1. Mathematics in Management Science
Review for Exam 1
Chinese Postman Problem
Traveling Salesman Problem
Network Problem
Critical Path Analysis
Mathematics in Management Science
Spring 2015

2. pencils and a valid UC ID .
Exam 1 – Wed Feb 11
Chapters 1 & 2.
Please bring
pencils and a valid UC ID .
Please know your
Section # and M # .
No phones allowed. 
Seats will be assigned – come early.

3. Please sit in every other seat starting in the front rows.
Exam 1 – Wed Feb 11
Please sit in every other seat starting in the front rows.
Please be certain to include your:
M # , Version Letter
Please have your ID out.
No phones out.

4. Example
What are critical paths? What is ECT?

5. Example
Spp task 6 time reduced by 4. What is new ECT? 3

6. Example
Spp task 7 time increased by 4. What is new ECT? 12

7. Example
What are critical paths and ECT for pictured job?

8. Chinese Postman Problem
Given connected graph: Find a minimal length circuit traversing entire graph.
If the graph has an Euler circuit, it is such a minimal length circuit. If not:
Eulerize the graph
Find an EC on new graph
Transfer back to original graph

9. Solving CPP – easy case
If the graph has an Euler circuit (or path), it is a minimal length circuit (or path) traversing the entire graph.
Euler’s Circuit/Path Theorems tells us exactly when a graph has EC/EP.
How do we actually find an Euler circuit or Euler path? (Fleury’s Algorithm)

10. Euler’s Theorem A graph has an Euler circuit if and only if
the graph is connected,
and,
all its vertices have even valence.

11. Fleury’s Algorithm
Check that the graph is connected and either has no odd vertices (for circuit), or has two odd vertices (for path).
Choose a starting vertex. For a circuit, can be any vertex; for path, it must be one of the two odd vertices.
At each step, if you have a choice, don’t choose a bridge of the yet-to-be-traveled part of the graph.

12. Solving CPP – harder case
What if there are no ECs or EPs?
No EC means: some odd vertex!
No EP means….(?)
No EC/EP: gotta retrace edge(s).
Eulerize the graph.
Find an EC (or EP) on new graph.
Transfer back to original graph.

13. Eulerizing Graphs – to get EC
If all vtxs have even, done (bcuz have an EC). Otherwise, have odd vtxs: mark them. (Must have an even number of odd vtxs!)
Pick an odd vtx. Duplicate an existing edge having this vtx as an endpoint.
Change degrees of two vtxs just connected. All degrees even – done; otherwise, repeat the above step.

14. Finding Good Eulerizations
Locate odd vertices. Try to duplicate any connecting edges between pairs of odd vertices.
Sometimes vertices are more than one edge apart; in this case, gotta duplicate edges between vertices.
Look for the fewest edges to duplicate to make all vertices even.

15. Solving a TSP TSP Brute Force Algorithm Nearest Neighbor Algorithm
Given complete weighted graph, find a minimum cost Hamiltonian circuit.
Brute Force Algorithm
Nearest Neighbor Algorithm
Repetitive Nearest Neighbor Algorithm
Sorted Edges Algorithm

16. Nearest Neighbor Algorithm
From a given vertex, go to the nearest vertex not already visited.
Repetitive Nearest Neighbor
Do NNA with each vertex of the graph as the starting vertex.
Of all the circuits obtained, keep the best one.
If there is a designated starting vertex, rewrite this best circuit using that vertex as the reference point.

17. Sorted Edges Algorithm
Pick the cheapest link (edge with smallest weight) available.
Continue picking next cheapest link available that does not
close a circuit which is not an HC, nor
create three edges coming out of a single vertex.
Done when get a Hamilton circuit.

18. CPP versus TSP
CPP
Given a connected graph, find a minimal length circuit (or path) traversing all edges of the graph.
TSP
Given a complete weighted graph, find a minimum weight Hamilton circuit (visiting all vtxs just once).

19. Find a minimal cost spanning tree in a given weighted graph.
Network Problem
Find a minimal cost spanning tree in a given weighted graph.
Of all possible spanning trees, the one whose sum of the associated weights is as small as possible is called a
minimum-cost spanning tree.

20. Use the edges, in order of increasing cost, so that
Kruskal’s Algorithm
Use the edges, in order of increasing cost, so that
no circuits get formed
Done when
all vertices on some edge
connected

21. Critical Paths
Look at all paths; each has an associated completion time (the sum of all processing times for tasks along the path).
A path with the longest processing time is called a critical path.
Can find these via backflow algorithm; we use eyeball approach.

22. Earliest Completion Time
The shortest possible time it takes to complete the overall project. You cannot do better than this.
A major factor in the overall cost.
Just the length of a critical path:
ECT=length of critical path