Optimization in crowd movement
    models via anticipation

 Dmitry Krushinsky, Alexander Makarenko
      Institute for Ap...
Contents
• Motivation
• Brief description of the basic model
• Anticipating pedestrians
• One-step anticipation and space ...
Why it is important?
    •   The movement of large–scale human crowds potentially can result in a variety of unpredictable...
Why it is important?
Chaotic      - hard to control & predict
behavior     - undesired phenomena: high “pressure”, shock w...
Overview of the models
                  Simple           Complex
                (physically     (with mentality
        ...
Basic model
         Data Layer                                      Routing Layer


                 P2
            P3   ...
Simplest model of anticipating
              pedestrian
Supposition: the pedestrians avoid blocking each other. I.e.
  a p...
Simplest model of anticipating
         pedestrian
                               Model-based prediction:

               ...
Spatial de-localization
Growth of the neighbourhood …




                                … and impact on performance
Multi-step prediction and temporal
          de-localization
Example           4
                          3 1
           ...
Multi-step prediction and temporal
          de-localization
           Bipartite matching                       “Greedy” ...
Finding optimal trajectories:
          network flow approach
                        G(T)                                ...
Finding optimal trajectories: neural
         network approach
  Example scenarios tree…                         ... and c...
Finding optimal trajectories:
network flow vs. neural network
• exact         • iterative
• sequential    • parallel
• …  ...
Conclusion:
          evolution of the model of pedestrian


MP(1,0)




                                                0...
Conclusion:
                                      performance
                  MP(1,0)



                            MP(...
Thank you!


     0   1   2     3    T
                 time




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         ?
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Optimization in Crowd Movement Models via Anticipation

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

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Optimization in Crowd Movement Models via Anticipation

  1. 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. 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. 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. 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. 5. Overview of the models Simple Complex (physically (with mentality inspired) accounting) microscopic - anticipation - lattice gas - decision making - billiards - etc. macroscopic ? - fluid dynamics
  6. 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. 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. 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. 9. Spatial de-localization Growth of the neighbourhood … … and impact on performance
  10. 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. 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. 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. 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. 14. Finding optimal trajectories: network flow vs. neural network • exact • iterative • sequential • parallel • … • …
  15. 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. 16. Conclusion: performance MP(1,0) MP(2,1) MP(R,1) evacuation time MP(R,T) ? MP(∞, ∞) … … … absolute global minimum
  17. 17. Thank you! 0 1 2 3 T time ? ? ?

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