Snow plows in Iowa city

1. Snow plows in Iowa city Graph theory at work

2. Project • Examine the procedure utilized by snow plows in Iowa City • Systemize and minimize routes • Review mathematical concepts involved • Look into how math concepts apply to this problem • Model • Apply to example section • Conclusions/Recommendations

3. Why this project • Winter of 2007-2008 • Unplowed areas • Results • Uneven roads • Unable to plow • Cracks and potholes

4. Importance • Environmental • Reduce Gas Consumption • Greenhouse gas emissions • Save money • Public • Complaints • City website • Safety

5. Importance • System • Current process • Downtown/Bus routes • Steep slopes • Flat secondary roads • Easy to teach • Little confusion

6. Math Background • Seven Bridges of Könisburg • Euler

7. Eulerian Circuits • Traverse each edge exactly once • Circuits exist if all vertices of even degree • Digraph: indegree equals outdegree for all vertices • Use here • If one exists, will be optimal route • More than one truck

8. Models • Multigraph • Vertices - intersections of roads • Edges – bidirectional streets • Directed arcs – one-way streets • For snow plows • Must traverse each lane of each road at least once • Digraph • Vertices – intersections of roads • Arcs – directed lanes

9. Chinese Postman Problem • Kwan Mei-Ko 1960’s • Goal: traverse every street in least distance • More general than bridge problem • If contains eulerian circuit, this is the shortest route • If not, solution can be found

10. Weighted Graphs • If not using city’s current priorities • Weights represent distance • Want Mininimum • Find degrees of all vertices in graph • Must be even number of vertices of odd degree • Handshake lemma • Find shortest weighted paths between these vertices • Draw duplicate edges along path • Will then have all even degrees • Create Eulerian Circuit

11. Weighted Graphs • If I choose to comply with current process • Assign weights to streets • Weights represent grade of street • Find maximal weighted paths first • Represent steep slopes • Follow by lower weighted paths • Flatter streets

12. Adjacency Matrix • Square matrix • Each row and column represents a vertex • ‘1’ in Xij if arc (edge) exists from i to j • ‘0’ otherwise • Will be used to find degree of vertex in multigraph • Sum of ones in vertex row/column (digraph)

13. Solution Methods • Traverse bus routes • Simple because already circuits • Divide city into sections (10) • Within each section, split roads into phases • Maintain city’s current priorities • Each phase • Create adjacency matrix • For vertices of odd degree, create connected graph with weights of shortest distance between • Find perfect matching • New edges along path • Find Eulerian circuit • Repeat for steep roads, flat roads

14. Example 8 Vertices 2,3,4,5,6,7 have odd degree 7 6 4 5 3 2 1

15. Example Assign weights (shortest distance) Find minimal matching 7 1 4 6 1 1 4 1 5 3 3 2

16. Example Duplicate red edges 7 1 4 6 1 1 4 1 5 3 3 2

17. Example 8 7 6 4 5 3 1 2

18. Example 8 7 6 4 5 3 1 2

19. Example 8 7 6 4 5 3 1 2

20. Example 8 7 6 4 5 3 1 2

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22. Example 8 7 6 4 5 3 1 2

23. Example 8 7 6 4 5 3 1 2

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25. Example 8 7 6 4 5 3 1 2

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27. Example 8 7 6 4 5 3 1 2

28. Example 8 7 6 4 5 3 1 2

29. Example 8 7 6 4 5 3 1 2

30. Conclusions/Recommendations • Divide the city into sections • Determine which streets fall into which phase • Determine distances between vertices • Create computer program • Takes in vertices/edges • Forms adjacency matrices • Finds degrees • Forms weighted matrix for vertices of odd degree • Minimizes matching • Duplicates these edges • Results in minimal distance path for each phase

31. Questions?