Oyama, Y., Hato, E., Scarinci, R., Bierlaire, M. (2017) Markov assignment for a pedestrian activity-based network design problem. The 6th symposium arranged by European Association for Research in Transportation (hEART), Haifa, Israel.
Coefficient of Thermal Expansion and their Importance.pptx
Yuki Oyama - Markov assignment for a pedestrian activity-based network design problem
1. 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
2. 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
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
3. Jury 29, 2016 3
Outline
Outline
1. Introduction
2. Pedestrian activity assignment
A) Methodology
B) Illustrative examples
3. Application to a network design
4. Conclusion
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
4. Jury 29, 2016 4
Introduction
Motivation
Describing the sequence of travels and activities in city centers
Time
Time-constraint
moving
staying
moving
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
Space
(About 1 km2 square)
5. Jury 29, 2016 5
Introduction
Motivation
Describing the sequence of travels and activities in city centers
Time
Time-constraint
moving
staying
stayingmoving
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
6. 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
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
7. 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)
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
Space
Time
A B
C
Space
(Network)
Time
8. 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
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
9. Jury 29, 2016 9
Outline
Outline
1. Introduction
2. Pedestrian activity assignment
A) Methodology
B) Illustrative examples
3. Application to a network design
4. Conclusion
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
10. Jury 29, 2016 10
Framework | network description
Spatial network: includes staying node/link where activities are performed
Pedestrian activity assignment
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
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
11. 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.
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
Pedestrian activity assignment Methodology
12. 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
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
Pedestrian activity assignment Methodology
13. 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
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
Pedestrian activity assignment Methodology
Time-space prism
14. 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
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
Pedestrian activity assignment Methodology
15. 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.
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
Pedestrian activity assignment Methodology
16. Jury 29, 2016 16
Network assignment
Assignment algorithm (forward)
kij
kij
kij
State flow:
Edge flow:
: flow of link (i,j)
Spatial link flow:
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
Pedestrian activity assignment Methodology
Input:
(Generating flow)
17. Jury 29, 2016 17
Outline
Outline
1. Introduction
2. Pedestrian activity assignment
A) Methodology
B) Illustrative examples
3. Application to a network design
4. Conclusion
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
18. 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:
Pedestrian activity assignment
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
Numerical example
19. 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
Pedestrian activity assignment
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
Numerical example
20. 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.
Pedestrian activity assignment
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
Numerical example
22. Jury 29, 2016 22
Outline
Outline
1. Introduction
2. Pedestrian activity assignment
A) Methodology
B) Illustrative examples
3. Application to a network design
4. Conclusion
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
23. 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
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
24. 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
Pedestrian activity-based network design
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
25. 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
Pedestrian activity-based network design
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
[Scarinci et al., 2017]
26. Jury 29, 2016 26
Acceptance criterion
existing
solution
new solution
accepted
new solution
rejected
Network update
(A neighborhood search)
Pedestrian activity-based network design
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
27. 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])
Pedestrian activity-based network design
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
28. 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
Pedestrian activity-based network design
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
29. Jury 29, 2016 29
Outline
Outline
1. Introduction
2. Pedestrian activity assignment
A) Methodology
B) Illustrative examples
3. Application to a network design
4. Conclusion
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
30. 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
Oyama, Y. et al. (UT&EPFL) September 12, 2017hEART 2017 in Israel
33. 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]
34. 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).
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Case study
Network design
Arc performance
Utility loss (gain) for identifying the worst (best) moving arc:
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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
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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)