 4, 7, 11] – choose node 5
* * * *

7. Network Flow Problems – Shortest Path Problem
Dijkstra’s labeling algorithm: 9 2
[3,1] 6
(10,5) 3 6 2 7 1
[0] 3
[5,2] 3 3 1 4 3 4
[4,1] 5
[7,4]
L(4) = [0, 3, 5, 4, 7, 10] – choose node 6
* * * * *

8. Network Flow Problems – Shortest Path Problem
Dijkstra’s labeling algorithm: 9 2
[3,1] 6
[10,5] 3 6 2 7 1
[0] 3
[5,2] 3 3 1 4 3 4
[4,1] 5
[7,4]
L(5) = [0, 3, 5, 4, 7, 10] – done
* * * * * *
Shortest Path:

9. Shortest Path Problem – Excel Solution

10. Shortest Path Problem – Cost Example
Equipment Replacement Problem:
d15
d14
d13
d35
d12
d23
d34
d45 1 2 3 4 5
 where Ki is purchase price in year i , Si is salvage value after i years in service, and ct is maintenance cost after t years of service.

11. Minimal Spanning Tree
Objective: Find the minimum distance such that all
nodes are visited once (i.e. no cycles). 9 2 6 3 6 2 7 1 3 3 3 1 4 3 4 5

12. Minimal Spanning Tree
One possible spanning tree, Z = 26 (note, no cycles). 9 2 6 3 7 1 3 4 3 4 5

13. Minimal Spanning Tree Possible applications:
Phone lines between cities.
Rail lines between cities.
Road networks.
Air conditioning ducts.
etc.. 9 2 6 3 7 1 3 4 3 4 5

14. Minimal Spanning Tree Algorithm:
Step 1 – Select an arbitrary node initially. Identify a node
that is closest and include the arc connecting these two nodes
in the spanning tree.
Step 2 – Out of the remaining nodes, determine the one that
is closest to a node already selected in the spanning tree.
Include the arc connecting these two nodes in the spanning
tree. Whenever there is a tie for the closest node, it is
broken arbitrarily.

15. Minimal Spanning Tree Start with node 1. Node 2 is closest. 9 2 6 3 67 1 3 3 3 1 4 3 4 5

16. Minimal Spanning Tree
Node 3 is closest to partial spanning tree {(1,2)}. Remove
any cycles. 9 2 6 3 6 2 1 3 3 3 1 4 3 4 5

17. Minimal Spanning Tree
Node 4 is closest to partial spanning tree {(1,2), (2,3)}.
Remove any cycles. 9 2 6 3 6 2 1 3 3 3 1 3 4 5

18. Minimal Spanning Tree
Node 5 is closest to partial spanning tree {(1,2), (2,3),(3,4)}.
Remove any cycles. Note, arbitrarily select (3,5) or (4,5). 9 2 6 3 6 2 1 3 3 1 3 4 5

19. Minimal Spanning Tree
Node 6 is closest to partial spanning tree {(1,2), (2,3),(3,4),(4,5)}.
Remove any cycles. Stop, no more nodes. 2 6 3 2 1 3 3 1 3 4 5
Minimal spanning tree is {(1,2), (2,3),(3,4),(4,5),(5,6)}. Z = 12.