Presentation escc 2016
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The document summarizes a presentation on a proposed hybrid approach to solve the multi-modal journey planning problem. The approach combines mathematical programming and heuristic methods like Dijkstra's algorithm. It develops a mixed integer linear program model to minimize travel time and environmental cost. Future work aims to improve the algorithm by reducing the model's dimensionality and constraints to enhance computational speed for online applications.
Presentation escc 2016
- 1. 3rd International Conference on “Energy, Sustainability and Climate Change” ESCC 2016 A presentation of the “Journey planning problem” By Dimitrios Rizopoulos for GreenYourMove team Email: dimrizopoulos@gmail.com 1 With the contribution of the LIFE programme of the European Union - LIFE14 ENV/GR/000611
- 2. Presentation structure General description of the multi-modal journey planning(MMJP) problem Characteristics of MMJP and previous work The proposed solution approach The mathematical programming model that our team has developed Future work 2
- 3. The multi-modal journey planning problem & similar problems The journey planning problem: The computation of an optimal, feasible and personalized journey from a starting point A to an ending point B, where A and B are nodes of a transportation network. Similar problems: • Shortest path problem • Earliest arrival problem • Range problem • Multi-criteria JP problem (environmental cost, CO2 emissions, financial cost, travel time, arrival time, comfort of travel) 3 The multi-modal journey planning problem: Mostly in public transportation networks, the multi-modal journey planning problem (MMJP) seeks for journeys combining schedule-based transportation (buses, trains) with unrestricted modes (walking, driving).
- 4. Characteristics of the MMJP problem Characteristics: • Increasing popularity due to strong practical interest & increasing availability of data. • General Transit Feed Specification(GTFS), which defines the file formats. (series of text file describing different aspects of the data) It is supported by Google and TriMet since 2005. • Many open source initiatives that help us deal with the MMJP problem. 4
- 5. Previous work Extensive work has been conducted for route planning in static networks: Solved using shortest path algorithms : A*, Dijkstra’s, Hierarchical techniques Most of the approaches are based on heavy precomputation of paths Modern MMJP applications need to use data from public transport, which are schedule based and dynamic networks(traffic) and calculate paths for different criteria. 5 Problems that occur with time-expanded graphs: - Need to do the vast precomputations that they are based on frequently - Need to do precomputations for each mode of transport and each criteria and then get to combine them
- 6. Proposed method We propose a hybrid approach where we get to combine mathematical programming with some heuristic methods in order to achieve the desired results both in terms of “paths” generated by the algorithm and speed of calculations. We get to combine a mixed integer-linear program with Dijkstra’s algorithm and graph partitioning(for unrestricted modes e.g. walking) and graph selection techniques. Dijkstra’s algorithm calculated the parts of the solution that are needed to be fast and are not characterized by big margins between the optimal and the heuristically calculated solution. MILP program is used to solve the MMJP problem. 6
- 7. Proposed method 7 The user inserts the starting and ending points as well as the departure time of his journey Dikjstra's algorithm is applied to find the closest public network node S (stop or station) to the starting point and the closest node T to the ending point2, creates a list of 3 points for S and T The mathematical model is built and solved in order to compute the optimal journey for all combinations of S and T The optimal journey minimizing both travel time and environmental cost is delivered to the user Selects sub-network according of stations ( ID & OSM)
- 8. The mathematical model 8 8 We use those indices to refer to make references between the different variables of the mathematical formulation: i Network’s stations j Network’s stations h Network’s stations k Mode of transport n Different itineraries Multi-dimensional constants of the formulation used to represent the data Ci,j,k Cost of transportation from i station to j station using mode k ΤΤi,j,k Travel time of the transportation from i station to j station using mode k ΤoDi,j,k,n Time of departure of the transportation from i station to j station using mode k and itinerary n Nomenclature of the single-dimension constants N Number of stations considered by the model M Number of modes of transport considered L Number of different sets of itineraries S Starting station S (user input) T Ending station T (user input) a Weight coefficient for cost b Weight coefficient for time DT User’s departure time AT User’s maximum arrival time WT1 Walking time 1 from starting point to entrance point in the network WT2 Walking time 2 from exit of the network to the final destination
- 9. The mathematical model 9 9 Decision variables Xi,j,k,n Binary decision variable, takes values 0 or 1, 1 when the transfer from station i to station j occurs , with mode k and itinerary n Si,j,k,n Positive integer decision variable, and is equal to the departure time of the transfer from i to j with k and n when it occurs minimize The objective function XTTC nkjikji N i N j M k L n kji ba ,,,,, 1 1 1 1 ,, *)**(
- 10. The mathematical model 10 10 Constraint Meaning We always need to start from starting point S We always need to go to the last station Transportation from one to another with any mode and any itinerary can happen only once 1 1 1 1 ,,, N j M k L n nkjSX 1 1 1 1 ,,, N i M k L n nkTiX Constraint Meaning There’s no need to visit S again, that is we make this decision unavailable. We never need to leave the final station T, so we make transfers from T unavailable. 0 1 1 1 ,,, N i M k L n nkSiX 0 1 1 1 ,,, N j M k L n nkjTX Tii N j M k L n nkjiX ,,1 1 1 1 ,,, Constraint Meaning When you visit a station you need to leave it too. This constraint does not apply to the starting and the ending station. TShh N j M k L n nkjh N i M k L n nkhi XX ,,,0 1 1 1 ,,, 1 1 1 ,,,
- 11. The mathematical model 11 11 Constraint Meaning When the transfer from i to j with k and n occurs then S variable needs to be equal to the corresponding ToD We make sure that if ToD is 0, which means that there is no available itinerary, transfer X can’t happen This constraint makes sure that there is time continuation in the problem. By using it we make sure that when you make a transfer in time from i to h with k and n, the next transfer from station h to j happens after the travelling time from i to h and h to j. By inserting this constraint into our mode we make sure that the departure time at the first node happens first in chronological order from the rest that are about to be calculated next nkjiM M X ToDSX nkji nkjinkjinkji ,,,),1(* )1(* ,,, ,,,,,,,,, nkjiToDX nkjinkji ,,,,,,,,, TShh N j M k L n nkjh N i M k L n nkhikhi N i M k L n nkhi S XTTS ,, )*( 1 1 1 ,,, 1 1 1 ,,,,, 1 1 1 ,,, SiM N j M k L n nkji N j M k L n nkji N j M k L n nkjS X SS ,*)1( 1 1 1 ,,, 1 1 1 ,,, 1 1 1 ,,,
- 12. The mathematical model 12 Constraint Meaning If there is no transfer between station i and station j with mode k and itinerary n then the corresponding variable S should be zero, too. The departure time from the semifinal station should be the biggest in a chronological order The departure time from starting station S should be bigger in a chronological order than the departure time of the whole trip plus the walking time WT1. Corresponding constraint for the last station the arrival time there, which equals departure time plus travelling time. We need the Arrival time at the destination to be bigger in chronological order than the walking time plus the arrival time at the last station. nkjiM XS nkjinkji ,,,* ,,,,,, Tj N i M k L n nkji N i M k L n nkTi SS ,0 1 1 1 ,,, 1 1 1 ,,, DTWT N j M k L n nkjSS 1 1 1 1 ,,, ATWT N i M k L n nkTikTi N i M k L n nkTi XTTS 2 )*( 1 1 1 ,,,,, 1 1 1 ,,,
- 13. Advantages & Future work Advantages Minimal pre-processing time Disadvantages Higher query times (for now of course) Future work Research is still ongoing for the improvement of our algorithm. The mathematical formulation is still under modification and there will changes in the algorithm. Replace high dimensional variables with variables of fewer dimensions Reduce the number of constraints Integrate decomposition techniques for the mathematical model Create a wrapper application that will serve as an API to the algorithm and will allow its use on online applications 13
- 14. Thank you for your attention! Any questions? Learn more at: http://www.greenyourmove.org/ #
Editor's Notes
- #3: NP hard (non deterministic-polynomial-time hard), at least as hard as np-complete
- #6: 1. Precomputation for each mode, precomputation for each criteria, precomputation often
- #8: If asked: Selection happens right now according to IDs of the route in GTFS & TRAINOSE DATA ALWAYS
- #10: Cijk is set to 0, or we could minimize SUMi,k,n of SiTkn
- #14: 10 minutes / 1.56 minutes