SMART Seminar Series: Improving Public Transport Accessibility via the Optimisation and...

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

,

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

,

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

,

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

,

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

1 0 0

1 0 0

0 1 0

1 0 0

0 1 0

0 0 1

x=

1

1

1

1

1

1

z = c1x1 + c2x2 + c3x3


Column Generation


1 0 0

1 0 0

0 1 0

1 0 0

0 1 0

0 0 1

x=

1

1

1

1

1

1

z = c1x1 + c2x2 + c3x3 Generate Column

1

0

1

0

0

1


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

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

z = c1x1 + c2x2 + c3x3 + c4x4 Generate Column

0

1

0

1

1

0


Column Generation


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

x=

1

1

1

1

1

1

z = c1x1 + c2x2 + c3x3 + c4x4 + c5x5


Column Generation


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

x=

1

1

1

1

1

1

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 + λcMr)

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 : λcMr +∑

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∈πti, Time Cost

cMr

=∑

i∈πmax{pi(cr−di),0}, MC.Cost.

s t

1

23

4

5

6


Subproblem








i∈π

(ti − wi)


i∈π(i)

ti ≤ ui, ∀i ∈ N .

∑

i∈π



Where: cr =∑

i∈πti, Time Cost

cMr

=∑


s t

1

23

4

5

6s

1


Subproblem








i∈π

(ti − wi)


i∈π(i)

ti ≤ ui, ∀i ∈ N .

∑

i∈π



Where: cr =∑

i∈πti, Time Cost

cMr

=∑


s t

1

23

4

5

6s

1

2


Subproblem








i∈π

(ti − wi)


i∈π(i)

ti ≤ ui, ∀i ∈ N .

∑

i∈π



Where: cr =∑

i∈πti, Time Cost

cMr

=∑


s t

1

23

4

5

6s

1

2

4


Subproblem








i∈π

(ti − wi)


i∈π(i)

ti ≤ ui, ∀i ∈ N .

∑

i∈π



Where: cr =∑

i∈πti, Time Cost

cMr

=∑


s t

1

23

4

5

6s

1

2

4

6


Subproblem








i∈π

(ti − wi)


i∈π(i)

ti ≤ ui, ∀i ∈ N .

∑

i∈π



Where: cr =∑

i∈πti, Time Cost

cMr

=∑


s t

1

23

4

5

6s

1

2

4

6 t

110101

Route Cost = cr + λcMr


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

1021

25

5

14

16

18

23

30

67

8

22

24

27

28

29

413

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)

Plot of the Vehicle Routing Solution(Obj: Minimise Total Travel Time and Missed Connection Cost): Using 6 vehicles.

9

11

17

67

8

22

24

27

28

29

2

3

1021

23

30

1

413

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

243

6

26

5

13

16

17214

11

1518

27

Total Travel Time Cost = 104,Missed Connections = 1,


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

14

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

14

7827





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

816 18

21

29

5

13

23

24

25

3

9

12

17

19

2

4 14

1528

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

315

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 Business0

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...

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