Vehicle Routing Problem for Journey planning

1. ESCC 2016
Vehicle Routing Problem
Dr. Georgios K.D. Saharidis
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
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14. Thank you for
your attention!
Any questions?
Learn more at:
http://www.greenyourmove.org/ #