1. Finding Optimal Refueling Policies Shieu-Hong Lin Biola University

2. The Optimal Refueling Problem

3. Finding Optimal Refueling Policies Optimal refueling in fixed routes Optimal refueling in transportation networks Conclusions

4. Transportation in Fixed Routes A fixed route selected from vertex A to vertex B. Cost of gas varies depending on the vertices. Gas consumption varies between vertices. Minimal and maximal fuel levels allowed: L and U $3.50$3.70$3.60$3.00 6 gallons 8 gallons 4 gallons 4 gallons A B [L,U]=[0, 12] Initial level=0

5. Optimal Refueling in Fixed Routes $3.50$3.70$3.60$3.00 To minimize the total refueling cost, for each vertex along the route, should we stop to refuel and if so, how much fuel to purchase? 6 gallons 8 gallons 4 gallons 4 gallons A B [L,U]=[0, 12] Initial level=0

6. Optimal Refueling Polices $3.50$3.70$3.60$3.00 6 gallons 8 gallons 4 gallons 4 gallons A B [L,U]=[0, 12] Initial level=0 An optimal refueling policy: minimizes the total refueling cost, represented as a vector of fuel amounts. Example: for the depicted case below, is the optimal policy.

7. A Greedy Algorithm for Finding Optimal Refueling Polices From the starting vertex in the route: Case (1): if no cheaper fuel is reachable ahead given a full tank of fuel, fill the tank and move to the next vertex; Case (2): otherwise, fill in the minimum amount to reach the closest vertex with a cheaper fuel price, move to that vertex (no refueling in between).

8. A Greedy Algorithm for Finding Optimal Refueling Polices $3.50$3.70$3.60 $3.00 6 gallons 8 gallons 4 gallons 4 gallons A B [L,U]=[0, 12] Initial level=0 Case I Case II Case II Case II

9. Complexity for Optimal Refueling in Fixed Routes For every vertex v in the route, determine the farthest vertex far(v) reachable ahead given a full tank of fuel, and the closest vertex low(v) ahead with a cheaper fuel price.  Obviously O(n 2 ) time is sufficient  Can be done in O(n) time. [Lin, etc. 2007]

10. Key Properties of Optimal Refueling Policies in Fixed Routes If we arrive at vertex v at the minimal fuel level L, we will then either refuel to attain the maximal fuel level U at v, or refuel and then proceed to reach another vertex u at the minimal fuel level L without refueling in between.

11. Key Transition Properties of Optimal Refueling Policies in Fixed Routes If we refuel at vertex v to the maximal fuel level U, we will then refuel in some intermediate vertex v’ once and then either reach another vertex u at the minimal fuel level L or attain the maximal fuel level U at v’.

12. LU Transition Property An optimal refueling policy along any fixed route can be equivalently viewed as a series of transitions between the minimal and the maximal fuel levels among vertices (L  L, L  U, U  U, U  L) with a refueling stop involved in each transition.

13. Optimal Refueling in Transportation Networks Finding optimal refueling policies for all pairs of vertices in a given transportation network. –Need to consider all routes between vertices. –Find the best route and use the optimal policy for it.

14. Optimal Refueling in Transportation Networks

15. LU Transition Property An optimal refueling policy along any fixed route = a series of transitions between the minimal and the maximal fuel levels among vertices (L  L, L  U, U  U, U  L) with a refueling stop involved in each transition.

16. LU Transition Graph LU Vertices: (v, L) and (v, U) where v is a vertex is the transportation network. LU Edges:,,, where u and v are vertices in the transportation network. Length of edges: the minimal transition costs of the LU transitions in the transportation network.

17. Reduction to the Shortest Path Problem The LU transition graph can be constructed in O(n 3 ) time where n is the number of vertices. Finding all-pairs optimal refueling routes is equivalent to finding all-pairs shortest paths in the LU transition graph.

18. Complexity of Optimal Refueling in Transportation Networks Through the reduction to the shortest-path problem O(n 3 ) time for all-pairs optimal refueling O( n 3 log D ) time for all-pairs optimal refueling with the constraint of no more than D refueling stops used.

19. Related Research Khueller [ESA 07]: (Independently) O(nlog n) time for the fixed- route optimal refueling problem O(n 4 ) time for all-pairs optimal refueling O( min(D 2 n 3, Dn 3 log n ) ) time for all- pairs optimal refueling with the constraint of no more than D refueling stops used.

20. Related Research [Lin, etc., Operations Research Letters 07] O(n) time for the fixed- route optimal refueling problem

21. Our Contributions [ Lin, AAIM 08] A reduction to the shortest-path problem O(n 3 ) time for all-pairs optimal refueling O( n 3 log D ) time for all-pairs optimal refueling with the constraint of no more than D refueling stops used.

22. Optimal Refueling in Transportation Networks An optimal route may have loops and may include up to Θ(n 2 ) vertices.