Minimizing the evacuation time of a crowd from a complex building using rescue guides
MMinimizing the evacuation time of a crowd from a complex building using rescueguides
Anton von Schantz ∗ , Harri Ehtamo Aalto University, Department of Mathematics and Systems Analysis, P.O. Box 111000, FI-00076 Aalto, Finland
Abstract
In an emergency situation, the evacuation of a large crowd from a complex building can become slow or evendangerous without a working evacuation plan. The use of rescue guides that lead the crowd out of the buildingcan improve the evacuation efficiency. An important issue is how to choose the number, positions, and exitassignments of these guides to minimize the evacuation time of the crowd. Here, we model the evacuating crowdas a multi-agent system with the social force model and simple interaction rules for guides and their followers.We formulate the problem of minimizing the evacuation time using rescue guides as a stochastic control problem.Then, we solve it with a procedure combining numerical simulation and a genetic algorithm (GA). The GAiteratively searches for the optimal evacuation plan, while numerical simulations evaluate the evacuation timeof the plans. We apply the procedure on a test case and on an evacuation of a fictional conference building. Theprocedure is able to solve the number of guides, their initial positions and exit assignments in a single althoughcomplicated optimization. The attained results show that the procedure converges to an optimal evacuationplan, which minimizes the evacuation time and mitigates congestion and the effect of random deviations inagents’ motion.
Keywords:
Multi-agent systems, Numerical simulation, Evacuation, Stochastic optimization, Geneticalgorithms c (cid:13) http://creativecommons.org/licenses/by-nc-nd/4.0/ ∗ Corresponding author
Email addresses: [email protected] (Anton von Schantz), [email protected] (Harri Ehtamo) a r X i v : . [ phy s i c s . s o c - ph ] S e p . Introduction Large complex buildings like airport terminals, high-rise buildings and subway stations need crowd man-agement to ensure safety. In an emergency situation, like a fire or a bomb threat, a well-planned operation isneeded to ensure a fast and safe evacuation. It is known that people are willing to follow authorities (Proulx,2002; Nishinari et al., 2004), and that the strategic use of trained safety personnel, or rescue guides, that leadthe crowd out of the building, improves evacuation efficiency (Hou et al., 2014). However, as the crowd size andcomplexity of the building increases, codes of practice and human intuition are not sufficient to form an optimalevacuation plan. Rather attention needs to be paid to mathematical optimization approaches (Haghani, 2020).In this paper, we are interested in what will happen in a complex building containing a large crowd, after thesound of a whistle or a fire alarm, in a serious situation, when there is enough yellow-coated rescue personnel,whose members have enough skill and authority to guide the crowd out of the building in minimum time. Evenmore so, how should these guides be coordinated so that they manage in their task in an optimal way. To ourknowledge, this problem has not been solved using a rigorous mathematical optimization approach.In the research field of crowd and evacuation dynamics, three distinct research streams can be distinguished:experimental research on pedestrian behavior, descriptive mathematical modeling of crowd movement andinteractions, and mathematical optimization of evacuation. A good source of references on crowd movementmodels and optimization approaches is (Vermuyten et al., 2016). The most popular crowd movement model dueto its ability to produce realistic physical movement is the microscopic agent-based social force model (Helbing& Moln´ar, 1995). In it the movement of individual agents are described by Newton’s equations of motion.A typical concern is that focusing mainly on physical movement in crowd evacuation modeling may poten-tially lead to an underestimation of the evacuation time, since evacuees can engage in a variety of activitiesthat do not move them towards the exits (Gwynne et al., 2016). We assume, that in an evacuation of a largecrowd controlled by guides, such activities can be important but rare. From a modeling perspective, thesebehavioral deviations are small compared to the systemic, deterministic part of motion (Helbing & Johansson,2013). Deviations from the usual rules of motion can be approximated by adding a Gaussian noise term to theequations of motion (Helbing & Moln´ar, 1995). So, in this study, our main focus is not to tackle various humanbehaviors, but rather to consider how the guides should be positioned and what kind of routes they should takein a minimum time evacuation, provided some simple assumptions about the crowd behavior in such a situationare made.There is a need for mathematical approaches for evacuation optimization (Kurdi et al., 2018; Haghani, 2020).Typically, the optimal guided evacuation has been studied by comparative analysis, where the evacuation issimulated and compared for different guide configurations. The optimal number of guides has been studiedin, e.g., (Pelechano & Badler, 2006; Wang et al., 2012; McCormack & Chen, 2014). Also, initial positioning ofguides has been studied in, e.g., (Aub´e & Shield, 2004; Wang et al., 2012; Cao et al., 2016). It has been suggested2hat embedded, peripheral, near exit and uniformly distributed positions improve the evacuation efficiency. Theimportance of coordination has been noted in (Hou et al., 2014; Cao et al., 2016). Actually, without strategicallyplanned initial positions and exit assignments of guides, their use can worsen the evacuation instead of improvingits efficiency.To our knowledge, there are only two noteworthy studies that use mathematical optimization to study theevacuation of a crowd using guides (Albi et al., 2016; Zhou et al., 2019). The lack of research is probably due tothe large state space caused by a moving crowd and the stochastic nature of the problem. In (Albi et al., 2016)the crowd evacuates from a simple building. Its members are assumed to be unfamiliar with the environment.Hence, their movement is a combination of random walk and following nearby agents. Invisible guides are addedto the crowd, i.e., the other agents in the crowd do not recognize them as guides. The minimum evacuationtime problem is formulated as a stochastic control problem, where the trajectories of the guides are solved.However, the number of guides is not an optimization variable. Albi et al. (2016) do not directly generate animplementable evacuation plan for rescue guides, but rather do a mathematical study on optimal herding ofa crowd to an exit. Still, with small changes, their modeling framework can be used for visible rescue guides.Nevertheless, we are skeptical that the solution methods they propose for the problem would be applicable fora more complicated building geometry with the crowd scattered over it.The study of Zhou et al. (2019) is a simultaneous but independent line of work with ours, and the two workswere done without knowledge of each other. Zhou et al. (2019) use a deterministic version of the social forcemodel to describe the crowd movement. At the start they optimize the number and initial positions of the guidesusing a suitable criterion. After that, they optimize their routes and exits given their starting positions. Thetwo optimization problems are independent of each other, and neither the criterion nor the starting positionsof the guides are corrected based on the information got from the second problem. It should be noted that inthe first problem any suitable criterion, not only that used by Zhou et al. (2019), can be used to fix the numberand starting positions of the guides. All they give a different value for the evacuation time; the minimum valuecan be obtained only by chance. The model of Zhou et al. (2019) returns an evacuation plan that does notminimize the evacuation time, but does something else.Furthermore, both (Albi et al., 2016; Zhou et al., 2019) use their modified versions of the social force model,which are in some sense more sophisticated than typically, but they can only be used for optimization andevacuation simulations in simple building geometries. In them, the follower agents might get lost and areunable to follow the guide, if the guide goes into another room, or even behind a corner (Li et al., 2016).In this paper, we formulate and solve the problem of minimizing the evacuation time of a crowd from acomplex building using rescue guides. The problem is formulated as a stochastic control problem, where theobjective is to minimize the expected evacuation time. The optimization variables are the number of guides,and their routes represented as origin-destination pairs. The state of the system is modeled with social forceequations and simple rules of interaction between guides and exiting agents. The resulting problem solution3pace is very large, and it cannot be solved with derivative-based optimization. Thus, we first reformulatethe problem as a scenario optimization problem (Calafiore & Campi, 2006), where the expected evacuationtime is replaced by its sampled mean. Then we apply a hybrid numerical simulation and genetic algorithm(GA) approach (Goldberg, 1989). In it, the GA iteratively searches for the optimal solution, and numericalsimulations evaluate their sample mean evacuation time and steers the randomized search process. We ensurethe accuracy and efficiency of the algorithm on a test case, and then apply it on a more complex conferencebuilding case. Our three main contributions are that our method applies for complex buildings, it takes intoaccount stochasticity and gives the number of guides, their initial positions and exit assignments needed tominimize the crowd evacuation time in a single optimization.The paper is structured as follows. In Sec. 2, we discuss the assumptions and mathematical details of thecrowd movement model. In Sec. 3, we formulate the optimization problem, and present the GA approach usedto solve it. The performance of the GA is analyzed on a test case in Sec. 4, and after that the GA is appliedon a more complex conference building case in Sec. 5. Then, in Sec. 6, we revisit and discuss the behavioralassumptions made. Sec. 7 is for implementation details and performance of our algorithm. Finally, Sec. 8 isfor conclusion. For more information about parameter values and exact mathematical expressions of the socialforce model, see Appendix A. And, for a detailed analysis on the effect of stochasticity on our problem, seeAppendix B.
2. Evacuation model with guides
The social force model was first presented by Helbing & Moln´ar (1995). In it the motion of a pedestrian isdescribed by Newton’s equations of motion. The model is based on a paradigm from social theory, where anaction taken by a pedestrian is understood as a measure of her personal motivation to perform that action. Thataction is taken according to her psychological processes of assessment of alternatives and utility maximization.In the case of pedestrian behavior, this motivation evokes the physical production of an acceleration force.One can say that a pedestrian acts as if she would be subject to external forces (Helbing & Moln´ar, 1995;Hoogendoorn & Bovy, 2003; Helbing & Johansson, 2013). Additional physical interaction forces, inspired fromdriven granular flows, are assumed to come into play, when pedestrians get so close to each other that theyhave physical contact.The social force model has been integrated into the Fire Dynamics Simulator with Evacuation (FDS+Evac)of the National Institute of Standards and Technology (NIST, 2020). We will use the version of the socialforce model found in the FDS+Evac user manual (Korhonen, 2018). But instead of approximating the hu-man body with three overlapping circles (Heli¨ovaara et al., 2012a), we use a single circle. Also, we make asmall modification to the social force term. In the original social force model, a distance-dependent term isassumed. However, Karamouzas et al. (2014) have analyzed a large amount of publicly available crowd data,from several outdoor environments and controlled laboratory settings, showing that the social force depends on4he time-to-collision between agents rather than their distance. We will use the social force term presented in(Karamouzas et al., 2014).
The evacuation model describes the evacuation of a crowd of agents N ∪ G from a space Ω ⊂ R after thealarm has been given. The space includes obstacles O ⊂
Ω and exits
E ⊂
Ω. The obstacles are line segments,or walls, with index set W . The crowd consists of exiting agents N and guide agents, or guides, G . Here N = { , ..., n } and G = { n + 1 , ..., n + m } are the index sets of the exiting agents and guides, respectively.Exiting agents represent regular people that do not have full knowledge of the exits in the building, and headto their familiar exits by default. The guides represent trained safety personnel that use routes instructed bythe evacuation planner. They are assumed to have enough authority to influence the route choices of exitingagents. More specifically, the first time an exiting agent is within the interaction range r guide from a guide,it receives information of the exit the guide is moving towards, and starts also heading there. However, in athreatening situation, we assume that an exiting agent wants to get out of the building as fast as possible. Thatis why, if an exit is within the visibility range r exit , it starts to head there instead, regardless of where it waspreviously heading. In all situations, the exiting agents move to the exits using a shortest path (for informationon its numerical computation see Sec. 7).It is assumed that a mixture of socio-psychological and physical forces influence the agents’ motion in thecrowd. At time t , agent i ∈ N ∪ G with mass m i and radius r i likes to move with a certain desired velocity v desi ( t ), where v desi ( t ) = v desi ( t ) e desi ( t ). Here, v desi ( t ) is its desired speed, and e desi ( t ) is a unit vector that pointsto the direction that gives the shortest path to the exit it is heading towards. Agent i attempts to change itsactual velocity v i ( t ) to v desi ( t ) with a reaction time τ reac .If agents i and j , j ∈ N ∪ G, j (cid:54) = i , are about to collide, they experience a repulsive social force f socij ( t ).When agent i is in contact with another agent j or wall w ∈ W , the physical contact forces f cij ( t ) or f ciw ( t ),respectively, arise. Additionally, we assume that agent i is affected by a small random fluctuation force ξ i ( t ).Typically, the social force model includes a random force term in each agent’s equation of motion. This forcerepresents random deviations in behavior; more precisely, situations where two or more behavioral alternativesare equivalent, i.e., whether to pass an obstacle from the left or right hand side. It can also be thought todescribe accidental or intentional deviations from the usual rules of motion (Helbing & Moln´ar, 1995). Theforce ξ i ( t ) is drawn from a truncated bivariate normal distribution with zero mean. However, for guides thisforce equals a zero vector . For the parameter values used and the exact mathematical expressions of theforces, see Appendix A.The change of velocity at time t for agent i is then given by the equation of motion: m i d v i dt = m i v desi − v i τ reac + (cid:88) j ( (cid:54) = i ) ( f socij + f cij ) + (cid:88) w f ciw + ξ i ; (1)5he change of position vector x i ( t ) is given by the velocity d x i ( t ) dt = v i ( t ) . (2)In addition, we assume the following interactions to take place in the evacuation: Assumption 1.
An exiting agent heads to its familiar exit by default.
Assumption 2.
The guides’ initial positions and target exits are optimization variables. A guide moves fromits starting position to its target exit using the shortest path.
Assumption 3.
If an exiting agent is moving towards its familiar exit, and a guide comes within the interactionrange r guide , it starts to move to the same exit as the guide. If multiple guides are within the r guide range, itstarts to follow the closest one. Assumption 4.
If an exiting agent follows a guide, it will not switch to follow another guide.
Assumption 5.
If an exit is within the visibility range r exit from an exiting agent, it heads there, whether itpreviously was following a guide or heading to its familiar exit. If multiple exits are within the r exit range, itheads to the closest one.
3. Optimization model
We are interested in minimizing the evacuation time of the crowd, or equivalently the evacuation time of thelast evacuated agent T last . If we denote the evacuation times of the agents in the crowd with t , ..., t n + m , themaximal element of the set is T last . Because of the random fluctuation force term ξ i ( t ) in Eq. (1), our problemis stochastic. Thus, as objective function we choose to minimize the expected value of T last with respect to ξ i ( t ) , t ∈ [0 , T last ] , ≤ i ≤ n . Recall that for i ∈ G the random fluctuation force term is equal to .We discretize the space Ω into square grid cells ω , so that Ω ⊂ ∪ ω = ¯Ω. Also, we denote the pointsassociated with an exit by ε ⊂ E , so that E = ∪ ε . Each guide g ∈ G = { n + 1 , ..., n + m } is associated with astarting grid cell ω g ⊂ ¯Ω, and a target exit ε g ⊂ E . The optimization variables are ( ω g , ε g ) , g ∈ G . We assumethat the number m of the possible guides is sufficiently large, so that one or more guides can remain idle in thesimulation. The idle guides are mapped to a dummy grid cell outside ¯Ω, and their evacuation time is defined tobe zero. Our focus here is either a fixed number of active guides used in optimization, or a set of active guidesobtained as a result of optimization.The initial position of a guide g , x g (0) ∈ ω g , is a prespecified point in its corresponding starting grid cell ω g .The end position of a guide g , x g ( t g ) ∈ ε g , is any point in its corresponding target exit ε g . The initial positionsof the exiting agents are prespecified x i (0) = x i, , i = 1 , ..., n . The exiting agents can evacuate using any of theavailable exits; thus it holds for the end positions x i ( t i ) ∈ E .6he agents move to the exits according to Eqs. (1), (2) and Assumptions 1-5 defined in Sec. 2. Theseequations constitute the constraint equations of this problem. Now we can formulate the problem of minimizingthe evacuation time of a crowd using rescue guides:min ( ω g ,ε g ) E (cid:2) T last | ξ i ( t ) , ≤ i ≤ n, t ∈ [0 , T last ] (cid:3) ; ω g ⊂ ¯Ω , ε g ⊂ E , g ∈ G (3)subject to Eqs. (1) , (2); Assumptions 1-5; x i (0) = x i, , x i ( t i ) ∈ E , i ∈ N ; x g (0) ∈ ω g , x g ( t g ) ∈ ε g , g ∈ G. To solve problem Eq. (3) we will use scenario optimization (Calafiore & Campi, 2006). The expected evacu-ation time of the last agent is replaced by its sampled empirical version: M independent identically distributedsamples of the random force ξ i , ≤ i ≤ n , are generated which we denote by δ (1) , ..., δ ( M ) , and then we definethe sample mean, ˆ T last = 1 M M (cid:88) k =1 T ( k ) last . (4)Here, T ( k ) last is calculated for given ( ω g , ε g ), g ∈ G , by numerically simulating the system equations Eqs. (1), (2)with a numerical integration scheme, given the sample vector δ ( k ) , Assumptions 1-5 and the initial and endconditions (see more implementation details in Sec. 7).The scenario optimization problem is solved with a genetic algorithm (GA) (Goldberg, 1989). The GAiteratively searches for the optimal solution, while the numerical simulations evaluate the fitness of the foundsolutions and steer the randomized search process. In our optimization problem, the fitness is the sample meanEq. (4), and a solution, or chromosome, is an evacuation plan { ( ω n +1 , ε n +1 ) , ..., ( ω n + m , ε n + m ) } . A single gene ina chromosome contains both the starting grid cell and target exit of a guide. Because a solution has a variablenumber of active guides, we apply the hidden genes GA (Abdelkhalik, 2013). In it each gene gets a tag, whichtells if the gene is active, or if it is hidden or idle. Only the active genes affect the evacuation simulation.However, the genetic information of the idle genes is carried on in the process.In the GA, a population of a predefined number of solutions is maintained in the consecutive iterations, orgenerations. At the initial iteration, a population of random solutions is generated. After that, the populationgoes through three operations, which are selection, crossover and mutation. In selection, the solutions withsmallest fitness values are chosen to undergo the next two operations. In this paper, we use elitist selection,where we replace a few of the worst solutions in the current generation with the best solutions of the previousgeneration.In the crossover operation, genetic information is combined to create new solutions. In this paper, we usethe single-point crossover operator, meaning that the offspring solutions of the two parent solutions contain half7f the genetic material of both parents. The crossover operator is applied with a certain probability, and it isapplied on both genes and the tags. If it is not applied, the offspring solutions have exactly the same geneticinformation as their parents. See Fig. 1 for a depiction of the mechanism; if the gene is colored gray, the geneis hidden, which means that the guide corresponding to the gene is not present in the evacuation, i.e., is idle.In the single-point crossover, the genetic material on the right side of the crossover point of the two parentchromosomes is changing places to create offspring chromosomes. Figure 1: The single-point crossover operation applied on the two parent chromosomes to create offspring solutions.
Then, a mutation operator is applied on the offspring solutions. The mutation operator is applied separately,with a certain probability, on each gene and tag. When applied on a gene, it can either alter the starting gridcell, or the target exit, or both of them. The mutation operator applied on a tag can switch the gene fromhidden to active, or vice versa. Finally, the GA goes back to evaluating the fitness values of the solutions ofthe new generation, i.e., the offspring solutions, and then proceeds to the selection operation, and the iterationcontinues. The algorithm has converged, when the best solution has not changed in a predefined number ofsuccessive iterations.
4. Evacuation of a hexagon-shaped area
Typically, to solve an optimization problem with a GA, the algorithm parameters are tuned manually in aproblem-specific manner. Hence, we will first apply the GA on a test case, of which we have some idea what theoptimal evacuation plan might be. The algorithm parameters are tuned, so that it converges as efficiently andaccurately as possible to the near-optimal solution. The same parameters are also used for the more complexcase presented in the next section.Here, we consider a hexagon-shaped area. We assume it to be an outdoor space with six exits. All theexiting agents have entered through one door, and are thus familiar only with that one. Also, we assume thisto be an unhurried situation, where people want to move out of the area, but do not sense such an urgency that8hey would by themselves try other exits. Thus, we do not use Assumption 5. The interaction range of a guideis set to r guide = 5 m.For exiting agents i ∈ N , the initial positions x i , radii r i , masses m i , and desired speeds v desi are fixed forall simulations. Before fixing the values, the parameters m i , r i , and v desi are drawn from a truncated normaldistribution with a cutoff at three times the standard deviation. The mean and standard deviations are 73 . . .
255 m and 0 .
035 m, and 1 .
25 m/s and 0 . m i , r i and v desi . On theother hand, for guide agents g ∈ G , we set the values of a typical male: m g = 80 kg, r g = 0 .
27 m, and v desg = 1 .
15 m/s. The reaction time is τ reac = 0 . (a) (b)(c) (d)Figure 2: Evacuation of a hexagon-shape area. (a) Initial situation. (b) Without guides, a jam forms after a while. (c) Feasiblestarting grid cells for guides. (d) The near-optimal evacuation plan. × .
85, the mutation probability to 0 . r guide range, and do not switch to follow another guide (Assumptions 3 and 4). As it was mentioned earlier, allof the exiting agents have already a single familiar exit, so it is enough to redirect the majority of the exitingagents to the other five exits, thus utilizing all six exits in the evacuation.One could have expected that the guides would be in symmetric positions close to their target exits. It isnot needed, because the evacuation time is affected both by the walking and the queuing time of the agents.Additionally, the flow at the exit is known to be a nonlinear function of the size of the crowd in front of it(Schadschneider et al., 2008). Thus, up to a certain point, the flow can be increased by increasing the size of thecrowd in front of the exit. However, after a certain point the flow will start to decrease. Due to these effects, thenear-optimal solution is not very sensitive to the starting positions of the guides, as long as the exit utilizationis fairly even, and each guide influences about 1 / r guide wouldbe increased, the positions of the guides would have to be chosen more carefully, so that the guides do not alsoinfluence exiting agents farther away. In Fig. 3 we see how the hidden genes GA converges to the near-optimalevacuation plan. Starting from the 11th generation, the best solution does not change for 15 generations, andthe GA is considered to have converged.We also try how the solution changes with a fixed number of active guides, i.e., using the GA withouthidden genes. We use a larger mutation probability 0 .
40 for the 30 first generations, and after that we changeit to only 0 .
05. We are able to have a clear exploration phase in the beginning of the algorithm. It is neededfor convergence accuracy, when the hidden genes are not used. Otherwise, we use the same settings as withthe hidden genes GA. In Fig. 4 we see a comparison of the sample mean evacuation time for the near-optimalevacuation plans with different numbers of guides. We see that adding guides speeds up the evacuation, up tofive guides. The slight increase in evacuation time with six guides is a statistical deviation rather than the sixth10
Generation E v a c u a t i o n t i m e ( s ) Figure 3: The sample mean evacuation time for the best solution of each generation of the GA, when applied to the test case. guide slowing the evacuation, as we will soon see from the depictions of the evacuation plan. Notice also thatthe marginal utility of adding guides is decreasing, i.e., adding the first guide cuts the sample mean evacuationtime almost in half, but increasing from four to five guides has only a 4 % benefit. g u i d e s g u i d e g u i d e s g u i d e s g u i d e s g u i d e s g u i d e s E v a c u a t i o n t i m e ( s ) Figure 4: The sample mean evacuation time of the near-optimal evacuation plan for different fixed numbers of guides.
The resulting near-optimal solutions can be seen in Fig. 5. If there is only one guide, it leads half of theexiting agents to the exit on the opposite side of their familiar exit. If there are two guides, the crowd is splitinto three parts heading towards three different exits. With three guides, the crowd is split into four partsheading towards four different exits. With four guides, the crowd is split into five parts heading towards fivedifferent exits. With five guides, we get almost the same solution as with the hidden genes GA, only the startingpositions of the guides are slightly altered. With six guides, the sixth guide is put aside to evacuate with thegroup that is heading to Exit 0. When adding the extra guide, the GA assigns a starting position and target11xit to it so that it does not interfere with the minimum time evacuation plan. (a) (b)(c) (d)(e) (f)Figure 5: The near-optimal evacuation plans with (a) one guide, (b) two guides, (c) three guides, (d) four guides, (e) five guidesand (f) six guides. . Evacuation of a conference building We have now tuned the GA parameters suitable for the test case, and will apply it to the evacuation of acrowd from a conference building in an emergency situation. The initial situation is depicted in Fig. 6.
Figure 6: The initial situation of the unguided evacuation of the conference building.
Initially, the crowd of 1100 exiting agents is split into subgroups in seven different rooms: Concert Hall A,Cafeteria, Piazza, Restaurant, Foyer, Concert Hall B and Orchestra Foyer. There are five exits in the building.All agents in a subgroup have the same familiar exit towards which they initially intend to head (depicted bythe blue arrows in Fig. 6). In the evacuation model, the agents move to their target exit using their shortestpath; note however that even in a small group the agents may take different (shortest) paths to their targetexit. This is the case for the group of agents initially in Piazza, which are heading to Exit 4 in Orchestra Foyer.Half of the group heads first to Concert Hall A and move along its left wall to Orchestra Foyer, while the otherhalf takes the route through Restaurant, Foyer and Concert Hall B. The development of one realization, orscenario, of the unguided evacuation is seen in Figs. 7(a) to 8(b).Almost immediately a large jam emerges at the bottom leftmost entrance of Concert Hall A. Half of theexiting agents from Piazza try to enter Concert Hall A, through the same entrance that the exiting agents inConcert Hall A use to enter Piazza. This jam is cleared out very slowly, and it is the main source of inefficiencyfor the evacuation.Here, all Assumptions 1-5 are used. We set r guide = 10 m. It is larger compared to the 5 m interactionrange used for the hexagon-shaped case. We could use the same value in both cases. The parameter seems tohave an effect on how close the guide should be positioned to the exiting agents under its influence, and howlarge these groups of exiting agents are. We here set it larger so that the guide could collect more exiting agentsfrom the large crowd. Perhaps, if it is set smaller here, more guides would be needed to have an effect on thecrowd. The exit visibility range is set smaller than the interaction range of the guide, r exit = 9 m. As a result of13 a)(b)(c)Figure 7: Snapshots of one realization of the unguided evacuation of the conference building at (a) t = 5 s; (b) t = 15 s; (c) t = 30s. a)(b)Figure 8: Snapshots of one realization of the unguided evacuation of the conference building at (a) t = 50 s; (b) t = 205 s. this, in some situations, the guides are able to influence exiting agents before they notice a nearby exit. So, therelation between the values of r guide and r exit can have an effect on the exiting agents’ exit choice in situationswhere both a guide and an exit are nearby. Otherwise, the same evacuation model parameter values are usedas in the hexagon-shaped case. In the GA, we set the number of genes in a chromosome to 10, meaning that the solution can have amaximum of 10 guides. When the GA has converged, all solutions in the final population have less than 10genes active, which strongly indicates that the optimal solution should have less than 10 guides. The feasiblestarting grid cells are obtained again by discretizing the building into sixty-two 10 m ×
10 m grid cells; seeFig. 9. The five exits in the conference building are the feasible target exits.As in the test case, the crossover probability is set to 0 .
85, the mutation probability to 0 .
10, and the15 igure 9: Feasible starting grid cells for the guides in the conference building. population size to 40, with 30 samples of each chromosome. The two worst individuals are always replaced withthe two best individuals of the previous generation. The near-optimal solution given by the hidden genes GAis illustrated in Fig. 10.
Figure 10: The near-optimal evacuation plan of the conference building.
The total number of active guides is eight (depicted by the yellow circles). They head to their target exits(Assumption 2) along the routes depicted by the red arrows. The exiting agents start to follow their closestguides, within the r guide range, and do not switch to follow another guide (Assumptions 3 and 4). One guide isset in the Orchestra Foyer to head towards Exit 2. Four guides are in Concert Hall A, to herd the large crowd.Out of those four, three are heading towards Exit 3, and one towards Exit 0. In the upper part of Piazza, thereis a guide, which leads half of the exiting agents in Piazza to Exit 0. The other half of the exiting agents inPiazza are initially free to move towards Exit 4 using the route through Restaurant, Foyer and Concert Hall B.16he exiting agents located in Cafeteria, Restaurant and Foyer move as they would without guidance. There isalso one guide in Foyer and another in Concert Hall B that lead half of the exiting agents from Concert Hall Bto Exit 2. The other half of the exiting agents in Concert hall B go to Exit 3 without guidance. Finally, thereis one guide located in the Orchestra Foyer, and it is heading towards Exit 2.The convergence pattern of the GA can be seen in Fig. 11. Starting from the 23rd generation, the bestsolution does not change for 15 generations, and the GA is considered to have converged. Generation E v a c u a t i o n t i m e ( s ) Figure 11: The sample mean evacuation time of the best solution of each generation of the GA, when applied to the conferencebuilding case.
In the unguided situation, a large jam forms at the bottom leftmost entrance to Concert Hall A. It is evidentthat it should be solved to increase efficiency. However, the question is where to redirect this part of the crowd,as not to create a jam somewhere else. In Figs. 12(a) to 13(b) the full evolution of one realization of thenear-optimal guided evacuation is illustrated with snapshots.There are many changes to the unguided evacuation. First and foremost, a big jam never emerges at thebottom leftmost entrance of Concert Hall A. This is because the crowd in Concert Hall A is taken to Exit 3through Orchestra Foyer, and the rest to Exit 0 via the rightmost door of Concert Hall A. Only a couple ofexiting agents from Concert Hall A are left out without a guide and move by themselves to their familiar exit,Exit 1.A large part of the crowd is heading towards Exit 3, and no guide is going towards Exit 4. However, guidingexiting agents to Exit 4 would be unnecessary, as about half of the exiting agents initially heading to Exit 3,once they are in the Orchestra Foyer, detect Exit 4 by themselves, and head there instead of Exit 3 (Assumption5). Now without guidance, the exiting agents from Piazza would be heading to Exit 4, which would cause too17 a)(b)(c)Figure 12: Snapshots of one realization of the near-optimal guided evacuation of the conference building at (a) t = 5 s; (b) t = 10s; (c) t = 15 s. a)(b)Figure 13: Snapshots of one realization of the near-optimal guided evacuation of the conference building at (a) t = 25 s; (b) t = 40s. much traffic there. However, in the near-optimal solution, a guide takes half of the exiting agents from Piazzato Exit 0 through Concert Hall A. The second half of the exiting agents in Piazza that head to Exit 4 throughRestaurant, Foyer and Concert Hall B are stopped at the Foyer and redirected to Exit 2.In a more complicated model, the guide could be set to wait in place for exiting agents crossing paths withit later on. Here, the optimization model circumvents this lack of feature, by setting guides to start movingfrom farther away, to cross paths with exiting agents just at the right time. This is illustrated by the guidefrom the upper right corner of Orchestra Foyer, that crosses paths with the exiting agents from Piazza at Foyer,and redirects them to Exit 2. To conclude, the main feature of a good guidance seems to be that all exits areutilized, while none of them are overutilized. Also, jams or counterflows should not either emerge at other partsof the building.Fig. 14 shows a histogram and a kernel density estimate of the evacuation time of both the unguided19nd near-optimal evacuation. For the unguided evacuation, the sample mean is 320 .
72 s, whereas the samplestandard deviation is 20 .
75 s. For the near-optimal evacuation, the respective values are 83 .
11 s and 3 . Evacuation time (s) P r o b a b ili t y Near-optimalguided evacuation Unguided evacuation
Figure 14: Histograms (staple diagrams) and kernel density estimates (solid lines) of the evacuation times of 30 scenarios.
Also, from another point of view, even though the initial positions of the agents are fixed for all simulations,as time goes on, due to the random force term, the positions will differ between scenarios. Thus, in some sense,the near-optimal evacuation plan is robust to reasonable variations in agents’ positions.
6. Behavioral considerations
To emphasize, what we have in mind with our model, or the basic question we ask is this: what will happenin a complex building containing a large crowd, after the sound of a whistle or a fire alarm in a serious situation,when there is enough trained yellow-coated rescue personnel, whose members have enough skill and authorityto guide the crowd out of the building as soon as possible. So, our main modeling question here is how tocoordinate the guides so that they manage with their task in an optimal way.20 typical concern in modeling crowd evacuation, is that the crude assumption of focusing only on physicalmovement may potentially lead to an underestimation of the time for the crowd to reach safety, since evacueesare likely to engage in a variety of activities during the operation (Gwynne et al., 2016). We believe that inour situation guided by trained personnel, such evacuation activities, although can be important, are very rare,since people in serious situations are willing to follow authorities (Proulx, 2002; Nishinari et al., 2004).Even more to the point, it does not matter if some individuals, at some time, would have turned to theright, or stopped for a while, since now guides more or less control their movement. In fact, even for a largegroup of people, a few guides are enough to control the whole crowd to their target (Dyer et al., 2009; Caoet al., 2016). For a more irregular building geometry, where the crowd is more scattered, more guides areneeded. In that case, the coordination between guides becomes very important to facilitate the evacuation.A useful quantitative information that a model should deliver in this case is a suitable macroscopic one, suchas the evacuation time of the crowd. It is in fact less prone to globally unnecessary details and fluctuationsthan microscopic information (Helbing & Johansson, 2013). Our paper solves the number of guides, theirinitial positions and exit assignments to minimize the crowd evacuation time in a single although complicatedoptimization problem.Nevertheless, we acknowledge the importance of taking into account behavioral aspects in crowd evacuationmodeling. It has been the focus of our previous research (Heli¨ovaara et al., 2012b, 2013; von Schantz & Ehtamo,2015, 2019). For example, in (von Schantz & Ehtamo, 2019) we model crowd behavior in an exit congestion.We couple a local decision-making model to the social force model. The decision-making model is based onbehavioral assumptions that are verified in our experiment with real humans (Heli¨ovaara et al., 2012b). Withour integrated treatment of behavioral and physical aspects, we are able to simulate when, why and how typicalphenomena of an evacuation through a bottleneck occur. Most importantly, we attain non-monotonous speedand kinetic pressure patterns near exits in threatening situations. These kind of behavioral phenomena areinteresting as such. However, to include them into the optimization would require us to increase the alreadyvery heavy computation.Instead of studying a single agent deviating from its usual movement, it would be interesting to study thepossibility of a larger group doing so. For example, it could happen like this: an agent decides to go to anotherexit and a large 20% part of the crowd decides to follow this agent. However, to add this behavior to a crowdmovement model takes careful calibration. It is not a simple task to alter the microscopic dynamics to generatethis effect. Moreover, a larger amount of Monte Carlo simulations would probably be needed to estimate themean of the evacuation time for a given evacuation plan. Perhaps, an easier way than for the model to generatethis effect, is to construct representative scenarios. They would describe all drastically different alternative waysthe evacuation could develop. In one of the scenarios, 20% of the crowd would go to another exit as opposedto where it goes in the other scenarios. Then, the optimal evacuation plan would be calculated over all thesescenarios. 21 . Implementation details and performance
Let us first start by discussing the implementation details of the evacuation model. In it, the shortestpaths that agents use to move to the exits were calculated using the detailed method presented in depth in(Kretz et al., 2011). The method involves solving the continuous shortest path to an exit. It is solved usingthe fast marching method (Sethian, 1999), which first discretizes the continuous space into a meshgrid. Thenthe method works almost like Dijkstra’s algorithm for finding shortest paths between nodes in a graph. Thesolution is a distance map from each point, or grid, in the building to the exit. The direction to which an agentshould head towards at each point can be calculated using the gradient direction. The distance maps, i.e., thedistances from each grid to every exit are calculated and stored before the optimization simulations.Interaction forces in the social force equations should be calculated between all agents, but to speed upthe computation, we subdivided the simulation domain with a blocklist algorithm with a cut-off distance of 3m (Yao et al., 2004). This should not affect the crowd dynamics, since the social force term is exponentiallydecaying.When using numerical simulations to calculate the sample mean evacuation time, we use a pseudorandomnumber generator to generate realizations of the random force terms. For each scenario, we store the seed ofthe pseudorandom number generator. Thus, the realizations of the random force terms are replicable. Then,for a specific seed, the social force equations are solved with the Velocity Verlet numerical integration schemewith the time step (cid:52) t = 0 .
01 s; see, e.g., appendix of (von Schantz & Ehtamo, 2019) for more details.The evacuation model is implemented in Python code. Some of the core parts of the code are written asNumba-decorated functions, which translates Python functions to optimized machine code at runtime. Numba-compiled numerical algorithms in Python can approach the speeds of C or FORTRAN (Oliphant et al., 2020).Whereas, the GA is implemented in Bash script that calls the crowd simulation scripts written in Python. Forreproducibility, all codes are published (von Schantz, 2020).We ran the numerical experiments on the Aalto University high-performance computing cluster Triton. Asingle generation of the GA was run in parallel on Triton using its computing nodes that are Intel Xeon X56502 .
67 GHz with 48 GB or 96 GB memory, and Xeon E5 2680 v2 2 .
80 GHz with 64 GB or 256 GB memory.One generation in the GA requires 1200 simulations, because a generation contains 40 solutions, each whichhave to be run for all 30 scenarios. Due to the user quotas set for Triton users, we only ran 300 simulations inparallel, which means that one generation had to be run in four parts. A single simulation of the evacuationof the hexagon-shaped space could take up to 5 min, while one simulation of the evacuation of the conferencecould take up to 1 h. Thus, the simulation of one generation took about 20 min and 4 h, respectively. We ranthe GA for 26 and 38 generations, for the test case and the conference building case, respectively. Thus, for thetest case, the GA converged to the near-optimal solution in 8 h 40 min, whereas, for the conference buildingcase it converged in 152 h, or 6 days 8 h. 22 . Conclusion
We have studied the problem of minimizing the evacuation time of a crowd from a complex building usingrescue guides. The problem is first formulated as a stochastic control problem, where the objective is theexpected evacuation time, and the optimization variables are the number of guides and their routes defined asorigin-destination pairs. The system equations are the equations of motion, given by the social force model,and the rules of interaction for exiting agents and guides. The problem is then reformulated as a scenariooptimization problem, which we solve with a hybrid numerical simulation and GA approach. With it, we areable to solve the number of guides, their initial positions and exit assignments to minimize the crowd evacuationtime in a single optimization.Typically, to solve an optimization problem with a GA, the algorithm parameters are tuned manually ina problem-specific manner. In our study, we do this by constructing a test case, the hexagon-shaped spacein Sec. 4, for which we know beforehand what the expected minimum evacuation time should be. Then weextensively try different parameter configurations to get the GA to converge as efficiently and accurately aspossible to the near-optimal solution. This GA parameter configuration is also used for solving the near-optimalevacuation plan for the conference building.Moreover, it is known that using GAs for problems with large solution space and large number of localminima, introducing noise to the fitness function, and evaluating it by taking multiple samples, improvesconvergence to the global minimum (Hammel & B¨ack, 1994). Our problem is inherently noisy, and we performmultiple Monte Carlo simulations to calculate the expected evacuation time for a given evacuation plan. Thus,this may assure that our GA does not get stuck in a local optimum.In both cases studied in this paper, the improvement with the near-optimal plan is dramatic, the samplemean evacuation time is only about 25 % of that of the unguided evacuation. Moreover, for the conferencebuilding case, we also compared standard deviations, and it was only about 15 % of that of the unguidedevacuation. This is probably due to the near-optimal evacuation plan solving all major jams, which decreasesthe nonlinear physical effects, and hence small deviations in agents’ movement does not affect the evacuationtime so much.It is interesting to see how the near-optimal solutions take into account physical phenomena like counterflows,jam formation and flow at bottlenecks being a nonlinear function of crowd size. There are computationallyfaster evacuation models, like cellular automata and network flow models that are good for planning large-scale evacuations (Løv˚as, 1998; Abdelghany et al., 2014). However, they cannot model the physics of crowdsevacuating from complex confined spaces.However, it should also be noted that the optimization procedure is computationally very slow. So, as such,this procedure could not be used for online optimization. Faster computation could be achieved by using animplicit integration scheme, where the step size can be set in some cases even 40 times larger than with an explicit23ntegration scheme (Karamouzas et al., 2017). Alternatively, a position-based dynamics approach could be used,which has been shown to produce real-time simulations (Weiss et al., 2019). Another interesting avenue forfuture research would be to use neural networks to deal with the large state space (Bertsekas & Tsitsiklis, 1996).In future research, other objective functions could be used as well, and studied how the optimal evacuationplans differ from that of ours. An evacuation plan should be both fast and safe (Løv˚as, 1995). If we want tosimultaneously take these two objectives into account, we can solve the problem with multi-objective optimiza-tion (Saadatseresht et al., 2009). As noted before, one form of risk are rare events that dramatically slow downthe evacuation. To account for this, we cannot only minimize the mean evacuation time, but we also have tominimize the variance or some other risk-related measure related to the evacuation time distribution. We couldalso include physical risks like pressures in the crowd or avoidance of dangerous areas in a building. On theother hand, if we are considering an unhurried evacuation, objectives related to the quality of service can beused, e.g., minimization of the average evacuation time or distance travelled by the evacuees, or time spent incongestions.
Acknowledgements
This study was first funded by a grant from the Foundation for Aalto University Science and Technology,and later with a grant from the Finnish Science Foundation for Technology and Economics. The calculations inthis study were performed using computer resources within the Aalto University School of Science ”Science-IT”project. We wish to thank our summer assistant Jaan Tollander de Balsch, who considerably helped us developthe simulation codes.
Declaration of interests
The authors declare that they have no known competing financial interests or personal relationships thatcould have appeared to influence the work reported in this paper.
Appendix A. Social force model parameters
The social force model parameters have been validated against data, and the collective phenomena observedwith the model are robust to reasonable parameter variations (Karamouzas et al., 2014; Korhonen, 2018). Inour study, the initial positions x i , radii r i , masses m i , and desired speeds v desi , for exiting agents i ∈ N , arefixed for all simulations. Before fixing them, the parameters m i , r i , and v desi are drawn from a truncated normaldistribution with a cutoff at three times the standard deviation. The mean and standard deviation are 73 . . .
255 m and 0 .
035 m, and 1 .
25 m/s and 0 . m i , r i and v desi . Whereas, forguide agents g ∈ G , we set m g = 80 kg, r g = 0 .
27 m, and v desg = 1 .
15 m/s. The reaction time is τ reac = 0 . f socij , i, j ∈ N ∪ G, i (cid:54) = j , follows closely that of (Karamouzas et al., 2014). There, it was inferred from a large publicdata set that the interaction energy associated with the repulsive social force follows a power law with a sharptruncation at large τ , E ( τ ) = k soc τ e − τ/τ . (A.1)Here, k soc is a constant that sets the units of energy. It is not that clearly documented, but by examining thecodes provided in the supplemental material of (Karamouzas et al., 2014), we deduce its value to be k soc = 1 . m i .The parameter τ is the interaction time horizon, and it is set to 3 s. The collision time of two agents i and j is denoted with τ . So, Eq. (A.1) defines the interaction energy of a pair of agents, which are on a collisioncourse. This energy is directly related to the social force f socij experienced by agent i due to the interaction withanother pedestrian j . In particular, f socij = −∇ x ij E ( τ ) . (A.2)At any given simulation step, we estimate τ by linearly extrapolating the trajectories of the pedestrians i and j based on their current velocities, v i and v j , and position vectors, x i and x j . Specifically, a collision is saidto occur at some time τ , if the corresponding circles of the pedestrians of radii r i and r j intersect. If no suchtime exists, the interaction force f socij is a zero vector . Otherwise, τ = b − √ da , where a = (cid:107) v ij (cid:107) , b = x ij · v ij , c = (cid:107) x ij (cid:107) − ( r i + r j ) , and d = b − ac . Here, x ij = x i − x j is their relative displacement, and v ij = v i − v j istheir relative velocity. By substituting τ into Eq. (A.2), the interaction force can be written explicitly as: f socij = − (cid:34) k soc e τ/τ (cid:107) v ij (cid:107) τ (cid:18) τ + 1 τ (cid:19)(cid:35) v ij − (cid:107) v ij (cid:107) x ij − ( x ij · v ij ) v ij (cid:113) ( x ij · v ij ) − (cid:107) v ij (cid:107) ( (cid:107) x ij (cid:107) − ( r ij ) ) . (A.3)Here, r ij = r i + r j . Furthermore, the force f socij is truncated from above to 2000 N.The original social force model (Helbing & Moln´ar, 1995) includes also a repulsive social force betweenagents and walls, f sociw . Because, even though the desired velocity v i does not point inside walls by construction,the actual velocity v i could do so, i.e., an agent could be pushed inside a wall by other agents in the crowd. Toavoid this, we use the approach from (Cristiani & Peri, 2017). In it, the desired velocity v i is constructed toheavily point away from a wall, when the agent gets close to it.Physical contact forces come into play, when agents i and j touch each other, r ij − (cid:107) x ij (cid:107) ≥ f cij = k ( r ij − (cid:107) x ij (cid:107) ) n ij + c d (cid:52) v nji n ij + κ ( r ij − (cid:107) x ij (cid:107) ) (cid:52) v tji t ij , (A.4)where n ij = ( n ij , n ij ) = x ij / (cid:107) x ij (cid:107) is the normalized vector pointing from agent j to i , t ij = ( − n ij , n ij ) is thetangential direction, (cid:52) v nji = − v ij · n ij is the normal velocity difference, and (cid:52) v tji = − v ij · t ij is the tangentialvelocity difference. The parameters k = 1 . · kg/s, c d = 500 kg/s, and κ = 4 . · kg/(m · s) are forceconstants. The first term in Eq. (A.4) represents a ’body force’ counteracting body compression, the secondterm a ’damping force, which reflects the fact that the collision of two humans is not an elastic one, and the25hird term is a ’sliding friction force’ impeding relative tangential motion. The physical agent-wall interactionforce f ciw is treated similarly and same parameter values are used.Finally, the components of the random force vector ξ i follow a truncated normal distribution with meanzero, standard deviation of 0 . m i m/s , and are truncated at three times of the standard deviation. Appendix B. The value of the stochastic solution
Let us evaluate the effect of stochasticity on our optimization problem using terminology from the literatureof stochastic programming (Birge, 1982). First, recall the mathematical definitions from Secs. 2 and 3. Then,let us denote ξ i ( t ) , ≤ i ≤ n, t ∈ [0 , T last ], with δ ( t ). And, because T last depends on ω g , ε g , g ∈ G and δ throughthe constraint equations, we define: φ ( ω g , ε g , g ∈ G ; δ ) := T last . (B.1)The problem of Eq. (3) is called the recourse problem , and using Eq. (B.1) it can be rewritten:RP := min ( ω g ,ε g ) E (cid:2) φ ( ω g , ε g , g ∈ G ; δ ) | δ (cid:3) ; (B.2) ω g ⊂ ¯Ω , ε g ⊂ E , g ∈ G, subject to the constraints defined in Eq. (3). We obtained the sample mean 83 .
11 s for RP, in the conferencebuilding case, using the detailed solution methodology presented in the paper.Next, we define the deterministic equivalent problem, or the expected value problem :EV := min ( ω g ,ε g ) φ (cid:0) ω g , ε g , g ∈ G ; E [ δ ] (cid:1) ; (B.3) ω g ⊂ ¯Ω , ε g ⊂ E , g ∈ G, subject to the constraints defined in Eq. (3). Recall that the random force terms ξ i ( t ) , ≤ i ≤ n, t ∈ [0 , T last ],are independent and have zero mean. Hence, E [ δ ] = . Furthermore, in the equation of motion Eq.(1), the random force appears as an additive linear term. Thus, the term disappears and we are left withthe deterministic social force equation. The deterministic equivalent problem is also solved with the solutionmethodology presented in the paper. The EV solution (ˆ ω g , ˆ ε g ) , g ∈ G is seemingly similar to the RP solution.Only the initial position of one of the guides is moved from Concert Hall A to Orchestra Foyer.The expected evacuation time for solution (ˆ ω g , ˆ ε g ) , g ∈ G is:EEV := E (cid:2) φ (ˆ ω g , ˆ ε g , g ∈ G ; δ ) | δ (cid:3) . (B.3)To calculate it, we perform Monte Carlo simulations. We obtain the sample mean of 85 .
59 s for EEV. Thus, the value of the stochastic solution is:VSS = EEV − RP = 83 . s − . s = − .
48 s . (B.5)26o, the sample mean of the evacuation time for the EV solution is close to that of the RP solution. Still, this isnot the only side to the matter. We experienced difficulties in getting the GA to converge to the optimum of thedeterministic equivalent problem. However, because we had solved the stochastic problem first, we knew whatwe were seeking. Thus, in the end, we made use of the stochastic problem solution, and performed an exhaustivelocal search to find the deterministic equivalent problem solution. This convergence issue seems to be a moregeneral concern in solving stochastic problems with their deterministic equivalent using a GA (Hammel & B¨ack,1994). For example, it might be that for a deterministic optimization problem having many local optima, butonly one global optimum, when adding stochasticity or averaging over many such deterministic problems makesthe problem landscape smoother with only a single optimum. References
Abdelghany, A., Abdelghany, K., Mahmassani, H., & Alhalabi, W. (2014). Modeling framework for optimalevacuation of large-scale crowded pedestrian facilities.
European Journal of Operational Research , , 1105–1118. doi: https://doi.org/10.1016/j.ejor.2014.02.054 .Abdelkhalik, O. (2013). Hidden genes genetic optimization for variable-size design space problems. Journal ofOptimization Theory and Applications , , 450–468. doi: https://doi.org/10.1007/s10957-012-0122-6 .Albi, G., Bongini, M., Cristiani, E., & Kalise, D. (2016). Invisible control of self-organizing agents leavingunknown environments. SIAM Journal on Applied Mathematics , , 1683–1710. doi: https://doi.org/10.1137/15M1017016 .Aub´e, F., & Shield, R. (2004). Modeling the effect of leadership on crowd flow dynamics. In P. M. A.Sloot, B. Chopard, & A. G. Hoekstra (Eds.), Cellular Automata (pp. 601–611). Springer. doi: https://doi.org/10.1007/978-3-540-30479-1_62 .Bertsekas, D. P., & Tsitsiklis, J. N. (1996).
Neuro-dynamic programming volume 5. Belmont, MA: AthenaScientific.Birge, J. R. (1982). The value of the stochastic solution in stochastic linear programs with fixed recourse.
Mathematical Programming , , 314–325. doi: https://doi.org/10.1007/BF01585113 .Calafiore, G. C., & Campi, M. C. (2006). The scenario approach to robust control design. IEEE Transactionson Automatic Control , , 742–753. doi: https://doi.org/10.1109/TAC.2006.875041 .Cao, S., Song, W., & Lv, W. (2016). Modeling pedestrian evacuation with guiders based on a multi-grid model. Physics Letters A , , 540–547. doi: https://doi.org/10.1016/j.physleta.2015.11.028 .Cristiani, E., & Peri, D. (2017). Handling obstacles in pedestrian simulations: Models and optimization. AppliedMathematical Modelling , , 285–302. doi: https://doi.org/10.1016/j.apm.2016.12.020 .27yer, J. R. G., Johansson, A., Helbing, D., Couzin, I. D., & Krause, J. (2009). Leadership, consensus decisionmaking and collective behaviour in humans. Philosophical Transactions of the Royal Society B: BiologicalSciences , , 781–789. doi: https://doi.org/10.1098/rstb.2008.0233 .Goldberg, D. E. (1989). Genetic Algorithms in Search, Optimization and Machine Learning . Boston, MA,USA: Addison-Wesley Longman Publishing Co.Gwynne, S. M. V., Hulse, L. M., & Kinsey, M. J. (2016). Guidance for the model developer on repre-senting human behavior in egress models.
Fire Technology , , 775–800. doi: https://doi.org/10.1007/s10694-015-0501-2 .Haghani, M. (2020). Optimising crowd evacuations: Mathematical, architectural and behavioural approaches. Safety Science , , 104745. doi: https://doi.org/10.1016/j.ssci.2020.104745 .Hammel, U., & B¨ack, T. (1994). Evolution strategies on noisy functions: How to improve convergence properties.In Y. Davidor, H.-P. Schwefel, & R. M¨anner (Eds.), Parallel Problem Solving from Nature–PPSN III (pp.159–168). Springer. doi: https://doi.org/10.1007/3-540-58484-6_260 .Helbing, D., Isobe, M., Nagatani, T., & Takimoto, K. (2003). Lattice gas simulation of experimentally studiedevacuation dynamics.
Physical Review E , , 067101. doi: https://doi.org/10.1103/PhysRevE.67.067101 .Helbing, D., & Johansson, A. (2013). Pedestrian, crowd, and evacuation dynamics. arXiv preprint , .ArXiv:1309.1609.Helbing, D., & Moln´ar, P. (1995). Social force model for pedestrian dynamics. Physical Review E , , 4282.doi: https://doi.org/10.1103/PhysRevE.51.4282 .Heli¨ovaara, S., Ehtamo, H., Helbing, D., & Korhonen, T. (2013). Patient and impatient pedestrians in a spatialgame for egress congestion. Physical Review E , , 012802. doi: https://doi.org/10.1103/PhysRevE.87.012802 .Heli¨ovaara, S., Korhonen, T., Hostikka, S., & Ehtamo, H. (2012a). Counterflow model for agent-based simulationof crowd dynamics. Building and Environment , , 89–100. doi: https://doi.org/10.1016/j.buildenv.2011.08.020 .Heli¨ovaara, S., Kuusinen, J.-M., Rinne, T., Korhonen, T., & Ehtamo, H. (2012b). Pedestrian behavior andexit selection in evacuation of a corridor–An experimental study. Safety Science , , 221–227. doi: https://doi.org/10.1016/j.ssci.2011.08.020 .Hoogendoorn, S., & Bovy, P. H. L. (2003). Simulation of pedestrian flows by optimal control and differentialgames. Optimal Control Applications and Methods , , 153–172. doi: https://doi.org/10.1002/oca.727 .28ou, L., Liu, J.-G., Pan, X., & Wang, B.-H. (2014). A social force evacuation model with the leadership effect. Physica A: Statistical Mechanics and its Applications , , 93–99. doi: https://doi.org/10.1016/j.physa.2013.12.049 .Huang, H.-J., & Guo, R.-Y. (2008). Static floor field and exit choice for pedestrian evacuation in roomswith internal obstacles and multiple exits. Physical Review E , , 021131. doi: https://doi.org/10.1103/PhysRevE.78.021131 .Karamouzas, I., Skinner, B., & Guy, S. J. (2014). Universal power law governing pedestrian interactions. Physical Review Letters , , 238701. doi: https://doi.org/10.1103/PhysRevLett.113.238701 .Karamouzas, I., Sohre, N., Narain, R., & Guy, S. J. (2017). Implicit crowds: Optimization integrator for ro-bust crowd simulation. ACM Transactions on Graphics , , 1–13. doi: https://doi.org/10.1145/3072959.3073705 .Korhonen, T. (2018). Fire dynamics simulator with evacuation: FDS+Evac. Technical Reference and User’sGuide . Technical Report VTT Technical Research Centre of Finland. URL: http://virtual.vtt.fi/virtual/proj6/fdsevac/documents/FDS+EVAC_Guide.pdf .Kretz, T., Grosse, A., Hengst, S., Kautzsch, L., Pohlmann, A., & Vortisch, P. (2011). Quickest paths insimulations of pedestrians.
Advances in Complex Systems , , 733–759. doi: https://doi.org/10.1142/S0219525911003281 .Kurdi, H. A., Al-Megren, S., Althunyan, R., & Almulifi, A. (2018). Effect of exit placement on evacuationplans. European Journal of Operational Research , , 749–759. doi: https://doi.org/10.1016/j.ejor.2018.01.050 .Li, W., Li, Y., Gong, J., & Shen, S. (2016). The Trace Model: A model for simulation of the tracing processduring evacuations in complex route environments. Simulation Modelling Practice and Theory , , 108–121.doi: https://doi.org/10.1016/j.simpat.2015.09.011 .Løv˚as, G. (1995). On performance measures for evacuation systems. European Journal of Operational Research , , 352–367. doi: https://doi.org/10.1016/0377-2217(94)00054-G .Løv˚as, G. (1998). Models of wayfinding in emergency evacuations. European Journal of Operational Research , , 371–389. doi: https://doi.org/10.1016/S0377-2217(97)00084-2 .McCormack, P., & Chen, T.-Y. (2014). Optimizing leader proportion and behavior for evacuating buildings.In Proceedings of the 2014 Symposium on Agent Directed Simulation
ADS ’14 (pp. 13:1–13:6). Society forComputer Simulation International. 29ishinari, K., Kirchner, A., Namazi, A., & Schadschneider, A. (2004). Extended floor field CA model forevacuation dynamics.
IEICE Transactions on Information and Systems , , 726–732.NIST (2020). Fire dynamics simulator. URL: https://pages.nist.gov/fds-smv/ accessed: 2020-03-16.Oliphant, T. et al. (2020). Numba: A high performance python compiler. URL: https://numba.pydata.org accessed: 2020-03-16.Pelechano, N., & Badler, N. I. (2006). Modeling crowd and trained leader behavior during building evacuation. IEEE Computer Graphics and Applications , , 80–86. doi: https://doi.org/10.1109/MCG.2006.133 .Proulx, G. (2002). Movement of people: the evacuation timing. In P. J. DiNenno et al. (Eds.), SFPE Handbookof Fire Protection Engineering (pp. 342–366). the National Fire Protection Association.Saadatseresht, M., Mansourian, A., & Taleai, M. (2009). Evacuation planning using multiobjective evolutionaryoptimization approach.
European Journal of Operational Research , , 305–314. doi: https://doi.org/10.1016/j.ejor.2008.07.032 .Schadschneider, A., Klingsch, W., Kl¨upfel, H., Kretz, T., Rogsch, C., & Seyfried, A. (2008). Evacuationdynamics: Empirical results, modeling and applications. arXiv preprint , . ArXiv:0802.1620.von Schantz, A. (2020). Minimizing the evacuation time of a crowd from a complex building using rescue guides– code (Version 1.2). Zenodo. doi: .von Schantz, A., & Ehtamo, H. (2015). Spatial game in cellular automaton evacuation model. Physical ReviewE , , 052805. doi: https://doi.org/10.1103/PhysRevE.92.052805 .von Schantz, A., & Ehtamo, H. (2019). Pushing and overtaking others in a spatial game of exit conges-tion. Physica A: Statistical Mechanics and its Applications , , 121151. doi: https://doi.org/10.1016/j.physa.2019.121151 .Sethian, J. A. (1999). Level set methods and fast marching methods: Evolving interfaces in computationalgeometry, fluid mechanics, computer vision, and materials science volume 3. Cambridge University Press.Vermuyten, H., Beli¨en, J., De Boeck, L., Reniers, G., & Wauters, T. (2016). A review of optimisation models forpedestrian evacuation and design problems.
Safety Science , , 167–178. doi: https://doi.org/10.1016/j.ssci.2016.04.001 .Wang, X., Zheng, X., & Cheng, Y. (2012). Evacuation assistants: An extended model for determining effectivelocations and optimal numbers. Physica A: Statistical Mechanics and its Applications , , 2245–2260.doi: https://doi.org/10.1016/j.physa.2011.11.051 .30eiss, T., Litteneker, A., Jiang, C., & Terzopoulos, D. (2019). Position-based real-time simulation of largecrowds. Computers & Graphics , , 12–22. doi: https://doi.org/10.1016/j.cag.2018.10.008 .Yao, Z., Wang, J. S., Liu, G. R., & Cheng, M. (2004). Improved neighbor list algorithm in molecular simu-lations using cell decomposition and data sorting method. Computer Physics Communications , , 27–35.doi: https://doi.org/10.1016/j.cpc.2004.04.004 .Zhou, M., Dong, H., Zhao, Y., Ioannou, P. A., & Wang, F.-Y. (2019). Optimization of crowd evacuationwith leaders in urban rail transit stations. IEEE Transactions on Intelligent Transportation Systems , ,4476–4487. doi: https://doi.org/10.1109/TITS.2018.2886415https://doi.org/10.1109/TITS.2018.2886415