Optimization in crowd movement models via anticipation

Dmitry Krushinsky, Alexander MakarenkoInstitute for Applied System Analysis,

NTUU “KPI”, UkraineBoris Goldengorin

University of Groningen, the Netherlands

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

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.

Terrorism

Technological

Natural cataclysms

Mass events

• 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.

disaster s

Why it is important?

simulation optimizationregulations, direction signs,…

assessmentoptimized infrastructure

Chaotic behavior

- hard to control & predict- undesired phenomena: high “pressure”, shock waves, etc.- poor performance (in emergency)

Determined behavior

- easy to control & predict- evenly distributed pedestrians- good performance (in emergency)

Overview of the modelsSimple

(physically inspired)

Complex(with mentality

accounting)

mic

rosc

opic

mac

rosc

opic

- lattice gas

- billiards

- fluid dynamics

- anticipation

- decision making

- etc.

?

P4

Basic modelData Layer

P1P3P2

Routing Layer

3 states per cell:

•Empty

•Obstacle

•Pedestrian

Cells contain directions that make up shortest exit path

Pk – probability of shift in k-th direction (k=1..4)

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.

P1P3

P2P4

kP )1( ,occkk PP ⋅−× α

Pk – probability of shift in direction k (k=1..4)Pk,occ – probability of k-th cell in the neighborhood being occupied (predicted)α – free parameter, expressing influence of anticipation

P2

P1

P4

P3

Simplest model of anticipating pedestrian

P3

P2P4

Model-based prediction:

∑+∑−∑=

≠≠≠=kjji

kjijjii

iiocck PPPPPPP

,

3

1,

Cells beyond elementary neighborhood are involved. Thus, the actual (extended) neighborhood has radius R=2.

Spatial de-localizationGrowth of the neighbourhood …

… and impact on performance

Multi-step prediction and temporal de-localization

Example scenarios tree…

… and corresponding graph G(T) (T=4, R=4)

X X XX

X

X

X

X

X

X XX

X

XX

1 1 11

1

1

1

1

1

1 11

1

11

22

2 2

2

2

2

2

2

22 2

2

22

3 33 3

3

3

3

3

3

3 33

3

33

4 4 4 4

4

4

4

4

4

4 4 4

4

445

5 5 5 5

5

5

5

5

5 5 5

5

55

Multi-step prediction and temporal de-localization

Bipartite matching “Greedy” tree

Sparse tree

...

P0P1P2P3P4

...

pede

stria

ns cells13

4 25X 13

42

5X

Finding optimal trajectories: network flow approach

)()()( ijij vqvqec −=edgetheofcapacity)(

functionquality"")(

V,,E),(edgesG(T)EverticesG(T)V

G(T)

−−

∈∈=−∈−∈

ij

i

jijiij

ecvq

vvvve

s t

G3(T)

s t

G1(T)

s t

G2(T)

auxi

liary

gra

phs

Gk(

T)

kP ))T(G()1( kk FP ⋅−+⋅ αα)T(Ginflow.max))T(G( kkF −

Finding optimal trajectories: neural network approach

Example scenarios tree… ... and corresponding perceptron

]1;0[

1

∈

= −∑ji

jk

kki

ji

P

PpP

= ∑ −

k

jkki

ji XwX 1σ

)x(σx

1

1p 01

p02

p03

p1 4

p15

p 25

p26p27

p 36p37p38

24P

25P26P27P28P

00X

w01

w02

w03

w14

w15

w25

w26w27

w36w37w38

24X

25X26X27X28X

Finding optimal trajectories: network flow vs. neural network

• exact• sequential• …

• iterative• parallel• …

Conclusion:evolution of the model of pedestrian

MP(1,0)

MP(2,1)

MP(R,1)

time0 1 2 3 T

MP(R,T)

MP(R,T) – model of pedestrianR – radius of (extended) neighborhood; T – time horizon of anticipation

Conclusion:performance

absolute global minimum

MP(1,0)

MP(2,1)

MP(R,1)

MP(R,T)

evac

uatio

n tim

e

MP(MP(∞∞, ∞), ∞)

?

… … …

Thank you!

? ?

?

time0 1 2 3 T