Yuki Oyama - Markov assignment for a pedestrian activity-based network design problem

Markov assignment for a pedestrian
activity-based network design problem
Yuki Oyama*,
Eiji Hato, Riccardo Scarinci and Michel Bierlaire
*Department of Urban Engineering
School of Engineering, The University of Tokyo
oyama@bin.t.u-tokyo.ac.jp / yuki.oyama@epfh.ch
September 12, 2017
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel

Jury 29, 2016 2
Outline
Outline
1. Introduction
2. Pedestrian activity assignment
A) Methodology
B) Illustrative examples
3. Application to a network design
4. Conclusion

Jury 29, 2016 3
Outline
Outline
1. Introduction
A) Methodology
4. Conclusion

Jury 29, 2016 4
Introduction
Motivation
Describing the sequence of travels and activities in city centers
Time
Time-constraint
moving
staying
moving
Space
(About 1 km2 square)

Jury 29, 2016 5
Introduction
Motivation
Describing the sequence of travels and activities in city centers
Time
Time-constraint
moving
staying
stayingmoving

Jury 29, 2016 6
Introduction
Focus of study
Activity-based network design
OD demand is not known a priori, but is the subject of responses in
user itinerary choices to infrastructure improvements.
[Kang et al., 2013]
Upper level network design problem
Lower level activity path choice problem

Jury 29, 2016 7
Introduction
Activity path choice problem
For daily household activity
ü Deterministic
ü Pre-trip (static)
ü Low-resolution (zone-
based)
For pedestrian in city centers
ü Probabilistic
ü Sequential (dynamic)
ü High-resolution (network-
based)
Space
Time
A B
C
Space
(Network)
Time

Jury 29, 2016 8
Contributions
1. Modeling pedestrian activity
- Combinatorial choice of route, location and duration
- Dynamic decision making in time-space network
2. Algorithms for complicated computation
- Applying Markovian (recursive) route choice model
- Network restriction based on the time-space prism
3. Application
- A pedestrian activity-based network design problem
- Bi-level and multi-objective programming
Introduction

Jury 29, 2016 9
Outline
Outline
1. Introduction
A) Methodology
4. Conclusion

Jury 29, 2016 10
Framework | network description
Spatial network: includes staying node/link where activities are performed
Pedestrian activity assignment
Methodology
Discretized time: with a constant interval
Activity path: the sequence of states
y
Space
x
tTime
y
t
x
: Node
(a) (b) (c)
: Link
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20
21 22 23 24 25
t=1
t=2
t=3
t=4
t=5
move
from 8 to 9
from 9 to 14
at 14
from 13 to 8
move
from 14 to 13
move
move
stay

Jury 29, 2016 11
Framework | assumptions
1. Travelers are homogeneous, move by only walk. Walking speed
is constant.
2. Based on Markov decision process, traveler’s state always
changes into a connected state at each discretized time t.
3. Traveler’s decision is restricted by time-constraint T. As the
result of sequential state transition from 0 to T, an activity path is
obtained.
4. Initial state (source, s0 = (0,o)) and final state (sink, sT = (T,d))
are always given.
Pedestrian activity assignment Methodology

Jury 29, 2016 12
Network restriction
1o
d
2 3
4 5
6 7
8 9 10
3
4 5
7
8 9
5
0/4
1/3
1/3
1/3
2/2
3/1
2/2
3/1
4/0
2/2
1o
d
2 3
4 5
6 7
8 9 10
3
4 5
7
8 9
5
9 10
Ex.）
STEP1:
STEP2: State existence condition
STEP3:
Time-space constraint
State connection condition
Topological distance

Jury 29, 2016 13
Network restriction
y
x1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20
21 22 23 24 25
x
t
T=5
*Initial and final states
*
0
Figure: Illustrations of constrained networks by the time-space prism
Time-space prism

Jury 29, 2016 14
Activity path choice model
Based on the Markov decision process
yy
？
yy
t
xx
: time discount factor
Forward-looking
decision
Individual at state chooses next state that maximizes the sum of
• State transition utility
• Discounted expected utility
Myopic decision
: time-space prism constraint
Transition probability

Jury 29, 2016 15
Maximum expected utility of the prism
Backward induction
1. Initialize , and
2. Set , and calculate with Eq. (*)
3. Finish the algorithm if , otherwise return to Step 2.
!(*)
*If the Bellman equation is non-linear, the same method can be applied.

Jury 29, 2016 16
Network assignment
Assignment algorithm (forward)
kij
kij
kij
State flow:
Edge flow:
: flow of link (i,j)
Spatial link flow:
Input:
(Generating flow)

Jury 29, 2016 17
Outline
Outline
1. Introduction
A) Methodology
4. Conclusion

Jury 29, 2016 18
Model specification
: a vector of coefficients
: travel time on arcs [min]
: sidewalk width [m]
: shopping street dummy variable
: deviated function of staying utility,
: diminishing marginal utility
Utility function:
Numerical example

Jury 29, 2016 19
Network setting
Department store Gintengai mall
Okaido mall
City hall
Department store
Park
1 2 3 4
5 6 7
8 9
10 11 12 13
14 15 16 17
18 19 20 21
1o
3o
4o
2o
n : Node for move/stay
n : Node for only move
o : Start/End node
100mN
Numerical example

Jury 29, 2016 20
Result | activity path choices
Figure: The most frequent paths departing from node 18 with deferent discount rates
1
2
3
4
5
6
7
8
910
11
12
1314
15
16
17
18
19
20
21
1
2
3
4
5
6
7
8
910
11
12
1314
15
16
17
18
19
20
21
(a) (b)
45min.
Finish (total: 81min.)
Finish (total: 45min.)
45min.
30min.
Numerical example

Jury 29, 2016 21
Result | assignment results
1 2 3 4
5 6 7
8 9
10 11 12 13
14 15 16 17
18 19 20 21
1 2 3 4
5 6 7
8 9
10 11 12 13
14 15 16 17
18 19 20 21
1 2 3 4
5 6 7
8 9
10 11 12 13
14 15 16 17
18 19 20 21
5
0
15
25
10
1 4 7 13 15 17 18 19 20 21 1 4 7 13 15 17 18 19 20 21 1 4 7 13 15 17 18 19 20 21
: 100
Link flow
: 250
: 500
: 1000
(min./person)
(a) (b) (c)
Staying node number
Activity duration
Figure: Activity assignment results with variable values of time constraints.
Table: Input flow pattern of initial and final states
Numerical example

Jury 29, 2016 22
Outline
Outline
1. Introduction
A) Methodology
4. Conclusion

Jury 29, 2016 23
Problem definition
Optimizing configuration of the pedestrian network
Decision variable:
: sidewalk width on moving links
s.t.,
: the minimum (current) width [m]
: the possible maximum width [m]
Activity assignment
Decision variable: : link flow at time t
Pedestrian activity-based network design

Jury 29, 2016 24
Multi-objective functions
1. Maximizing total duration time of district [min.]
2. Minimizing total increases of sidewalk area [m2]
: time discretization unit
: edge flow
: moving link set
: link length
: widening width

Jury 29, 2016 25
Solution methodology
Initial solution & network configuration
STEP1: Activity assignment
STEP4: Network Update
STEP3: Acceptance identification
If new solution is accepted, we add it to
the set of Pareto front solution
STEP2: Evaluation of
1) Objective function
2) Arc perfoemance value
Finish iteration
Yes
No
[Scarinci et al., 2017]

Jury 29, 2016 26
Acceptance criterion
existing
solution
new solution
accepted
new solution
rejected
Network update
(A neighborhood search)

Jury 29, 2016 27
Result | Pareto front solutions
82800 82900 83000 83100 83200
01000020000300004000050000
Total duration time [min.]
Totalareaofwidenedsidewalk[m2]
Accepted
Rejected
A
Figure: Trade-off curve between sojourn time and widened sidewalk area (CPU time: 5599.69 [s])

Jury 29, 2016 28
Result | example solution
Figure: Variation of (a) a network configuration of an example solution A (b) activity
flow in case of the network
1 2 3 4
5 6 7
8 9
10 11 12 13
14 15 16 17
18 19 20 21
1 2 3 4
5 6 7
8 9
10 11 12 13
14 15 16 17
18 19 20 21
+4
+4
+4+4+4
+4+5
+3 +1+8+8
(a) (b)
1 4 7 13 15 17 18 19 20 21
5
0
15
10
Staying node number
Arc flow
Activityduration
[min./person]
: 100
: 250
: 500
: 1000

Jury 29, 2016 29
Outline
Outline
1. Introduction
A) Methodology
4. Conclusion

Jury 29, 2016 30
Conclusion
• Modeling pedestrian behavior
– A probabilistic and dynamic activity path choice model is
proposed based on the Markov decision process.
– Time-constraint and time discount factor are significant
parameters for pedestrian activities in city centers.
• Computable algorithm
– Markovian assignment is equivalent to the MNL model but does not
require path enumeration.
– Time-space prism-based network restriction removes
unreachable states in advance and reduces the size of path set.
• Network design
– A pedestrian activity-based network design is presented.
Conclusions

Jury 29, 2016hEART 2016 in Delft 31
Thank you for attention.

Jury 29, 2016 32
Network restriction
Methodology
Table: Restricted path set (24 paths)

Jury 29, 2016 33
Case study
Network standardization
(a)
(b)
1
1
2
1 2
1
: arc length [m]
: walking speed [m/s]
: interval of time discretization [s]
: minimum duration time of staying node [s]

Jury 29, 2016 34
Case study
Network design
Network Update
Remove-Random-Width
Add-Random-Width
Remove-Worst-Width
Add-Best-Width
Remove a unit width from an arc randomly selected:
s.t.,
s.t.,
Add a unit width from an arc randomly selected:
Likewise,
where the worst and best are defined with arc performance value (in the next slide).

Jury 29, 2016 35
Case study
Network design
Arc performance
Utility loss (gain) for identifying the worst (best) moving arc:

Jury 29, 2016 36
Case study
Network design
Parameters
: unit removal/additional width [m]
Solution
• Neighborhood structure of Network Update is selected randomly.
• Initial solution is full-equipped network.
: unit capital cost [Yen/m2]
: Iteration number

Jury 29, 2016 37
Case study
Pareto front search
0 200 400 600 800 1000
8280082900830008310083200
Iteration number
Totalsojourntime
0 200 400 600 800 1000
01000020000300004000050000
Iteration number
Totalareaofwidenedsidewalk
Figure: Variation of (a) total sojourn time and (b) total area of widened sidewalk in
iteration process.
(a) (b)

Yuki Oyama - Markov assignment for a pedestrian activity-based network design problem

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