3rd Conference on Sustainable Urban Mobility26-27 May, 2016, Volos, Greece

A Hybrid Approach to the Problem of Journey Planning with the Use of Mathematical Programming and Modern

Techniques

With the contribution of the LIFE programme of the European Union - LIFE14 ENV/GR/000611

Authors• Georgios K.D. Saharidis• Dimitrios Rizopoulos• Afroditi Temourtzidou• Antonios Fragkogios• Nikolaos Cholevas• Asimina Galanou• George Emmanouelidis• Chrysostomos Chatzigeorgiou• Labros Bizas

Department of Mechanical Engineering

Polytechnic School University of Thessaly

Presentation structure• GreenYourMove Project

• The journey planning problem

• Proposed method

• Conclusion

GreenYourMove Project• GreenYourMove (GYM) is a European Research Project co-funded by

LIFE, the EU financial instrument for the environment - LIFE14 ENV/GR/000611.

• GreenYourMove’s main objective is the development and promotion of a co-modal journey application to minimize GHG emission in Europe. GreenYourMove develops a multi-modal transport planner (both routing & ticketing system) considering all kinds of urban public transportation (urban and sub-urban buses, metro, tram, trolley, trains), where the user gets alternative routes combining more than one transport modes if necessary. The routes are the environmentally friendliest ones, since emissions are calculated for different scenarios

• Target: Creation of a web platform and smartphone app.

GreenYourMove Project• Partners:– University of Thessaly– AVMAP– CHAPS (Czech Republic)– EMISIA– PLANNERSTACK (The Netherlands)– TRAINOSE

• Total Budget: 1,245,052 €– Start: September 2015– Finish: August 2018

The journey planning problem• The computation of an optimal,

feasible and personalized journey from a starting point A to an ending point B. Generally, the important task is to calculate the path from a point A to another point B, such that the total distance/time travelled from A to B will be the minimum.

The journey planning problem• Multi-Modal Journey Planning (MMJP) – 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).

The journey planning problem• Multi-Modal Journey Planning (MMJP) – Earliest Arrival Problem (EAP): Given a source stop A, a

target stop B and a departure time T, the problem asks for a journey that departs from A, no earlier than T, and arrives at B as early as possible.

– Range Problem (RP): Takes as input a time range (e.g. 6-9am) and asks for the journey with the least travel time that depart within that range.

The journey planning problem• Multi-Modal Journey Planning (MMJP) – Multi-Criteria Problem (MCP): Different optimization

criteria:• The number of transfers • The total journey cost• Ecological footprint (Environmental MMJP): Minimization of

GreenHouse Gas (GHG) emissions • Etc.

Journey, which is the most environmental friendly

Proposed Method• Hybrid approach, because it is a combination of:– Dijkstra’s algorithm– Mathematical model for the MMJP

Proposed MethodThe 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 point

The mathematical model is built and solved in order to compute the optimal journey between S and T

The optimal journey minimizing both travel time and environmental

cost is delivered to the user

Proposed Method

Dijkstra’s algorithm

Mathematical Model

Proposed Method• Dijkstra’s Algorithm– The mostly known Shortest Path algorithm, which is a

label setting algorithm introduced by Dijkstra in 1959.

Source Target

Total distance: 28

Proposed Method• Dijkstra’s Algorithm– Scan nodes near Source and Target until you hop on a

stop of the Public Transportation network.

Proposed Method• Mathematical model for the MMJP– Mixed Integer Linear Program (MILP) in order to

compute the optimal journey between the departure and arrival stops of the public network.

– In-between those two stations, the model prompts the user to use up to a number of different modes of transport, depending on his/her input. While in the network, the user follows an optimal journey that minimizes the travel time and the environmental cost.

Proposed Method• Mathematical model for the MMJP– Indices• i, j, h Nodes of the network.• k Modes or transport.• n Trips.

Proposed Method• Mathematical model for the MMJP

– Data• Ci,j,k Environmental cost of moving from i to j with mode k.• TTi,j,k Time of transfer from i to j with mode k.• ToDi,j,k Time of departure of trip n with mode k from i to j.• N Number of nodes.• M Number of modes of transport.• L Maximum Number of trips in all available modes.• S The departure station. • T The arrival station.• a Coefficient in the objective function.• b Coefficient in the objective function.• DT Departure time of the user from the starting point.• AT Latest arrival time of the user to the ending point.• WT1 Walking Time from starting point to S.• WT2 Walking Time from T to ending point.

Proposed Method• Mathematical model for the MMJP– Decision variables• Xi,j,k,n Binary Variable used to represent whether a

transfer is made from i to j with mode k and trip n. X is equal to 1 when transfer is made and 0 when it is NOT.• Si,j,k,n Non-negative continuous variable used to

represent the departure time from i to j with k and n. If the transfer is NOT made S is equal to 0

Proposed Method• Mathematical model for the MMJP– Constraints

, , ,1 1 1

1N M L

S j k nj k n

X

1. Connection from point S to a next point happens

, , ,1 1 1

0N M L

i S k ni k n

X

2. Once a journey has departed from S, it will never go through S again

, , ,1 1 1

1N M L

i T k ni k n

X

3. A connection between any point and T is made

Proposed Method• Mathematical model for the MMJP– Constraints

4. The journey reaches the target node and it never departs from it again

, , ,1 1 1

0N M L

T j k nj k n

X

, , ,1 1 1

1, ,N M L

i j k ni k n

i i TX

5. The journey goes through each node at most once

, , , , , ,1 1 1 1 1 1

0, , ,N M L N M L

i h k n h j k ni k n j k n

h h S TX X

6. Whatever node of the network we visit, we have to leave from it as well

Proposed Method• Mathematical model for the MMJP– Constraints

7. When X is 1, then S gets equal to ToD. The constraint relaxes, when X is 0

8. The departure that corresponds to X will not happen if ToD is 0, meaning there is no transfer available

9. Time continuity in the problem

, , , , , ,, , , , , ,*(1 ) *(1 ), , , ,

i j k n i j k ni j k n i j k nM M i j k nS ToDX X

, , , , , ,, , ,i j k n i j k n

i j k nToDX

, , , , ,, , , , , ,1 1 1 1 1 1 1 1 1

( * ) , ,N M L N M L N M L

i h k i h k ni h k n h j k ni k n i k n j k n

h h S TS STT X

Proposed Method• Mathematical model for the MMJP– Constraints

10. The departure time from the first node is the smallest

11. It initializes S to 0 if there is no transfer between i and j with mode k and trip n

12. The time of departure from the node before T is the largest time of departure in the journey delivered

, , ,, , , , , ,1 1 1 1 1 1 1 1 1

(1 )* ,N M L N M L N M L

i j k nS j k n i j k nj k n j k n j k n

M i SS S X

, , ,, , ,* , , ,i j k ni j k n

M i j k nS X

, , , , , ,1 1 1 1 1 1

0,N M L N M L

i T k n i j k ni k n i k n

j TS S

Proposed Method• Mathematical model for the MMJP– Constraints

13. Uses the solution from Dijkstra’s algorithm, which we have run before the construction of the mathematical model.

It makes sure that the first departure from a node of the network happens after the departure time of the traveler plus the walking time from the starting point to the

station S

, , ,1 1 1

1N M L

S j k nj k n

WT DTS

Proposed Method• Mathematical model for the MMJP– Objective Function

Minimization of 2 criteria, the total environmental cost and the total travel time of the journey, which is proposed to the user. Coefficients a and b are predefined by the user.The Environmental Cost Ci,j,k is pre-computed for each arc i-j and mode k of the public transportation network. This pre-computation is made using emission calculation models, that take into consideration several parameters, such as the type of fuel (gasoline, diesel, electricity etc.) and the fuel consumption, which concern the vehicle of the public means of transport. Other parameters concern the trip, which the vehicle follows, such as the distance and the gradient between the stops.

, , , , ,, ,1 1 1 1

( * * )*N N M L

i j k i j k ni j ki j k n

Min z a bC TT X

Conclusion• Novel approach for solving the Environmental Multi-

Modal Journey Planning problem. The proposed hybrid algorithm combines the Dijkstra’s algorithm with a Mixed Integer Linear Program (MILP) in order to deliver the journey with the least travel time and environmental cost.

• Research is still ongoing for the improvement of our algorithm. The mathematical formulation is still under modification and there may be slight changes in the algorithm.

Conclusion• Future work– The objective function shall be split into two, one for the

minimization of the travel time and one for the minimization of the environmental cost. Thus, the computation of a Pareto set of optimal journeys will be possible so that the user has more options to consider.

– Addition of more objective functions for the minimization of number of transfers, the total fare of the journey etc.

Conclusion• Future work– A decomposition method, such as Benders

Decomposition method, shall be implemented on the MILP so that it will be easier to solve for big data sets.

– Finally, the integration of the algorithm in an online platform having data from the public transportation network of Greece will make possible the wide use of it by the passengers with a large benefit both for them and the environment

Questions

Thank you for your attentionLearn more at:

http://www.greenyourmove.org/