2nd ISM-ZIB-IMI MODAL Workshop on Mathematical Optimization and Data Analysis

1. ⋇ Graduate School of Mathematics, Kyushu University
† Institute of Mathematics for industry, Kyushu University
Akira Tanaka⋇
Katsuki Fujisawa †
September 26, 2017
2nd ISM-ZIB-IMI MODAL Workshop
Practical Approach to Evacuation Planning
Via Network Flow and Deep Learning

2. Evacuation Map
Yodogawa Area in Osaka City
#nodes : 2,933
#edges : 8,924
#evacuees : 50,000 ~ 80,000
#shelters
with capacity : 36 (total volume:35,549)
Without capacity : 50 (each volume:∞)
Based on
midnight population
Higher Grounds
Volume: ∞
A Public Housing
Volume : 77
Higher Grounds
Tall and Strong Buildings
shelter with Capacity
shelter without Capacity
Osaka City Yodogawa Area

3. Lexicographically Quickest Flow (LQF)
• Minimizes the evacuation time Θ∗
• Greedily maximize the cumulative
number of evacuees who have already
completed evacuation at every time
𝜃 in the order of 𝜃 = 1,2, . . , Θ∗s
Definition LQF
Quickest Flow
Algorithm Time
EvacuationCompletion(%)
• Computing the maximum flow for time-
expanded graphs repeatedly
• Not practical(7h for the target area)
• Size of time-expanded graph is huge
• Compute the maximum flow repeatedly

4. Visualization of LQF
Movement of evacuees are expressed
with HSV color model
• Hue ↔ direction of movement
• Saturation ↔ speed of movement
• Value ↔ congestion of arcconstant
Spots will vanish
when the corresponding
evacuees reach shelters

5. Proposed Model
The number of evacuees
moving in each basic direction
Evacuation
Completion
Time(ECT)
Movements of
Each Evacuee
OutputInput
𝟑. 𝟕𝟏 × 𝟏𝟎−𝟒
𝐬𝐞𝐜𝐨𝐧𝐝𝐬
(Emergency)
Preprocessing
Techniques
Deep Learning
(Convolutional
Neural
Network)
・
Population
Distribution
Evacuation
Plan
(including
ECT)
Time-Expanded
Graph
LQF
Training
(Normal Time)
① ②
③

6. Preprocessing Techniques
Movement of
Each Evacuee
𝑥𝑖, 𝑦𝑖:Coordinate of evacuee 𝑖
𝜃𝑖: Direction of movement
of an evacuee 𝑖
Deep Learning Input
𝑓: 𝐶, 𝐻, 𝑊 ∈ ℕ+
3
→ ℝ
Resolution of Vector
Geometry Modification
𝜃𝑖
𝑥𝑖, 𝑦𝑖
𝑥𝑖, 𝑦𝑖, 𝜃𝑖
C:Channel↔Direction
H:Height
W:Width
↔ Region
𝑓 𝑐, ℎ, 𝑤 :
Number of evacuees
moving in direction c
on (h,w) region
2 preprocessing Techniques

7. Preprocessing Techniques
Movement of
Each Evacuee
𝑥𝑖, 𝑦𝑖:Coordinate of evacuee 𝑖
𝜃𝑖: Direction of movement
of an evacuee 𝑖
Deep Learning Input
𝑓: 𝐶, 𝐻, 𝑊 ∈ ℕ+
3
→ ℝ
Resolution of Vector
Geometry Modification
𝜃𝑖
𝑥𝑖, 𝑦𝑖
𝑥𝑖, 𝑦𝑖, 𝜃𝑖
C:Channel↔Direction
H:Height
W:Width
↔ Region
𝑓 𝑐, ℎ, 𝑤 :
Number of evacuees
moving in direction c
on (h,w) region
2 preprocessing Techniques

8. Resolution of Vector(D=4)
𝜃𝑖
Original Vector
Object
• Avoid cancellations of vectors
• Compact the movements of evacuees
①
Basic Vectors
𝜋
2
1
②
𝜃𝑖
1
Original Vector
Resolved Basic Vectors
Normalized Basic Vectors
④ 1
Resolution
of Vectors
Without
Resolution
of Vectors
Before After
③
1
D=4
𝜃𝑖 ≤
𝜋
2

9. Geometry Modification
Object: Reflect the geometric information delicately
𝑅3 = 𝑽 𝟏× 𝐑 ∩ 𝐑 𝟑 ÷ 𝑹
𝐑 ∩ 𝐑 𝟑
𝐕𝟏
R
Original
Vector
Normalized Basic Vectors
R
𝑅1
𝑅2
𝑅3 𝑅4
Original Vector
Normalized Basic Vectors
① ②
③ ④
𝑅3

10. How to Make Deep Learning Input
LQF_T
=1,500
LQF_T: The remaining time for the evacuation completion by seconds
Movements of
Each Evacuee
Preprocessing
Techniques Model Input
Deep Learning Input(Numbers of evacuees moving in specific directions)
𝜃𝑖
𝑥𝑖, 𝑦𝑖
HSV Image
HSV Color Model
Resolution of Vector

11. Main Points of Preprocessing Techniques
Resolution of Vector
Geometry Modification
Reflect the geometric
information delicately
Compacting the movements of
evacuees without reducing the
amounts of information

12. Convolutional Neural Network(CNN)
Convolutional Layer Max Pooling Layer Fully Connected Layer
𝑯
𝑪
𝑾
The number of evacuees
moving in each basic direction
Input Output
=
Evacuation
Completion
Time(ECT)
4 Convolutional and
2 Fully Connected Layers

13. Optimizer and Output of CNN
Predicting:2636
Error:80(2.9%)
Predicting:2036
Error:36(1.3%)
Euclidean Loss (cost function)
1
2𝑁
𝑖=1
𝑁
𝑦𝑖
1
− 𝑦𝑖
2 2
𝑦𝑖
1
:Accuare output(LQF_T)
𝑦𝑖
2
:Predicting output
𝑁:data size
Adam(optimizer)
Predicting:1741
Error:41(1.5%)
Predicting:1038
Error:38(1.4%)
𝑦𝑖
2
• First-order gradient-based
optimization
→ feasible to compute in practice for
high-dimensional data
• Storing exponentially decaying
average of past gradient and
squared gradient
→Large updates for infrequent and
smaller updates for frequent
→accelerate learning
Accurate
=2,716
Accurate
=2,000
Accurate
=1,700
Accurate
=1,000
Error := |Accurate – Predicting| (Error / 2,751)
Note: 2,751 is the average evacuation time

14. Results of Our Model
Average Error in
Several Stages of Evacuation
Cumulative Probabilities
Average error(55s 2%) is quite a small
Data
#Training Data: 43,200
#Validation Data: 5,400
90% probability that we make
prediction within 3.6% error
90%
3.6%
Prediction for the early stage is better than
that for the final stage
6.5%
99%

15. Summary
Previous Model(LQF)
Not practical(7h)
• Compute max-flow algorithm
repeatedly for time-expanded graph
• Time-expanded graph is huge
Proposed Model
Practical (less than 1s)
• Combing LQF and CNN
• Predict evacuation
completion time
immediately(less than 1s)
and almost accurately(2%)