SMART Seminar Series: Improving Public Transport Accessibility via the Optimisation and Synchronisation of Schedules for Key Transport Modes

Synchronisation of Key Travel Modes
within a Transport Hub
Dr Michelle Dunbar
SMART Infrastucture Facility,
University of Wollongong
May 26, 2015
Michelle Dunbar, UoW Synchronisation ofTravel Modes within a Transport Hub 1/35

Outline
Motivation for improving synchronisation in multi-modal transport.
Variations of the Vehicle Routing Problem (VRP).
A mathematical formulation for the VRPTW with heterogeneous
travellers.
Preliminary results.
Future directions + Application.

Motivation for Improved Synchronisation
In modern cities, transport infrastructure has typically developed according to a radial
pattern, in response to urban-sprawl.
Figure: Heatmap of population density in Sydney. Source: RP Data.
Population density increase may lead to inaccessibility to transportation services.
Infrastructure has traditionally developed separately and sequentially =⇒ lack of
complementarity and synchronisation between services at Hubs (△).

Passengers increasingly required to
make a number of interchanges at
Hubs, between diﬀerent transport
modes.
Excessive waiting-times, infrequent
feeder services =⇒ poor connectivity.
Long-term planning and coordination: A
key driver for environmentally and
ﬁnancially sustainable transport
development (Transport for NSW, 2012).

In areas without an existing transport infrastructure (such as an existing rail line),
buses are typically used to service the population.
- May be undesirable: ﬁxed routes, infrequent services =⇒ increased car usage.
One approach commonly used around the
world, is that of a Dial-a-Ride shuttle-bus
system. (e.g. SkyBus in Melbourne)
Mobile technology has allowed for ease-of-use
and uptake for services to major Transport
Hubs.

The Vehicle Routing Problem
The Vehicle Routing Problem: Visit each node exactly once in minimal time.
Source: http://neo.lcc.uma.es/dynamic/vrp.html

Existing Vehicle Routing Approaches
Vehicle Routing Problem (VRP), VRPTW (Vehicle Routing with Time Windows) and
DARP (Dial-a-Ride Problems).
1 Typically assume passengers/items are homogeneous w.r.t importance/priority,
2 Minimise total route time, cost or number of vehicles,
3 Solution approaches have typically utilised a combination of exact and heuristic
techniques (eg. tabu search),
4 Ignore the potential multi-modal aspect of a passenger’s trip.

Existing Vehicle Routing Approaches
Recent extensions include:
1 Multi-zone, multi-trip VRPTW to and from a
one or more depots (Crainic et al., 2012).
2 Heterogeneity of items and route-cost factors:
weight, volume, distance and number of stops.
(Cesseli et al., 2009.)
3 Exact solution techniques: Column generation.
(Ceselli et al., 2009)
4 Customer perceptions of quality of service:
waiting time at pick up node, trip length.
(Pacquette et al., 2013).
Figure 1: Example of a multi-zone, multi-trip solution.
vs
Figure 2: Non-Perishable vs Perishable items.

Contributions of Our Model
We extend the ideas of Ceselli et al. and Pacquette et al. Our approach:
1 Incorporates passenger heterogeneity with respect to value-of-time and importance of
outbound connection,
2 Minimises the time cost and missed connection cost,
3 Solved exactly via Column Generation.

Route-Based Formulation for the VRPTW
The VRPTW model may be formulated as a route-based model.
- Each route corresponds to a column of the coeﬃcient matrix and has an
associated decision variable.
xr =



1, If route r is chosen,
0, otherwise.
(1)
Example:








1
0
1


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

,

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
0
1
0








Passenger 1
Passenger 2
Passenger 3
Route 1 Route 2
s t1
2
3

xr =



0, otherwise.
(1)
Example:








1
0
1








,

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0
1
0








Passenger 1
Passenger 2
Passenger 3
Route 1 Route 2
s t1
2
3x1 x2

xr =



0, otherwise.
(1)
Example:








1
0
1



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



,

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
0
1
0








Passenger 1
Passenger 2
Passenger 3
Route 1 Route 2
s t1
2
3x1 x2
s 1
3
t
Route 1

xr =



0, otherwise.
(1)
Example:








1
0
1








,

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0
1
0








Passenger 1
Passenger 2
Passenger 3
Route 1 Route 2
s t1
2
3x1 x2
s
2
t
Route 2

Column Generation
Master Problem Subproblem

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1 0 0
1 0 0
0 1 0
1 0 0
0 1 0
0 0 1




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

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
x =





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
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
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







1
1
1
1
1
1



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





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

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



z = c1x1 + c2x2 + c3x3

Column Generation

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1 0 0
1 0 0
0 1 0
1 0 0
0 1 0
0 0 1








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
x =

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


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


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







1
1
1
1
1
1

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z = c1x1 + c2x2 + c3x3 Generate Column


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
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1
0
1
0
0
1
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

Column Generation

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
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


1 0 0 1
1 0 0 0
0 1 0 1
1 0 0 0
0 1 0 0
0 0 1 1






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
x =

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




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





1
1
1
1
1
1


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

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



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


z = c1x1 + c2x2 + c3x3 + c4x4

Column Generation




















1 0 0 1
1 0 0 0
0 1 0 1
1 0 0 0
0 1 0 0
0 0 1 1




















x =




















1
1
1
1
1
1













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



z = c1x1 + c2x2 + c3x3 + c4x4 Generate Column












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0
1
0
1
1
0
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

Column Generation

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


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



1 0 0 1 0
1 0 0 0 1
0 1 0 1 0
1 0 0 0 1
0 1 0 0 1
0 0 1 1 0






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

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
x =








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
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



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1
1
1
1
1
1

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


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
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
z = c1x1 + c2x2 + c3x3 + c4x4 + c5x5

Column Generation



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












1 0 0 1 0
1 0 0 0 1
0 1 0 1 0
1 0 0 0 1
0 1 0 0 1
0 0 1 1 0






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




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

x =




















1
1
1
1
1
1


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
z = c1x1 + c2x2 + c3x3 + c4x4 + c5x5
Terminate if we can ﬁnd no other
beneﬁcial columns.
(i.e. All reduced costs, cj ≥ 0)

Solving the VRPTW using Column Generation
Objective is to obtain a minimal time-cost assignment of available vehicles to
passenger pick-ups, ensuring each passenger is picked up within their speciﬁed
time-window.
Master Problem
Minimise :
r∈R
crxr
Subject to :
r∈R
airxr = 1 ∀i ∈ N
r∈R
xr ≤ N, xr ∈ {0, 1}
Subproblem
Generate a feasible vehicle
route (satisfying time window
and duration constraints).
Append to the set:
R = set of all possible routes.

Route-Based VRPTW with Passenger Heterogeneity
Objective is minimise both time cost and missed (outbound) connection cost
at the Transport Hub, whilst ensuring each passenger is picked up within their
speciﬁed time window.
Master Problem
Minimise :
r∈R
cr + λcM
r xr
Subject to :
r∈R
airxr = 1 ∀i ∈ N
r∈R
xr ≤ N, xr ∈ {0, 1}
Subproblem
Generate a feasible vehicle
route (satisfying time window
and duration constraints).
Append to the set:
R = set of all possible routes.

Subproblem
A Label-Setting algorithm is used to determine the path with minimal reduced cost:
- Let π denote a path from source to sink.
- Let ti be the time-cost incurred by travelling from the predecessor node π−(i) to node i.
- Let wi be the dual multiplier for node i.
- Let li and ui denote the lower and upper bounds of the time window for node i.
Subproblem Formulation
Minimise : λcM
r +
i∈π
(ti − wi)
Subject to: li ≤
i∈π(i)
ti ≤ ui, ∀i ∈ N.
i∈π
ti ≤ Tmax, ∀i ∈ N
π is a path from s to t.
Where: cr = i∈π ti, Time Cost
cM
r = i∈π max{pi(cr−di),0}, MC.Cost.
s t
1
2
3
4
5
6

Subproblem
Minimise : λcM
r +
i∈π
(ti − wi)
Subject to: li ≤
i∈π(i)
ti ≤ ui, ∀i ∈ N.
i∈π
cM
s t
1
2
3
4
5
6s
1

Subproblem
Minimise : λcM
r +
i∈π
(ti − wi)
Subject to: li ≤
i∈π(i)
ti ≤ ui, ∀i ∈ N.
i∈π
cM
s t
1
2
3
4
5
6s
1
2

Subproblem
Minimise : λcM
r +
i∈π
(ti − wi)
Subject to: li ≤
i∈π(i)
ti ≤ ui, ∀i ∈ N.
i∈π
cM
s t
1
2
3
4
5
6s
1
2
4

Subproblem
Minimise : λcM
r +
i∈π
(ti − wi)
Subject to: li ≤
i∈π(i)
ti ≤ ui, ∀i ∈ N.
i∈π
cM
s t
1
2
3
4
5
6s
1
2
4
6

Subproblem
Minimise : λcM
r +
i∈π
(ti − wi)
Subject to: li ≤
i∈π(i)
ti ≤ ui, ∀i ∈ N.
i∈π
cM
s t
1
2
3
4
5
6s
1
2
4
6 t




1
1
0
1
0
1




Route Cost = cr + λcM
r

Preliminary Numerical Results
We randomly generated 4 different datasets with characteristics reflecting ‘likely’
passenger compositions according to time-of-day requests.
- School commute (≈ 8am/3pm),
- Balanced number of requests (≈ 11am/2pm),
- Inter-city commute (≈ 7am/5pm),
- Business commute (≈ 6am/6pm).
For each of these datasets, we simulated 10 random instances with different passenger
outbound connection departure times at the Hub, to determine the effectiveness of our
algorithm.
Each dataset consists of 30 passengers, with the restriction that vehicles must return to
the Hub in <= 20 mins.

Preliminary Numerical Results
In the slides that follow, we compare:
- The Base Case (Min TTT): Objective is to minimise Total Travel Time (TTT).
- Our Model (Min TTT+MC): Objective is to minimise Total Travel Time and Missed
Connection Cost.
We compare the following quantities:
- Time Cost,
- Missed Connection Cost,
- Time Cost + Missed Connection Cost (weighted),
- Total Cost (includes additional vehicle cost ($40/20 min), if required).

Results: School (5,5,15,5)
−5 −4 −3 −2 −1 0 1 2 3 4 5
−5
−4
−3
−2
−1
0
1
2
3
4
5
Distance (kms)
Distance(kms)
Plot of the Vehicle Routing Solution
(Obj: Minimise Total Travel Time): Using 6 vehicles.
1
2
3
10
21
25
5
14
16
18
23
30
6
7
8
22
24
27
28
29
4
13
15
19
9
11
17
26
1220
Total Travel Time Cost = 89,
Missed Connections = 4,
Missed Connection Cost = 310,
Weighted Cost Sum = 244.
Hub Aircraft Connection: 5 Inter−city Train Connection: 5 Bus Connection: 15 Walk Connection: 5
−5 −4 −3 −2 −1 0 1 2 3 4 5
−5
−4
−3
−2
−1
0
1
2
3
4
5
Distance (kms)
Distance(kms)
(Obj: Minimise Total Travel Time and Missed Connection Cost): Using 6 vehicles.
9
11
17
6
7
8
22
24
27
28
29
2
3
10
21
23
30
1
4
13
15
19
1220
25
5
14
16
18
26

Results: Balanced (10,10,10,0)
−5 −4 −3 −2 −1 0 1 2 3 4 5
−5
−4
−3
−2
−1
0
1
2
3
4
5
Distance (kms)
Distance(kms)
1
6
12
19
28
29
3
11
15
17
18
21
26
5
14
25
30
4
13
16
27
2
20
9
10
22
23
24
7
8
Hub Aircraft Connection: 10 Inter−city Train Connection: 10 Bus Connection: 10
−5 −4 −3 −2 −1 0 1 2 3 4 5
−5
−4
−3
−2
−1
0
1
2
3
4
5
Distance (kms)
Distance(kms)
141
12
19
28
29
7
25
30
2
20
8
9
10
22
23
24
3
6
26
5
13
16
17
214
11
15
18
27
Weighted Cost Sum = 124(164).

Results: Inter-city Commute (5,20,5,0)
−5 −4 −3 −2 −1 0 1 2 3 4 5
−5
−4
−3
−2
−1
0
1
2
3
4
5
Distance (kms)
Distance(kms)
6
12
18
30
9
11
15
16
26
2
14
21
22
23
13
17
25
3
5
10
28
1
4
78
1920 24
27
29
Weighted Cost Sum = 348.5.
Hub Aircraft Connection: 5 Inter−city Train Connection: 20 Walk Connection: 5
−5 −4 −3 −2 −1 0 1 2 3 4 5
−5
−4
−3
−2
−1
0
1
2
3
4
5
Distance (kms)
Distance(kms)
1920 24
29
3
5
10
17
13
25
28
2
14
21
22
23
9
11
15
16
26
6
12
18
30
1
4
78
27

Results: Business (15,10,5,0)
−5 −4 −3 −2 −1 0 1 2 3 4 5
−5
−4
−3
−2
−1
0
1
2
3
4
5
Distance (kms)
Distance(kms)
1
6
10
22
27
8
16 18
21
29
5
13
23
24
25
3
9
12
17
19
2
4 14
15
28
7
1120
26
30
Hub Aircraft Connection: 15 Inter−city Train Connection: 10 Bus Connection: 5
−5 −4 −3 −2 −1 0 1 2 3 4 5
−5
−4
−3
−2
−1
0
1
2
3
4
5
Distance (kms)
Distance(kms)
1
6
26
810
22
27
2
4 14
18
29
3
15
21
28
5
13
23
24
25
1120
30
9
12
17
19
7
16

Average Costs for each Dataset over 10 Random Instances
Dataset
TotalCost($)
A Comparison of the Average Total Costs between Min TTT and Min TTT+MC over 10 Instances for each Dataset
School Balanced Inter−City Business
0
50
100
150
200
250
300
350
400
450
Time Cost (Min. TTT)
Missed Connection Cost (Min. TTT)
Additional Vehicle/Driver Cost (Min. TTT)
Time Cost (Min. TTT+MC)
Missed Connection Cost (Min. TTT+MC)
Additional Vehicle/Driver Cost (Min. TTT+MC)

Results: Average Percentage Improvement
Average percentage improvement of Min TTT+MC over the Min TTT approach.
Travel Cost MC Cost Weighted Sum Total Cost
School -1.35 100.00 55.67 55.67
Balanced -11.16 94.75 59.38 45.40
Inter-City -8.49 100.00 57.10 36.87
Business -6.705 95.41 73.76 57.37
Average -6.93 97.54 61.48 48.83

Remarks and Conclusions
Using the same number of vehicles, it is possible to obtain a route with a 100%
decrease in missed connection cost, for only a 1.3% increase in time cost.
Over all 4 dataset types, the average reduction in missed connection cost was
between 94 − 100%.
Over all 4 dataset types, the Min TTT+MC approach outperformed the Min TTT
approach, even when (≤ 2) additional vehicles costs are accounted for, by an
average of 48.83%

Future Directions
Inclusion of additional passenger-centric measures.
Application to perishable-good delivery problem (eg. just-in-time delivery).
Figure: Routes for the delivery of spare parts from CP to Drop-points for Sydney Network.
Incorporation of real-time (offline) traffic data for specific time-of-day.
Extend to include delay information.

Application to Vehicle Logistics Company: DropPoint
How to reduce distribution time from CP → DPs?

Sydney Network.

Subproblem: Minimise reduced-cost, subject to:

Incorporate a variable link travel-time reﬂecting time-of-day information.

A step-function approximation for given data granularity.

How does our model know which time-of-day dataset to use?

We use a linearised model of a Heaviside step function.

For example, if the current time is 1 : 30pm, but have discretisations of 1
hour:

This results in:

This will be used to select the correct travel-time across a link, on-the-ﬂy.

Conclusion
Thank you!

SMART Seminar Series: Improving Public Transport Accessibility via the Optimisation and Synchronisation of Schedules for Key Transport Modes

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