The probabilistic graphical model. - Input: Guidance; Output: Crowd flow rate;. - Conditions: Fire status and path capacities. - Links: Conditional probability ...
Holistic Modeling and Optimization of Crowd Guidance in Building Emergency Evacuation
Peng Wang, Peter B. Luh, Shi-Chung Chang, Jin Sun
Building Emergency Evacuation l
Building emergency evacuation is of growing concern – Emergencies include fire, chemical spills, etc.
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Effective crowd guidance can improve egress efficiency and occupant survivability Existing guidance facilities – Guidance: exit signs, audio instructions – Static guidance versus dynamic fires and crowd movement 2 /28
Building Emergency Evacuation l
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To reduce the egress time the potential disasters such as stampeding or blocking should prevented. Can guidance help? – Traditional guidance is almost static, and does not consider how hazard event dynamics affects people’s behavior and cannot effectively prevent blockings in emergencies – Our method:
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Building Emergency Evacuation l
A model which can capture blockings is needed – The model predicts the potential blockings in the future based on current information of fires, egress and crowd movement – Guidance is updated to mitigate or prevent blockings
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Building Emergency Evacuation l
Existing models – Crowd movement in egress is captured by a network-flow model where stampeding or blocking events cannot be captured – Helbing’s social-force model captures blocking events (faster-is-slower), but focuses on one-room scenarios, not in an egress network 5 /28
Difficulties in Modeling l
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To capture blocking events it is necessary to combine Helbing’s model with the network-flow model? Gap exists between the two models Helbing’s model - A microscopic model - Equations for individual behaviors - Individuals ↔ Particles
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Network-flow model - A macroscopic model - Equations for collective behaviors - Crowd ↔ Flow
To bridge the gap – Crowd flow model is built up to translate Helbing’s microscopic model to new macroscopic model (Presented last time) 6 /28
Table Contents
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A probabilistic graphical model l
The probabilistic graphical model – Input: Guidance; Output: Crowd flow rate; – Conditions: Fire status and path capacities – Links: Conditional probability distribution
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A probabilistic graphical model l
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d if Qe Ce 1 exp Qe Ce d Pr(Qe | Qe , Ce ) exp Blc if Q Q e e Qd C e e
Faster-is-slower scenario is achieved in this block If Qde Ce , then Qe Qde with probability 1
If Q de Ce the probability of blocking increases as the difference of Qde and Ce increases
where 0 Question: Where is Qed from? 9 /28
A Probabilistic Graphical Model l
Conditional Probability Distribution Pr (Qed| we, sF)
The desired flow rate Qed(t): the number of people desiring to move out during [ t , t t ]
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If fire becomes closer, people become more impatient, and the desired flow rate Qed increases in probability sense
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A probabilistic graphical model xv: the number of people in the area v
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Conditional Probability Distribution: Pr (we | ue, xv) – Crowd response we: The number of occupant who will follow the guidance at time [ t , t t ] – Suppose each individual will follow the guidance ue with a certain probability (Trust Probability) – The probability reflects people's inclination to use an familiar exit 11 /28
A probabilistic graphical model l
The probabilistic graphical model incorporates two psychology factors: impatience and trust – Impatience is the cause of blocking events – Trust on guidance reflects how guidance changes crowd behaviors
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Table Contents
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Egress Networks l
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To incorporate the probabilistic graphical model into the traditional network-flow model let us review traditional networkflow models first Review egress network – Each area is represented by a node, and the area capacity is ignored because the bottleneck for crowd movement lies in the path capacity – Each path from one area to another is represented by a directed arc with specified capacity (persons per time unit) 14 /28
Crowd Flow Equation l l
Review crowd flow equation The crowd flow equation is a linear state equation
A simple example x 1 (t)
– State: The number of people at each area x(t) = [x1(t), x2(t), x3(t)]'
Q1 (t)
x 2 (t) Q3 (t)
Q2 (t)
x 3 (t)
– Flow: The movement of people over each arc Q(t) [Q1(t), Q2 (t), Q3 (t)]'
– Egress dynamics: Flow balance equation x(t 1) x(t) BQ(t)
1 1 0 with egress matrix B 1 0 1 0 1 1 15 /28
Crowd Flow Equation l
Incorporate the probabilistic graphical model into the crowd flow balance equation the new balance equation is given by x(t 1) x(t) B Q (t)
x ( t 1) x ( t ) B Q (u ( t ) | s F ( t ), C) l
The system dynamics involves two stochastic processes:
Fire process Crowd movement process
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Table Contents
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The Constraints and Objective Function l
Constraints for guidance – Never guide crowd to an area currently on fire or to be on fire in near future. – Guidance constraint is obtained based on prediction of fire propagation
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The objective function to be maximized is Cumulative number of people evacuated during [0, T]
The total number of people evacuated
J t 1 t ( x exit ( t 1) x exit ( t )) c T x exit (T ) T
t 0 x exit ( t ) (c T T) x exit (T) T 1
xexit(t): The number of people in exit areas at time t cT: The weight for terminal time; cT>>T 18 /28
The Objective Function l
Due to uncertainty both mean and semi-variance is calculated where the semi-variance is a risk measurement T 1
T 1
t 0
t 0
– Cumulative term: R 1 E[ x exit ( t )] c ins varsemi [ x exit ( t )] – Terminal term: l
R 2 E[ x exit (T )] c avg varsemi [ x exit (T )]
Objective Function: Maximize J, with J R 1 (c T T )R 2
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Solution Methodology l
In our problem the computation complexity is a challenge – State space is large (the number of occupant in every area) – The complexity is mainly from the huge state space
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To reduce the computation time Lagrangian Relaxation (LR) is applied – decompose overall way-finding problem into subproblems by grouping occupants
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Solution Methodology - LR l
How Lagrangian Relaxation (LR) works in our problem – The joint constraints is the path capacities shared by groups – Decompose the overall problem by relaxing joint constraints – Coordinate the solutions of the decomposed problems through Lagrangian multipliers Shared capacity
Decomposition
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Lagrangian Multipers (Coordination) 22 /28
Solution Methodology - LR l
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Joint constraints exist in shared path capacities and are embedded in the probabilistic graphical model. Approximation method is used to relax the joint constraint
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Solution Methodology - LR l
Approximation method is used to relax the joint constrain of the path capacities – First, the original holistic graphical model is separated by making path capacities go to infinity for subproblems – Second, a constraint is added to the outputs in the separate graphical models.
~ Step 1: C e Step 2:
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(i ) E [ Q e ( t )] C e i 1
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Solution Methodology - SDP l
As a result Lagrangian Relaxation can be applied and the subproblem can be solved by the stochastic dynamic programming (SDP) with time steps as stages – SDP looks backward – SDP yields a NP hard problem – In our problem the computation complexity is mainly from the large size of state space – SDP guarantee the optimality 25 /28
Solution Methodology - Rollout l
Rollout algorithm with one-step look ahead policy – Rollout algorithm looks forward from current state – The states inaccessible from the current state are not included in computation, thus the state space for computation is reduced – The tradeoff is the optimality of the solution
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Solution Methodology - Rollout l
Rollout algorithm for a single group way-finding problem
~ u ( k ) arg max E{g k [ x ( k ), u ( k ), w ( k )] Jk 1[f k [ x ( k ), u ( k ), w ( k )]]} *
u(k)
– The heuristic policy: Decision by heuristic Distance to Exit arg min u Movement Speed Distance to Exit arg min u Flow Rate
where flow rate is calculate by the graphical model 27 /28
Table Contents
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Thank you
