AACIMP 2009 Summer School lecture by Alexander Makarenko. "Mathematical Modelling of Social Systems" course. 5th hour. Part 5.

1. Optimization in crowd movement
models via anticipation
Dmitry Krushinsky, Alexander Makarenko
Institute for Applied System Analysis,
NTUU “KPI”, Ukraine
Boris Goldengorin
University of Groningen, the Netherlands

2. Contents
• Motivation
• Brief description of the basic model
• Anticipating pedestrians
• One-step anticipation and space “de-
localization”
• Multi-step anticipation and time “de-
localization”
• Conclusions

3. Why it is important?
• The movement of large–scale human crowds potentially can result in a variety of unpredictable
phenomena: loss of control, loss of correct route and panics, that make groups of pedestrians
block, compete and hurt each other.
Mass events Technological
disasters
Natural cataclysms Terrorism
• So, it is evident that special management during such accidents is necessary. Moreover, well-
founded plans of evacuation based on realistic scenarios and risk evaluation must be designed.
This will either prevent harmful consequences or, at least, alleviate them.

4. Why it is important?
Chaotic - hard to control & predict
behavior - undesired phenomena: high “pressure”, shock waves, etc.
- poor performance (in emergency)
optimized infrastructure
simulation assessment optimization
regulations, direction signs,…
- easy to control & predict Determined
- evenly distributed pedestrians behavior
- good performance (in emergency)

5. Overview of the models
Simple Complex
(physically (with mentality
inspired) accounting)
microscopic
- anticipation
- lattice gas
- decision making
- billiards
- etc.
macroscopic
?
- fluid
dynamics

6. Basic model
Data Layer Routing Layer
P2
P3 P1
P4
3 states per cell: Cells contain directions that
make up shortest exit path
•Empty
•Obstacle
Pk – probability of shift
•Pedestrian in k-th direction (k=1..4)

7. Simplest model of anticipating
pedestrian
Supposition: the pedestrians avoid blocking each other. I.e.
a person tries not to move into a particular cell if, as he
predicts, it will be occupied by other person at the next
step.
P2
P3 P1 Pk Pk × (1 − α ⋅ Pk ,occ )
Pk – probability of shift in direction k (k=1..4)
P4
Pk,occ – probability of k-th cell in the neighborhood being
occupied (predicted)
α – free parameter, expressing influence of anticipation

8. Simplest model of anticipating
pedestrian
Model-based prediction:
3
Pk ,occ = ∑ Pi − ∑ Pi P j + ∑ Pi P j Pk
i= 1 i≠ j i≠ j,
P2
j≠ k
P4
P3 P1 P3
P2
P4
Cells beyond elementary
neighborhood are involved. Thus,
the actual (extended) neighborhood
has radius R=2.

9. Spatial de-localization
Growth of the neighbourhood …
… and impact on performance

10. Multi-step prediction and temporal
de-localization
Example 4
3 1
X 2 4
3 1
X 4 3 X
1
4 3 1 4
scenarios tree… 5 5 2 5
2
5
2
X 5 3 1
2
X
4 3 1 4 3 1 4 4
5 X 2 5 X 5 3 1 5 3
2 X 2 2 1
X
4 1 4
5 3 X 5 1
2 3 X
2
3
X 1 X 3 1 X 1 X
4 5 2 4 5 4 3 5 4 1
2 2 3 5
2
… and corresponding graph G(T)
(T=4, R=4)

11. Multi-step prediction and temporal
de-localization
Bipartite matching “Greedy” tree
P0
pedestrians
P1
P2 cells
P3
P4
...
...
Sparse tree
3
X 1 X 3 1
4 5 2 4 5
2

12. Finding optimal trajectories:
network flow approach
G(T) s
t
V ∈ G(T) − vertices
E ∈ G(T) − edges
auxiliary graphs Gk(T)
eij = (vi , v j ) ∈ E , vi , v j ∈ V G1(T)
q (vi )− " quality" function s
t
c(eij ) − capacity of the edge
c(eij ) = q (v j ) − q (vi ) G2(T)
s
t
Pk α ⋅ Pk + (1 − α ) ⋅ F (G k (T))
F (G k (T )) − max . flow in G k (T ) G3(T)

13. Finding optimal trajectories: neural
network approach
Example scenarios tree… ... and corresponding perceptron
2
p1 4 P4 w14 X2
4
p15 w15
p 01 w 01
2
1 P52 X0
0 X5
p 25 w 25
p02 w 02
p2 6 w26 2
p0 p2 P62 w w2 X6
3 7 03 7
p 36
P72 w 36
w 37
2
X7
p 37
p38 w38
2
P82 X8
 
Pi = j
∑ p ki Pk j− 1
X i j = σ  ∑ wki X kj − 1 
 
k  k 
Pi j ∈ [0;1] 1
σ (x)
x

14. Finding optimal trajectories:
network flow vs. neural network
• exact • iterative
• sequential • parallel
• … • …

15. Conclusion:
evolution of the model of pedestrian
MP(1,0)
0 1 2 3 T
time
MP(R,T)
MP(2,1)
MP(R,1)
MP(R,T) – model of pedestrian
R – radius of (extended) neighborhood; T – time horizon of anticipation

16. Conclusion:
performance
MP(1,0)
MP(2,1)
MP(R,1)
evacuation time
MP(R,T)
?
MP(∞, ∞)
… … …
absolute global minimum

17. Thank you!
0 1 2 3 T
time
? ?
?