Developing a Mathematical Model of River Travel

1. Developing a Mathematical Model of River Travel Gavin Stewart and Benjamin Spitz under Charles Stoddard

2. Definitions • k - The least number of possible next moves for a boat, unless the boat can reach the end of the river from an earlier campsite • Y - The number of campsites on the river

3. Our Problem • Based on the 2012 MCM problem • Maximize the number of boats traveling down a river • Constraints: • 6-18 day trip time • 12 hours of travel in a day (8AM-8PM) • Only one boat at each campsite • Known values: • 8 mph boat speed • 180 day boating season • Y campsites • River 225 miles long

4. Assumptions • There exists a path from the start to the end of the river with each stop being at most 96 miles from the last. • Ensures that any boat can traverse the river (96 is the number of miles a boat can travel in an optimal 12 hour traveling period). • Except during the first and last six days, an equal number of boats are on the river at all times. • Conservation of flow allows for network modeling techniques.

5. Generating Functions • Take a sequence and change it into a polynomial • Coefficients represent the presence of boats • 1 for occupied • 0 for empty • Exponent represents 0-indexed stop number • Powers of x greater than or equal to Y represent boats which have finished.

6. Generating Functions as Schedules • Two operations • Multiplying by x shifts boats over one stop • Adding one to a polynomial with a zero constant term moves a boat from the start of the river to the first stop. Day 1 Day 2 Day 3 1 + 0x + 0x2 … 1 + 1x + 0x2 … 1 + 1x + 1x2 …

7. Digraphs Directed Graph Type of Multigraph Each edge has a direction of travel. Graph Theory Multigraphs • Have vertices • Edges connect certain vertices • Any number of edges can connect vertices. Graphs • Type of Multigraph • Has only one edge between any two vertices.

8. The River as a Digraph • Call the digraph representation of the river R. • Beginning, end, and each stop are vertices. • Indices start at 0 and end at Y+1 • The edge (a,b) exists if a is at most 96 miles before b. • The river is considered solvable if: • There is at least one path from the beginning to the end.

9. Bottlenecks • Two types: • River Configuration • Time Requirement • The strictest bottleneck determines the flow of boats down the river.

10. Upper Bound When Ignoring Distance • All conditions identical, except we assume a boat can travel from any stop to any other stop • At most Y boats can finish the river in six days • If more than Y boats finish, multiple boats must occupy the same stop

11. Upper Bound When Ignoring Distance • Use generating functions • Form: • Initial Term: • First term moves boats on river • Second term moves boats from start onto river

12. Upper Bound When Ignoring Time • All constraints the same, except boats can finish in any number of days • Let k be the minimum outdegree of a stop which has no stops before it connecting to the end of the river. • At most k boats can finish the river in a day. • Approach: • Give each vertex of a river graph unit weight. • Construct a flow network with node splitting. • By max-flow min-cut, at most k boats can travel down the river

13. 1 1 1 0 2 2 Calculation of k

14. Upper Bound When Ignoring Time

15. General Upper Bound • Actual problem has same restrictions when ignoring time and distance • Must have the same upper bounds as the two less constrained problems • Max number of boats finishing in six days: • Consider two types of rivers:

16. Travel time restriction stricter than flow restriction • Max flow in six days is Y boats • First six days no boats finish • Stops spaced evenly enough to warrant generating function approach

17. Intuitive Development: Uniformly Spaced Stops • Stops divide river into segments • Each segment miles long • Boats travel stops per day

18. Intuitive Development: Uniformly Spaced Stops • Generating Function: • Sends boats down when is not a multiple of six • Sends more boats every sixth day when is not a multiple of six

19. Intuitive Development: Uniformly Spaced Stops • Schedule is legal • Boats take six days to finish • Equality Always Holds • Boats do not travel over 96 miles in a day • Farthest traveled:

20. Intuitive Development: Uniformly Spaced Stops • Schedule is Optimal • Y per six day period • 29Y in a season

21. Non-Uniformly Spaced Stops • Previous schedule may not work • May send out too many boats in a single day • Spread out these boats over multiple days

22. Non-Uniformly Spaced Stops • Schedule Legal • Boats take 6 days to finish • Proof similar to the one for uniform • Summation of evaluates to Y • Boats always travel no more than k stops • When 6 divides k, • Otherwise,

23. Non-Uniformly Spaced Stops • Schedule is optimal • In six days, Y boats travel down the river, with the exception of the first six days • 29Y boats finish

24. Flow restriction stricter than travel time restriction • Max flow in six days 6k boats • Requires graph theory

25. Divide river into subgraphs so exactly one boat can finish each day after the first six • No vertices except the start and end shared • The largest set of these river subgraphs contains exactly k subgraphs. • Cannot contain more than k; removing k vertices makes the river unsolvable. • We can construct a set of size k indexed by r • Edges are in the subgraph whenever they are in the parent graph

26. Efficiency • Boats can complete the river in six days • Boats travel over 96 miles in two days • River is only 225 miles • Boats do not finish in the first six days • Boats finish for 174 of the 180 days • 174k boats can finish in a season

27. Acknowledgements • Charles Stoddard, Mentor and Resident Advisor • Xcel Energy Foundation and 1984 Alumni, Sponsors • The Kinder Morgan Foundation and the Spitz Family, Sponsors • Nicholas Horianopoulos, Advisor and Instructor • Krystal Meisel, Writing Advisor • Lori Ball, FSI Program Director • The Math and Science Teaching Institute (MAST) • Maurice Irving Woods III and Anthony Gandara, Instructors • Karen Allnutt, Resident Advisor • Other FSI and MAST Staff • Meghan Combs, poster assistance and co-mentee