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Network redesign for efficient crowd flow and evacuation
Accepted Manuscript
Network redesign for efficient crowd flow and evacuation Lakshay Taneja , Nomesh B. Bolia PII: DOI: Reference:
S0307-904X(17)30548-6 10.1016/j.apm.2017.08.030 APM 11942
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
23 December 2016 6 August 2017 22 August 2017
Please cite this article as: Lakshay Taneja , Nomesh B. Bolia , Network redesign for efficient crowd flow and evacuation, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.08.030
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Highlights Proposed ways for efficient crowd flow and evacuation during mass gatherings. In order to achieve a desired state, network optimization model is presented. Model aims at optimal capacity change for efficient flow through network redesign Developed model can also help in finding the suitability of a venue for event.
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ACCEPTED MANUSCRIPT Network redesign for efficient crowd flow and evacuation
Lakshaya,*, Nomesh B. Bolia a a
Abstract
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Department of Mechanical Engineering, Indian Institute of Technology Delhi, Hauz Khas, New Delhi-110016, India * Corresponding author – [email protected] Phone: +91-8375038818 Fax: +91-11-2658-2053
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Over the past few years, many crowd related accidents have happened at places of mass gatherings such as religious places, venues of sports, political and social events, and railway stations due to poor crowd management or poor system design. To mitigate the impact of such tragedies, developing adequate crowd management strategies is essential. A critical component of crowd management strategies in such emergencies is an evacuation plan. A proactive measure for evacuation strategies is to make a plan to utilize the available capacity by efficient network redesign. Redesigning the network helps in influencing the demand by enhancing or limiting the use of certain routes, without the use of force or explicit guiding mechanisms. A bi-level network based optimization model is proposed for network redesign. This approach aims at an optimal capacity change for an efficient crowd flow and evacuation during mass gatherings. To determine the optimal capacity change required to achieve the desired evacuation state, an Interior Point Algorithm (IPA) is applied. Further, to increase the computational efficiency, a heuristic is developed that finds a good starting point for IPA. Finally, the robustness of the redesigned network with respect to evacuation demand is examined along with alternate measures that further help in reducing the evacuation time.
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Keywords: Crowd management, Network redesign, Mass gatherings, Emergency evacuation
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1. Introduction
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Mass gatherings happen at a number of places for various reasons such as festivals, sports and concerts. Large crowds at these gatherings pose a challenge for infrastructural capacity. Over the past few years, many accidents have happened at these places due to overcrowding. In September 2015, around 2,177 people were reported dead due to overcrowding in the Hajj stampede (The Associated Press, 2015). A representative sample of similar events where many lives were lost is presented in Table 1. It is presumed that these accidents happen due to irrational behavior of the crowd. However, there are several other reasons related to design, information and management which may be responsible for these accidents (Still, 2016). Therefore, an appropriate design of the facility and adequate crowd management strategies are crucial for enabling a safe movement in and out of the venue. A critical component of crowd management strategies is emergency management.
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1990 2008
2010 2013 2015
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Year 1883 1989
Table 1. Past incidents of crowd related accidents (Still, 2016) Place Reason Casualties Victoria Hall (Britain) Stampede due to free toys distribution 180 Hillsborough Stadium Local police decided to open stadium gates 249 (Britain) to an already full stadium Hajj (Mina, Mecca) Overcrowding 1,426 Chamunda devi Stampede due to false rumors of bomb 249 temple, Jodhpur, Rajasthan (India) Phnom Penh Capacity of suspension bridge was exceeded 450 (Cambodia) Ratangarh temple Stampede on bridge 89 (India) Hajj (Mina, Mecca) Overcrowding 2,177
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In case of an emergency, individuals look for the exits desperately. Due to the collective movement of individuals, there is an unusual increase in demand, and hence supply (of infrastructural capacity) becomes scarce. A well-designed system is required to keep a sound balance between increased demand and supply. The system should focus on managing the flow of the crowd to move it quickly out of danger by avoiding congestion and delays. Various strategies, control approaches and voluminous studies (Chen & Hung, 1993; Edward, 1973; Feng & Miller-Hooks, 2014a; Helbing, Farkas, Molnár, & Vicsek, 2002; Shende, Singh, & Kachroo, 2011) have been reported for improving the process of evacuation. Broadly, methods for improving the efficiency of evacuation include staging the process of evacuation, controlling the flow of people optimally, and providing information and route guidance to the evacuees.
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To improve the evacuation process, it is essential to know how it evolves. For this purpose, prediction of pedestrian movement is required and various mathematical models have been developed. These include the Kinetic model (Bellomo, Clarke, Gibelli, Townsend, & Vreugdenhil, 2016), Operatorial method (Fabio Bagarello, Gargano, & Oliveri, 2015), Social force model (Xue, Jia, Jiang, & Shan, 2016) and Cellular automata model (F. Bagarello, Di Salvo, Gargano, & Oliveri, 2016; Klüpfel, 2003). These models predict the movement of people during both normal and panic situations. Table 2 lists some of these models along with their purposes. These models can predict or replicate (depending on the purpose of modelling) the movements of a crowd during an emergency. These models help in quantifying the process of the evacuation. As per Zheng, Zhong, & Liu (2009) these evacuation models can be categorized into three broad categories viz., fluid based models, simulation based models and network flow based models. Fluid dynamics based models (Hughes, 2000) involve partial differential equations obtained through section wise conservation of the pedestrians‟ mass to determine the time-based evolution of the crowd density. Due to the computational complexities involved in these models, they usually work well only for small-scale problems.
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ACCEPTED MANUSCRIPT On the other hand, simulation based models use a set of rules to evaluate specific strategies for evacuation. A review of simulation based models can be found in Zheng et al. (2009). These models can be used merely for evaluating and assessing already determined strategies and not for generating new evacuation strategies. Table 2. Models to predict pedestrian movement Model Developed
Xue, Jia, Jiang, & Shan (2016)
Social force model
Löhner (2010)
Purpose Effects of communication on evacuation efficiency
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Reference
Prediction of pedestrian movement through a passage flow Evolution of the evacuation through a non-linear differential equation. State is described by partial distribution function People‟s reaction when an alarm occurs
Kinetic model
Bagarello, Gargano, & Oliveri (2015)
Operatorial model
F. Bagarello, Di Salvo, Gargano, & Oliveri (2016)
Quantum dynamics
Change in local density of a given population within a room
Cristiani & Peri ( 2015)
Swarn optimization model
Evacuation time reduction by adding obstacles at the optimal location
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Bellomo, Clarke, Gibelli, Townsend, & Vreugdenhil (2016)
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Flow based models usually deploy some mathematical form of the network flow. These models try to find optimal ways through which the crowd should be evacuated in case of an emergency. Simulation based models are generally based on „what-if‟ scenarios while flow based models indicate „what-to-do‟ in developing evacuation plans. In network flow based models, a network of edges and nodes (based on the actual structure) represents the spatial layout of the infrastructure. Various objective functions have been used for finding the optimal routes over the network, i.e., minimum cost network flow (Yamada, 1996), quickest path flow (Chen & Hung, 1993), and universally maximum flow problem (Edward, 1973). A model based on this approach, Evacnet+ (Taylor, 1996) finds the minimum time required to evacuate a building by following an optimal plan for exit usages. It uses a coarse network (assuming rooms as nodes) and evacuation time is calculated for the capacitated network. Similar to this model, Takahashi, Tanaka, & Satoshi (1989) developed a model to find the optimal exit to be taken from a room in order to achieve a given objective. Although a lot of work has been done on modeling the crowd movement during evacuation, little has been done on controlling the process of evacuation, i.e., directing the evacuation towards the desired state. Zheng & Liu (2010) worked in this direction and developed a model to forecast the time dependent distribution of the pedestrians in place. This distribution is guided towards the desired state by controlling the splitting rates at intersections. Similarly, Shende, Singh, & Kachroo (2011) and Wadoo & Kachroo (2010) worked on a control strategy by using feedback control theory to find the optimum velocity in different sections to achieve a given evacuation time. Both these models push the flow through routes in a controlled way to
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ACCEPTED MANUSCRIPT achieve the required performance. However, implementing these controls in real life is a challenging task. An alternative to these strategies can be redesigning of the network itself.
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Redesigning is a proactive measure to utilize the available capacity effectively. Redesigning the network helps in influencing the demand by either enhancing or limiting the use of certain routes. There are several ways in which redesigning can be done such as changing capacity of the routes, closing or opening new routes, use of temporary barriers, and changing illumination of routes (Feng & Miller-Hooks, 2014a). However, a complete redesign of the network based on the surge demand is financially not feasible. Therefore, it is required to determine the location and extent of capacity change needed to achieve the desired evacuation state.
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The network should be redesigned in such a way that it supports the goals of both the crowd manager and individuals. A crowd manager wants the total system evacuation time to be minimized. Accordingly, some individuals may have to sacrifice their individual goals for the benefit of the system. In reality, individuals behave in a selfish manner (Feng & MillerHooks, 2014a) and want to choose a route which they presume is the best for them. To accomplish the goals of both, a bi-level network based optimization modeling approach has been proposed in this paper. This model aims at an optimal capacity change for an efficient crowd management during mass gatherings. Feng et al. (2014) have worked in a similar direction. However, the model proposed by them does not provide enough flexibility to individuals to choose an exit of their choice. The output of our bi-level model is the capacity change vector (of the pathways) required to meet the desired goals. In the proposed bi-level model, the upper level focuses on network redesign using a mathematical program. Details of the model are presented in section 3. The lower level model incorporates the responses of individuals to network redesign in terms of their choice of route. More clarity on the approach of the lower level problem is presented next.
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The lower level problem is a route choice problem. There is a set of available paths from which individuals have to choose a path depending on the criteria considered. Various route choice behavior models have been used in the past ranging from the simple shortest path (Yamada, 1996) to dynamic route choice behavior (Kemloh Wagoum, Seyfried, & Holl, 2012). The criterion for route choice used in this article is „utility maximization‟. The underlying assumption in utility maximization models (Hoogendoorn & Bovy, 2004) is that the preference of individuals for paths can be expressed in terms of their utility. This utility depends on a number of factors such as time, distance, safety and physical effort. Among these, walking time and walking distance are the most influencing attributes for an individual‟s route choice (Fisk, 1980). Hence, in this paper, the utility of a path is assumed to be a function of the path length and the travel time on that path. Individuals select the path that provides the highest utility to them. It is assumed that individuals choose a path based on their perceived knowledge of route performance rather than perfect knowledge. The lower level problem finds the path flow based on stochastic user equilibrium assignment (Fisk, 1980).
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ACCEPTED MANUSCRIPT The rest of this paper is structured as follows: section 2 explains how individuals choose a path from given alternatives. The mathematical formulation and solution methodology for the bi-level model is explained in Section 3. The model is non-linear in nature, therefore a heuristic for finding the initial feasible solution has been proposed in section 4. The solution methodology has been applied on a hypothetical network and the results obtained are reported in section 5. Section 6 gives the discussion and insights of developed models. Conclusions drawn and proposed future work have been presented in section 7.
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2. Pedestrian Assignment Model
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Pedestrian assignment problem attempts to determine the route that pedestrians take from the given options. The physical structure offers the set of paths from which individuals select their preferences. This physical structure can be converted to a network graph of nodes and edges by applying graph theory. In the network, a path can be defined as a sequence of joined edges from the origin to the destination. The characteristics of a path used in the model are its length, travel time and utility. The network is allowed to have multiple origins and destinations. Let O be the set of all origins, and D be the set of all destinations. Let K be the set of all paths, and E the set of all edges. Then the length Lk of path k from origin „o‟ to destination „d‟ is the summation of the length of all edges in that particular path, i.e.,
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Lk leodk ,e ,
o O; d D; k K ,
(1)
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where le is the length of edge e and odk ,e is a binary parameter, which is 1 if path k contains
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the edge e, and 0 otherwise.
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Similarly, travel time on a path can be computed as the sum of travel times on all the edges in k that particular path. Let Todk (bod ) be the travel time on path k, between origin „o‟ and
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destination „d‟, and bodk be the corresponding flow. Then,
Todk (bodk ) teodk ,e ,
o O; d D; k K
(2)
eE
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where te is the travel time on edge e. The edge travel time te can be computed using the „BPR based travel time function‟ (Branston, 1976). It is modelled as a function of the flow and capacity of an edge. The function considers congestion effect through the intensity factor (Ie) and is given by: f (3) e E te ( f e ) te0 (1 I e ( ce )2 ), e
where te0 ,fe and ce are the free flow travel time, flow and capacity of an edge e, respectively. As stated earlier, the preference of individuals for a particular path can be expressed by the path utility. Individuals are assumed to be utility maximizers, i.e., they choose a path which helps them in maximizing their utility. Let and be the sensitivity parameters (Daamen, Bovy, & Hoogendoorn, 2005) for individual preference over distance and time respectively. k Then the utility uod for a path k between origin „o‟ and destination „d‟ is modelled as:
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ACCEPTED MANUSCRIPT k k uod Lk Todk (bod ) ln( psk ),
o O; d D; k K ,
(4)
where psk is the path size factor of path k and is the corresponding sensitivity parameter. The path size factor is a correction term added to incorporate the correlation between overlapping routes. It is computed using equation 5 (Frejinger, Bierlaire, & Ben-Akiva, 2009): l 1 (5) psk t , tk Lk N t
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where Nt is the number of paths that pass through edge t. Fisk (1980) reports that Kuhn-tucker conditions of the utility maximization model generate equilibrium flows on each path that follow a logit-based assignment. The procedure for computing equilibrium flows on the edges, i.e., the lower-level problem, is presented next.
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During the evacuation, individuals may choose any exit they want. Therefore, it is a multi-destination problem which is computationally complex. An approach to convert this into a single destination problem (Han, Yuan, Chin, & Hwang, 2006) is as follows: All exits are connected to a single destination called super-sink (S) through dummy edges which have infinite capacity and zero travel time. The problem of evacuation from a given source to multiple exits is equivalent to evacuation from the source to the super-sink.
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For the resultant single destination problem, the equilibrium flow on paths can be computed quickly. The procedure for the same is shown in figure 1. The first step is to find out the set of paths from all the origins to the super-sink (S). Individuals will choose a path that will maximize their utility. Successive average with logit assignment (Feng et al., 2014) is used for finding the equilibrium flows. The logit assignment is performed based on the initial utility and then the corresponding path flows are calculated.
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k Let uoS be the utility of path k, between the origin „o‟ and super-sink S; the scaling
k exp( uoS ) p , exp( uoSr ) k oS
o O; k K
(6)
rRoS
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parameter of the utility function; RoS the set of path from origin „o‟ to super-sink S. Then the k probability ( poS ) of selecting path k from the origin „o‟ to „S‟ is given by:
Pedestrians are assigned to paths according to the computed probabilities. Let moS be the number of individuals to be evacuated from the origin „o‟ to S. Then the flow on path k ( bodk ) is given by:
k k boS mo S poS ,
o O; k K
(7)
Edge flow fe over edge e can be calculated after computing path flows using equation 8:
fe
b
oO kKoS
k oS
.oSk ,e ,
(8)
where oSk ,e is a binary parameter which is 1 if path k contains edge e, and 0 otherwise.
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Figure 1. Steps for computing equilibrium flows on paths
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Based on the current edge flows, utility values need to be updated. The logit assignment is applied again to find the new edge flow based on the current utility. This iterative process is repeated until convergence is reached.
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3. Mathematical Modelling and Formulation To assess the process of evacuation, evacuation time is used as a measure. Evacuation time is defined as the time by which the entire crowd in the hazardous zone is evacuated to safe destinations. Network redesign is used as a method to control the evacuation time. As indicated in section 1, a bi-level model is developed which takes into consideration the goal of both the crowd manager and individuals. The individual goal is achieved through their utility maximization as described in section 2. Crowd managers fulfil their goal by influencing the demand through capacity change. Individuals then try to maximize their utility for that particular capacity change. The outcome of the model is the optimal change in the network capacity required to achieve a given evacuation time. The model incorporates
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ACCEPTED MANUSCRIPT possible ceilings on the capacity change of an edge and attempts to minimize the cost of network redesign.
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The following assumptions have been made in developing the models: All the origins and their expected demand are assumed to be known and constant. Individuals have a flexibility to choose an exit of their choice. Individuals are assumed to be utility maximizers. Monetary cost for all edges are known and assumed to be constant. There is an upper limit to the increase in the capacity of an edge. The herding behavior of individuals is not considered while calculating their route choice. Let ce be the capacity of an edge e, ceup the upper limit on capacity of an edge e, ge the
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monetary cost for unit capacity change, i.e., cost incurred in changing the capacity of an edge by one unit and we the change in capacity of an edge e. Then the total budget (B) of network redesign is given by:
B g e we
(9)
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The bi-level model can then be formulated as mathematical program P1. P1:
min B
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f ,w
subject to
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e
0
ToSk Tgiven
k K ; o O
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w
e E
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0 ce we ceup
(10)
(11) (12) (13)
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The objective function (equation 10) minimizes the budget for capacity change. The budget incurred is a function of the edge flow vector (f) and capacity change vector (w). The constraints for the model are given in equations 11-13. Equation 11 puts an upper limit on the increase in the capacity of an edge. This is done to make the problem more realistic. The value of the parameter ceup comes from the relevant stakeholders such as network planners and users. Equation 12 ensures that the total space before and after redesigning is the same so that the context of the problem does not change. The objective function, i.e., the budget for the network change makes sense only when it is used in conjunction with a target evacuation time. This feature of the model is incorporated in equation 13, which shows that the travel time ToSk on the chosen path k is less than the target time Tgiven. Travel time on any path can be computed using equation 3. In calculating ToSk , the flow on the edges (fe) is required which is determined in the lower level problem explained in section 2.
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It can be clearly seen that the program P1 is a non-linear constrained optimization problem. Obtaining global minima for this program is extremely challenging. As an alternative strategy, an IPA (Byrd, Gilbert, & Nocedal, 2000) with global search is adopted to get a solution. Figure 2 describes the detailed procedure for finding a solution to the program using this methodology. The global search algorithm works by generating trial points on which an IPA is applied. All the solutions are stored in a global search object. The solution having minimum value of the objective function in the global search object is chosen as the solution of P1.
Figure 2. Solution methodology of bi-level model
Apart from calculating the capacity change vector required to bring the evacuation time less than or equal to the available time (Tgiven), the crowd manager may also be interested to know the least possible evacuation time achievable by network redesign. Evacuation time ( T ) is the maximum of travel time across all paths, i.e. T = max( ToSk ) k K ; o O . The least possible evacuation time can be obtained by solving the mathematical program P2.
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ACCEPTED MANUSCRIPT P2: min T
(14)
b, w
subject to 0 ce we ceup
0
b
moS
e
eE
kK
k oS
(15) (16)
o O
(17)
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w
e E
The objective function given by (14) seeks to find the flow vector and capacity change vector that minimize the evacuation time ( T ). The constraints 15 and 16 for program P2 are the same as constraints 11 and 12 of the program P1. The constraint 17 is for conservation of flow over different paths. Heuristic for Starting Point
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4.
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While solving a non-linear program, a good starting point often increases computational efficiency significantly. To leverage this behavior, a heuristic is developed to find a good starting point for IPA. The steps involved in the heuristic are summarized in figure 3. The initial step is to perform a logit-based assignment on the network. In the logit-based assignment, the path having the maximum travel time is recorded as Pmax. The edge having the maximum travel time among all edges on Pmax is selected. The capacity of the selected edge needs to be increased to reduce T . As stated earlier, it is ensured that the total space
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before and after redesigning is the same. Therefore, if we increase the capacity of the selected edge by , there must be an edge where the capacity needs to be reduced by the same amount .
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In order to find the edge on which the capacity needs to be reduced, the path having the least travel time, recorded as Pleast, is chosen. The capacity of the edge having the least travel time among all edges on Pleast is reduced. After this initial redesigning of the network, T is
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computed again and compared with the target evacuation time Tgiven. If T is less than Tgiven, the current capacity change vector and flow vector serve as a starting point for IPA. If it is more than Tgiven, the process is repeated until T becomes less than Tgiven. 5. Illustrative Example 5.1. Test Network To test the ability of the developed model, the solution methodology has been applied to a hypothetical network as shown in figure 4. The test network consists of 20 nodes and 30 0 up edges. Each edge of the network is characterized by le, Ie, ce, te , ce and ge. Table 3 shows
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ACCEPTED MANUSCRIPT the values of these parameters for the test network. The values of Ie and ceup are chosen
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according to other studies in the literature (Proulx, 2002).
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Figure 3. Heuristic proposed for starting point solution
Figure 4. Illustrative example network
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ACCEPTED MANUSCRIPT Various evacuation scenarios are considered for analyzing the impact of network redesign on the evacuation time. The evacuation time depends on the number of people evacuating, i.e., the evacuation demand. Therefore, to begin with, an initial scenario is considered in which the effect of evacuation demand on evacuation time has been analyzed. For this purpose, the evacuation time is calculated for different values of evacuation demand. Table 3. Parameters value of test network ce
ge
ceup
Edge
le
te0 (s)
Ie
ce
ge
ceup
49.30
8E-04
30
5
50
16
100
70.42
8E-04
30
5
50
50
35.21
8E-04
10
5
50
17
100
3
100
70.42
8E-04
30
5
50
18
200
4
100
70.42
8E-04
30
5
50
19
200
5
100
70.42
8E-04
30
5
50
20
200
6
100
70.42
8E-04
30
5
50
21
200
7
100
70.42
8E-04
30
5
50
22
225
8
100
70.42
8E-04
30
5
50
23
144
9
300
211.27
8E-04
30
5
50
24
100
10
260
183.10
8E-04
30
5
50
25
11
260
183.10
8E-04
30
5
50
12
260
183.10
8E-04
30
5
50
13
100
70.42
8E-04
20
5
14
100
70.42
8E-04
30
15
100
70.42
8E-04
30
70
2
70.42
8E-04
30
5
50
140.85
8E-04
20
5
50
140.85
8E-04
30
5
50
140.85
8E-04
30
5
50
140.85
8E-04
30
5
50
158.45
8E-04
10
5
50
101.41
8E-04
10
5
50
70.42
8E-04
20
5
50
100
70.42
8E-04
20
5
50
26
100
70.42
8E-04
20
5
50
27
100
70.42
8E-04
20
5
50
50
28
100
70.42
8E-04
20
5
50
5
50
29
100
70.42
8E-04
20
5
50
5
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t
50
30
100
70.42
8E-04
30
5
50
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le (m)
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Ie
Edge
0 e (s)
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For further analysis and better clarity of exposition, a specific value of evacuation demand is chosen (3,000). For this demand, the evacuation time ( T ) is calculated using the pedestrian assignment model detailed in section 2. To control this process of evacuation, the impact of
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network redesign on T is further analyzed. The bi-level model, i.e., program P1, is applied
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on the test network to achieve the target evacuation time. To increase the computational efficiency of the model, the heuristic developed in section 4 is applied to provide a good starting point for the IPA. The model gives the minimum budget required for different values of the desired state. 5.2. Results
The evacuation scenario is as follows: due to some emergency, all the people need to be evacuated to a safe destination. It is assumed that people shall evacuate spontaneously (i.e., without any guidance mechanism, and as soon as signaled) attempting to maximize their utility. This self-guided evacuation behavior is key aspect to model real life behavior. In all scenarios, nodes 1 and 6 are considered as origins while nodes 16 and 20 are the safedestinations. This is a multi-destination problem, which needs to be converted to a single destination problem by connecting all the destinations to S as shown in figure 5.
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Figure 5. Super-sink added to the test network
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All paths from the origins to S are recorded. Table 4 shows the list of these paths. During the evacuation, free flow speed on all the links should be known in advance. Then the parameter
te0 is calculated by taking the ratio of the length of an edge to its free speed. As the free flow
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speed increases, link travel time decreases (by equation 3) which in-turn decreases the path travel time. Evacuation time is calculated as the maximum travel time over all the paths. The path travel time is given by equation 2. To find the effect of free speed on evacuation time, a series of experiments is performed. The experiments reveal that as the free flow speed increases, evacuation time reduces and varies almost linearly with the free flow speed. For the purpose of illustration, a free flow speed of 1.42 m/s (Thalmann, Musse, & Braun, 2007) is assumed for all the links. The results for this entire analysis are presented next in sections 5.2.1-5.2.2.
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5.2.1. Evacuation Time without Network Redesign
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The evacuation time can be calculated once flows on the edges are known. The method to calculate the equilibrium edge flows is presented in section 2. As stated earlier, the evacuation time depends on the evacuation demand. In the test network, various values of the total demand, distributed equally over the origins are considered. Figure 6 summarizes these results. It can be seen that T increases with an increase in the total evacuation demand, as expected.
Next, the total demand is considered to be 3,000, and it is distributed equally across the origins. For this scenario, T is 1016.3 seconds. Table 5 shows the equilibrium edge flows in the column „original network flow‟. This evacuation time is without any network redesign. The crowd manager wants this time to be reduced to a specific value, i.e., the target evacuation time, Tgiven. In effect, the crowd manager wants to control the process of evacuation, i.e., directing the evacuation towards a desired state by redesigning the network configuration constrained through equation 11 and 12 of program P1. The rest of this section
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ACCEPTED MANUSCRIPT describes the results of applying this model to a variety of scenarios developed using different values of Tgiven. The details and results are presented in section 5.2.2. Table 4. List of paths from origins (1 and 6) to super-sink (S) S No.
S No.
1-S
1
1 - 2 - 3 - 4 - 5 - 10 - 15 - 16 – S
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1 - 7 - 8 - 9 - 10 - 15 - 16 - S
2
1 - 2 - 3 - 4 - 5 - 10 - 15 - 20 – S
15
1 - 7 - 8 - 9 - 10 - 15 - 20 - S
3
1 - 2 - 3 - 4 - 5 - 10 - 16 - S
16
1 - 7 - 8 - 9 - 10 - 16 - S
4
1 - 2 - 3 - 4 - 9 - 10 - 15 - 16 - S
17
1 - 7 - 8 - 13 - 14 - 15 - 16 - S
5
1 - 2 - 3 - 4 - 9 - 10 - 15 - 20 - S
18
1 - 7 - 8 - 13 - 14 - 15 - 20 - S
6
1 - 2 - 3 - 4 - 9 - 10 - 16 - S
19
1 - 7 - 8 - 13 - 14 - 19 - 20 - S
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5.2.2. Evacuation Time with Network Redesign Recall that network redesign can be used as a method to control the evacuation time, i.e., to achieve the target evacuation time (Tgiven). Redesigning the network helps in influencing the demand by enhancing or limiting the use of certain routes without the use of force. The bilevel model of program P1 has been applied on the test network to achieve the desired evacuation time. For various values of the target evacuation time, minimum budget required is calculated. The solution method used is IPA with global search and the starting point heuristic presented in section 4. More details are available in section 3. Figure 7 shows the budget required for various values of Tgiven and a total evacuation demand of 3,000. From the figure, the crowd manager can determine the budget required to achieve a particular evacuation time. It can also be observed from the figure that the minimum evacuation time Tmin that can be achieved by network redesigning is 774.86 seconds Tmin is obtained by solving Program P2 explained in section 4. The corresponding budget for Tmin is
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2,489.176. Table 5 shows the optimal capacity change vector (we) and the corresponding edge flow vector (fe) to achieve Tmin (in the column „min. achievable time‟ of the table).
Figure 6. Variation of evacuation time with total evacuation demand
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It is also observed from table 5 that the system moves towards a uniform exit usage in the redesigned network as Tgiven is decreased. For the original network, 1,721 individuals select destination 20, while destination 16 is selected by 1,279 individuals and the total evacuation time is 1,016.3 seconds. Consider the scenario where Tgiven is chosen to be 895 seconds. It is seen that the flows at the destinations become 1,529 and 1,471 (as indicated in the column „particular time‟ of table 5), i.e., more uniform than the original flow of 1,721 and 1,279, respectively. We further reduce Tgiven and observe the trend of destination flows getting more uniform. Lastly, as indicated above, for Tgiven equal to Tmin , the destination flows of 1,490
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and 1,510 are much more uniform than the original network. It can be interpreted from these results that the crowd manager is able to achieve the system goals of evacuees changing their exit route preferences to make destination flows more uniform, albeit without using any force.
Figure 7. Budget verses evacuation state required
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ACCEPTED MANUSCRIPT Table 5. Values of Edge flow (fe) and Capacity change (we) obtained for various scenarios original network flow particular time min. achievable time
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fe 1055 1055 792 630 8 406 611 772 445 263 161 630 469 502 479 381 205 1492 48 58 97 1305 1023 1038 1118 1217 15 80 99 273 1490 1510
we 9.46 34.06 7.52 13.55 -29.85 -9.33 -2.08 6.26 3.38 -9.49 -17.94 9.33 -4.37 -5.91 -7.26 -13.96 -18.65 26.19 -23.78 -23.75 -21.27 39.90 20.75 20.56 28.11 29.85 -17.68 -14.55 -14.39 -14.67 ---
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we 0.01 12.02 0 -0.51 -17.04 0.05 0 0 0.58 -1.64 -1.3 0 -0.04 -0.07 0 -0.01 0.02 0.01 -0.01 0 0 11.38 -1.36 0 0.01 0.87 -0.19 -1.61 -1.15 -0.03 ---
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fe 968 968 704 490 728 805 662 876 532 263 214 490 495 747 987 851 747 772 455 407 642 724 277 480 647 783 203 167 136 746 1529 1471
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fe 776 776 619 469 919 977 630 780 724 158 150 469 306 757 1060 911 882 581 665 504 852 397 275 490 691 840 215 201 149 882 1721 1279
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Edges 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 20-S 16-S
To increase the computational efficiency, the developed heuristic is applied to generate a good starting point for IPA. The steps involved in the heuristic are shown in figure 3. The details, applied to a specific instance, are as follows. Consider the scenario of achieving a target evacuation time of 956 seconds, i.e. Tgiven=956 seconds. The heuristic is then applied to find a good starting point solution for IPA. Initially a logit-based assignment is performed on the test network and it is seen that path 9 has the maximum travel time. The edges on path 9 are 1, 2, 10, 7, 8, and 22 as seen in table 4. The
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Following the steps shown in the figure 3, it is determined that the capacity of edge 22 needs to be increased by 30. To keep the total space fixed, the capacity of edges 27, 28 and 29 should be reduced by 10 each. The resultant evacuation time is 950.36 seconds, which is less than the required 956.15 seconds. This satisfies the termination condition of the heuristic in figure 3. Hence, this point is selected as a starting point for the IPA. Table 6 compares the computational time (cT) required to find the solution from the model using global search, i.e., the standard library of MATLAB, to that with using our heuristic for the IPA corresponding to various values of target evacuation time (from the values already considered for budget calculations i.e. figure 7). It can be seen from the table that the budget computed is comparable for both the methods. However, the cT is significantly less when the heuristic is used. Thus, the heuristic increases the computational efficiency of the solution.
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Table 6. Budget and computational time using global search and using heuristic for IPA Budget using Global Search 121.28
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(cT): 11203 sec
(cT): 135.41 sec
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5.2.3. Algorithm Scale with Number of Edges With an increase in the number of edges and nodes, the number of alternate paths (from which individuals have to make a decision) increases as well. Due to increase in the number of paths, the computational time increases. Simulations are performed for the hypothetical network by running scenarios to find minimum achievable time for different network configurations. Each time four nodes are increased at the end of the network, seven edges corresponding to these four nodes are added. Figure 8a shows the part of the network increased during the different simulations. In each simulation, a part shown in the figure is added to the network.
Figure 8a. Additional part to be added to test network
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The evacuation demand of 3,000 is considered for all these simulations. Further, to find the computational time for each simulation of different network configurations, an initial starting point is given as the upper bound of the capacity for all the links. The results show that with an increase in number of edges and nodes, computation time increases non-linearly. Figure 8b shows the computation times for different simulations. Further, a polynomial trendline of order 2 is fitted to these results and its R-squared value comes out to be 0.9524.
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Figure 8b. Computational time for different number of nodes in the test network
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5.3. Robustness Analysis
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We further note that the evacuation demand may vary with time. Therefore, it is important for the redesigned network to be robust with respect to the evacuation demand. To check the robustness of our proposed solution, the network redesigned for a particular demand should be used for other values of the demand. If the evacuation time is still less than the original evacuation time (i.e., the evacuation time without any network redesign), it indicates the robustness of the proposed approach. Consider the scenario with a total demand of 3,000 and a target evacuation time of 895 seconds. The solution of the model gives the location and the extent of the capacity change required to achieve Tgiven. Network is redesigned according to the solution. Now consider the redesigned network with a total demand of 3,500. The evacuation time for this demand turns out to be 1020.8 seconds, which is less than the original evacuation time of 1,156.8 seconds. Figure 9 summarizes the evacuation time for different values of the total demand and Tgiven corresponding to the network optimally redesigned for a demand of 3,000. The results indicate that the model solution is robust for upto 15% variation in total demand (500 out of 3,000).
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Figure 9. Effect of change in demand on evacuation time 5.4. Alternate Measures
5.4.1. Location of Additional Emergency Exit
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The evacuation time can also be reduced if an emergency exit, a new edge etc. is added to the physical network. In the test network, there are four locations where an emergency exit can be built as shown in table 7. As computed in section 5.2.1, the evacuation time without adding an emergency exit is 1016. 3 seconds. The evacuation time at different locations after adding an emergency exit is reported in table 7.
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It can be seen from the table that adding an emergency exit at location 3 is the most useful. In addition, it can be observed from table 7 that the evacuation time could increase if an emergency exit is added to location 2. That happens because more congestion builds up near exits. Therefore, it is important to know the location(s) where an emergency exit should be added and the methodology proposed in the paper can determine the same.
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Further, a scenario has been considered where evacuation from only one exit is possible. Specifically, the exit at node 16 is not accessible. The two aims of considering this scenario are: i) to find the evacuation time when only one destination exists, and ii) to check the robustness in the identification of the emergency exit location. The evacuation time in this scenario comes out to be 1172.9 seconds which is greater, as expected, than that of the original test network as crowding and congestion increase due to a single exit. In addition, the effect of adding an additional exit is also studied in this scenario. We check all four alternatives for the emergency exit locations. It is observed that in this scenario, similar to the scenario with only two exits, adding an emergency exit at location 3 is the most useful (table 7, column „1D Evac time‟).
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5.4.2. Additional capacity
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While developing the bi-level model, it has been assumed that the total space before and after the redesign is the same. However, in some situations it might be possible to increase the total capacity. The method proposed in the paper can be used to estimate the benefit in those situations as well. To do so, the right hand side of equation 12 is increased from zero. As an illustration, table 8 presents a comparison when the right hand side is made two instead of zero. As expected, the required budget in the former case is lesser and the exact values are reported in the table. Table 8. Effect on budget of additional capacity Tgiven Budget (sumwe=0) Budget (sumwe= <=2) 835.2901 640.64 634.02 895.7201 251.715 240.47 956.1501 121.28 115.3 1016.3 0 -
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To make the evacuation process efficient, it is required to keep a sound balance between increased demand and the available network supply. To do so, different ways can be used including controlling the flow of the people, and determining the optimal velocity for different sections.
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However, implementing these controls to make evacuation efficient may be a challenging task. Therefore, an alternative is to manage the supply itself, i.e., redesign the network configuration, if possible. For this purpose, mathematical models have been developed. The results of these models help the crowd manager in the following ways:
Planning for a crisis: This model can find ways which control the process of an evacuation. Increasing safety: The models determine the modification required in the venue design for enhanced safety. Venue suitability: The model can help in finding the suitability of the venue for a given number of attendees of an event. Specifically, even after modifications, if the
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evacuation time cannot be reduced to an acceptable level, then the venue is not suitable for the event. Uniform exit usage: It has been demonstrated by the results that due to redesign, the system moves towards a more uniform usage of alternative exits, as desired by the crowd manager. Location of emergency exits: Different locations can be compared based on the evacuation time and the best ones can be chosen. Emergency staff training: The model can also help in finding the areas that will be most congested. This can help the crowd manager in training the emergency staff appropriately. Thus, we can conclude that the models can help the emergency manager make several key decisions for emergency management. 7. Conclusions and Way Forward
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Appropriate crowd management strategies are essential at places of mass gatherings for safe and smooth movement of pedestrians. These strategies must include planning for the evacuation in emergency situations that may occur at these places. A proactive measure that can be adopted for evacuation strategies is to make a plan to utilize the available capacity by an efficient network redesign. Redesigning the network helps in influencing the demand by enhancing/limiting the use of certain routes without the use of force or an explicit guiding mechanism. To accomplish the redesigning, a bi-level model is presented which determines the location and extent of change in the capacities required to achieve a desired evacuation state. The Interior Point Algorithm (IPA) with global search has been applied to determine the optimal capacity change vector to achieve the target evacuation time. Practical ways to redesign are changing the capacity of routes, closing or opening new routes, use of temporary barriers, and changing illumination of routes. A heuristic to compute a good starting point for better computational efficiency is also presented. The robustness of the redesigned network with respect to the evacuation demand is reported. Additional alternatives are proposed to further decrease the evacuation time, and the effects of implementing them using the proposed model are presented.
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Apart from developing crowd management strategies, this model may also help in determining the suitability of a venue. If the minimum evacuation time cannot meet the strategic evacuation time even by redesigning using program P2, then that venue may not be suitable. The developed model is tested on a hypothetical network and the results are encouraging. The results demonstrate that appropriate strategies can be formulated (made more efficient using the proposed heuristic), and the network redesign is indeed robust with respect to the evacuation demand. Further, the effect of alternate strategies is quantified to assist better decision making. It is observed that through network redesign, the evacuation process evolves towards a uniform use of all exits as desired from a management perspective. Thus, the system goals are achieved by influencing individual decisions and not by forcing them.
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References
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Bagarello, F., Di Salvo, R., Gargano, F., & Oliveri, F. (2016). ( H, ρ)-induced dynamics and the quantum game of life. Applied Mathematical Modelling, 43, 15–32. https://doi.org/10.1080/00221687609499673 Bagarello, F., Gargano, F., & Oliveri, F. (2015). A phenomenological operator description of dynamics of crowds: Escape strategies. Applied Mathematical Modelling, 39(8), 2276– 2294. https://doi.org/10.1016/j.apm.2014.10.038 Bellomo, N., Clarke, D., Gibelli, L., Townsend, P., & Vreugdenhil, B. J. (2016). Human behaviours in evacuation crowd dynamics: From modelling to “big data” toward crisis management. Physics of Life Reviews, 18, 1–21. https://doi.org/10.1016/j.plrev.2016.05.014 Branston, D. (1976). Link capacity functions: A review. Transportation Research, 10(4), 223–236. https://doi.org/10.1016/0041-1647(76)90055-1 Byrd, R. H., Gilbert, J. C., & Nocedal, J. (2000). A trust region method based on interior point techniques for nonlinear programming. Mathematical Programming, 89(1), 149– 185. https://doi.org/10.1007/s101070000189 Chen, G. H., & Hung, Y. C. (1993). On the quickest path problem. Information Processing Letters, 46(3), 125–128. https://doi.org/10.1016/0020-0190(93)90057-G Cristiani, E., & Peri, D. (2015). Handling obstacles in pedestrian simulations: Models and optimization. Applied Mathematical Modelling, 45, 285–302. https://doi.org/10.1016/j.apm.2016.12.020 Daamen, W., Bovy, P. H. L., & Hoogendoorn, S. P. (2005). Influence of changes in level on passenger route choice in railway stations. Transportation Research Record, (1930), 12– 20. https://doi.org/10.3141/1930-02 Edward, M. (1973). Maximal , Lexicographic , and Dynamic Network Flows. Operations Research, 21(2), 517–527. Feng, L., & Miller-Hooks, E. (2014). A network optimization-based approach for crowd management in large public gatherings. Transportation Research Part C: Emerging Technologies, 42, 182–199. https://doi.org/10.1016/j.trc.2014.01.017 Feng, L., Miller-Hooks, E., & Brannigan, V. (2014). Mathematical Modeling of Commandand-Control Strategies in Crowd Movement. Transportation Research Record: Journal of the Transportation Research Board, 2459(1), 47–53. https://doi.org/10.3141/2459-06 Fisk, C. (1980). Some developments in equilibrium traffic assignment. Transportation Research Part B: Methodological, 14(3), 243–255. https://doi.org/10.1016/01912615(80)90004-1 Frejinger, E., Bierlaire, M., & Ben-Akiva, M. (2009). Sampling of alternatives for route choice modeling. Transportation Research Part B: Methodological, 43(10), 984–994. https://doi.org/10.1016/j.trb.2009.03.001 Han, L. D., Yuan, F., Chin, S. M., & Hwang, H. (2006). Global optimization of emergency evacuation assignments. Interfaces, 36(6), 502–513. https://doi.org/10.1287/inte.1060.0251 Helbing, D., Farkas, I. J., Molnár, P., & Vicsek, T. (2002). Simulation of pedestrian crowds in normal and evacuation situations. Pedestrian and Evacuation Dynamics, 21, 21–58.
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Hoogendoorn, S. P., & Bovy, P. H. L. (2004). Pedestrian route-choice and activity scheduling theory and models. Transportation Research Part B: Methodological, 38(2), 169–190. https://doi.org/10.1016/S0191-2615(03)00007-9 Hughes, R. L. (2000). The flow of large crowds of pedestrians. Mathematics and Computers in Simulation, 53(July), 367–370. https://doi.org/10.1016/S0378-4754(00)00228-7 Kemloh Wagoum, A. U., Seyfried, A., & Holl, S. (2012). Modeling the Dynamic Route Choice of Pedestrians To Assess the Criticality of Building Evacuation. Advances in Complex Systems, 15(7), 1–22. https://doi.org/10.1142/S0219525912500294 Klüpfel, H. L. (2003). A Cellular Automaton Model for Crowd Movement and Egress Simulation. Tese de Doutorado, Duisburg-Essem University. Löhner, R. (2010). On the modeling of pedestrian motion. Applied Mathematical Modelling, 34(2), 366–382. https://doi.org/10.1016/j.apm.2009.04.017 Proulx, G. (2002). Proulx, G. Movement of People: The Evacuation Timing. SFPE HandBook of Fire Protection Engineering, 3, 342–366. Shende, A., Singh, M. P., & Kachroo, P. (2011). Optimization-based feedback control for pedestrian evacuation from an exit corridor. IEEE Transactions on Intelligent Transportation Systems, 12(4), 1167–1176. https://doi.org/10.1109/TITS.2011.2146251 Still, G. K. (2016). Crowd Disasters. Retrieved May 1, 2016, from http://www.gkstill.com/ExpertWitness/CrowdDisasters.html Takahashi, K., Tanaka, T., & Satoshi, K. (1989). An Evacuation Model for Use in Fire Safety Design of Buildings. Fire Safety Science, 2, 551–560. https://doi.org/10.3801/IAFSS.FSS.2-551 Taylor, I. R. (1996). A Revised Interface for Evacnet+. In Proceedings of 7th International Fire Science and Engineering Conference, Interflam. Thalmann, D., Musse, S. R., & Braun, A. (2007). Crowd simulation. John Wiley & Sons, Inc. The Associated Press. (2015, October). Saudi Arabia Hajj disaster death toll rises. Retrieved from http://america.aljazeera.com/articles/2015/10/19/hajj-disaster-death-toll-over-twothousand.html Wadoo, S. A., & Kachroo, P. (2010). Feedback control of crowd evacuation in one dimension. IEEE Transactions on Intelligent Transportation Systems, 11(1), 182–193. https://doi.org/10.1109/TITS.2010.2040080 Xue, S., Jia, B., Jiang, R., & Shan, J. (2016). Pedestrian evacuation in view and hearing limited condition: The impact of communication and memory. Physics Letters, Section A: General, Atomic and Solid State Physics, 380(38), 3029–3035. https://doi.org/10.1016/j.physleta.2016.07.030 Yamada, T. (1996). A Network Flow Approach to a City Emergency Evacuation Planning. International Journal of Systems Science. https://doi.org/10.1080/00207729608929296 Zheng, X., & Liu, M. (2010). Forecasting model for pedestrian distribution under emergency evacuation. Reliability Engineering and System Safety, 95(11), 1186–1192. https://doi.org/10.1016/j.ress.2010.07.005 Zheng, X., Zhong, T., & Liu, M. (2009). Modeling crowd evacuation of a building based on seven methodological approaches. Building and Environment, 44(3), 437–445.
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Network redesign for efficient crowd flow and evacuation Lakshay Taneja , Nomesh B. Bolia PII: DOI: Reference:
S0307-904X(17)30548-6 10.1016/j.apm.2017.08.030 APM 11942
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
23 December 2016 6 August 2017 22 August 2017
Please cite this article as: Lakshay Taneja , Nomesh B. Bolia , Network redesign for efficient crowd flow and evacuation, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.08.030
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Highlights Proposed ways for efficient crowd flow and evacuation during mass gatherings. In order to achieve a desired state, network optimization model is presented. Model aims at optimal capacity change for efficient flow through network redesign Developed model can also help in finding the suitability of a venue for event.
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Lakshaya,*, Nomesh B. Bolia a a
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Department of Mechanical Engineering, Indian Institute of Technology Delhi, Hauz Khas, New Delhi-110016, India * Corresponding author – [email protected] Phone: +91-8375038818 Fax: +91-11-2658-2053
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Over the past few years, many crowd related accidents have happened at places of mass gatherings such as religious places, venues of sports, political and social events, and railway stations due to poor crowd management or poor system design. To mitigate the impact of such tragedies, developing adequate crowd management strategies is essential. A critical component of crowd management strategies in such emergencies is an evacuation plan. A proactive measure for evacuation strategies is to make a plan to utilize the available capacity by efficient network redesign. Redesigning the network helps in influencing the demand by enhancing or limiting the use of certain routes, without the use of force or explicit guiding mechanisms. A bi-level network based optimization model is proposed for network redesign. This approach aims at an optimal capacity change for an efficient crowd flow and evacuation during mass gatherings. To determine the optimal capacity change required to achieve the desired evacuation state, an Interior Point Algorithm (IPA) is applied. Further, to increase the computational efficiency, a heuristic is developed that finds a good starting point for IPA. Finally, the robustness of the redesigned network with respect to evacuation demand is examined along with alternate measures that further help in reducing the evacuation time.
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Keywords: Crowd management, Network redesign, Mass gatherings, Emergency evacuation
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1. Introduction
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Mass gatherings happen at a number of places for various reasons such as festivals, sports and concerts. Large crowds at these gatherings pose a challenge for infrastructural capacity. Over the past few years, many accidents have happened at these places due to overcrowding. In September 2015, around 2,177 people were reported dead due to overcrowding in the Hajj stampede (The Associated Press, 2015). A representative sample of similar events where many lives were lost is presented in Table 1. It is presumed that these accidents happen due to irrational behavior of the crowd. However, there are several other reasons related to design, information and management which may be responsible for these accidents (Still, 2016). Therefore, an appropriate design of the facility and adequate crowd management strategies are crucial for enabling a safe movement in and out of the venue. A critical component of crowd management strategies is emergency management.
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1990 2008
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Year 1883 1989
Table 1. Past incidents of crowd related accidents (Still, 2016) Place Reason Casualties Victoria Hall (Britain) Stampede due to free toys distribution 180 Hillsborough Stadium Local police decided to open stadium gates 249 (Britain) to an already full stadium Hajj (Mina, Mecca) Overcrowding 1,426 Chamunda devi Stampede due to false rumors of bomb 249 temple, Jodhpur, Rajasthan (India) Phnom Penh Capacity of suspension bridge was exceeded 450 (Cambodia) Ratangarh temple Stampede on bridge 89 (India) Hajj (Mina, Mecca) Overcrowding 2,177
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In case of an emergency, individuals look for the exits desperately. Due to the collective movement of individuals, there is an unusual increase in demand, and hence supply (of infrastructural capacity) becomes scarce. A well-designed system is required to keep a sound balance between increased demand and supply. The system should focus on managing the flow of the crowd to move it quickly out of danger by avoiding congestion and delays. Various strategies, control approaches and voluminous studies (Chen & Hung, 1993; Edward, 1973; Feng & Miller-Hooks, 2014a; Helbing, Farkas, Molnár, & Vicsek, 2002; Shende, Singh, & Kachroo, 2011) have been reported for improving the process of evacuation. Broadly, methods for improving the efficiency of evacuation include staging the process of evacuation, controlling the flow of people optimally, and providing information and route guidance to the evacuees.
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To improve the evacuation process, it is essential to know how it evolves. For this purpose, prediction of pedestrian movement is required and various mathematical models have been developed. These include the Kinetic model (Bellomo, Clarke, Gibelli, Townsend, & Vreugdenhil, 2016), Operatorial method (Fabio Bagarello, Gargano, & Oliveri, 2015), Social force model (Xue, Jia, Jiang, & Shan, 2016) and Cellular automata model (F. Bagarello, Di Salvo, Gargano, & Oliveri, 2016; Klüpfel, 2003). These models predict the movement of people during both normal and panic situations. Table 2 lists some of these models along with their purposes. These models can predict or replicate (depending on the purpose of modelling) the movements of a crowd during an emergency. These models help in quantifying the process of the evacuation. As per Zheng, Zhong, & Liu (2009) these evacuation models can be categorized into three broad categories viz., fluid based models, simulation based models and network flow based models. Fluid dynamics based models (Hughes, 2000) involve partial differential equations obtained through section wise conservation of the pedestrians‟ mass to determine the time-based evolution of the crowd density. Due to the computational complexities involved in these models, they usually work well only for small-scale problems.
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Xue, Jia, Jiang, & Shan (2016)
Social force model
Löhner (2010)
Purpose Effects of communication on evacuation efficiency
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Prediction of pedestrian movement through a passage flow Evolution of the evacuation through a non-linear differential equation. State is described by partial distribution function People‟s reaction when an alarm occurs
Kinetic model
Bagarello, Gargano, & Oliveri (2015)
Operatorial model
F. Bagarello, Di Salvo, Gargano, & Oliveri (2016)
Quantum dynamics
Change in local density of a given population within a room
Cristiani & Peri ( 2015)
Swarn optimization model
Evacuation time reduction by adding obstacles at the optimal location
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Bellomo, Clarke, Gibelli, Townsend, & Vreugdenhil (2016)
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Flow based models usually deploy some mathematical form of the network flow. These models try to find optimal ways through which the crowd should be evacuated in case of an emergency. Simulation based models are generally based on „what-if‟ scenarios while flow based models indicate „what-to-do‟ in developing evacuation plans. In network flow based models, a network of edges and nodes (based on the actual structure) represents the spatial layout of the infrastructure. Various objective functions have been used for finding the optimal routes over the network, i.e., minimum cost network flow (Yamada, 1996), quickest path flow (Chen & Hung, 1993), and universally maximum flow problem (Edward, 1973). A model based on this approach, Evacnet+ (Taylor, 1996) finds the minimum time required to evacuate a building by following an optimal plan for exit usages. It uses a coarse network (assuming rooms as nodes) and evacuation time is calculated for the capacitated network. Similar to this model, Takahashi, Tanaka, & Satoshi (1989) developed a model to find the optimal exit to be taken from a room in order to achieve a given objective. Although a lot of work has been done on modeling the crowd movement during evacuation, little has been done on controlling the process of evacuation, i.e., directing the evacuation towards the desired state. Zheng & Liu (2010) worked in this direction and developed a model to forecast the time dependent distribution of the pedestrians in place. This distribution is guided towards the desired state by controlling the splitting rates at intersections. Similarly, Shende, Singh, & Kachroo (2011) and Wadoo & Kachroo (2010) worked on a control strategy by using feedback control theory to find the optimum velocity in different sections to achieve a given evacuation time. Both these models push the flow through routes in a controlled way to
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Redesigning is a proactive measure to utilize the available capacity effectively. Redesigning the network helps in influencing the demand by either enhancing or limiting the use of certain routes. There are several ways in which redesigning can be done such as changing capacity of the routes, closing or opening new routes, use of temporary barriers, and changing illumination of routes (Feng & Miller-Hooks, 2014a). However, a complete redesign of the network based on the surge demand is financially not feasible. Therefore, it is required to determine the location and extent of capacity change needed to achieve the desired evacuation state.
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The network should be redesigned in such a way that it supports the goals of both the crowd manager and individuals. A crowd manager wants the total system evacuation time to be minimized. Accordingly, some individuals may have to sacrifice their individual goals for the benefit of the system. In reality, individuals behave in a selfish manner (Feng & MillerHooks, 2014a) and want to choose a route which they presume is the best for them. To accomplish the goals of both, a bi-level network based optimization modeling approach has been proposed in this paper. This model aims at an optimal capacity change for an efficient crowd management during mass gatherings. Feng et al. (2014) have worked in a similar direction. However, the model proposed by them does not provide enough flexibility to individuals to choose an exit of their choice. The output of our bi-level model is the capacity change vector (of the pathways) required to meet the desired goals. In the proposed bi-level model, the upper level focuses on network redesign using a mathematical program. Details of the model are presented in section 3. The lower level model incorporates the responses of individuals to network redesign in terms of their choice of route. More clarity on the approach of the lower level problem is presented next.
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The lower level problem is a route choice problem. There is a set of available paths from which individuals have to choose a path depending on the criteria considered. Various route choice behavior models have been used in the past ranging from the simple shortest path (Yamada, 1996) to dynamic route choice behavior (Kemloh Wagoum, Seyfried, & Holl, 2012). The criterion for route choice used in this article is „utility maximization‟. The underlying assumption in utility maximization models (Hoogendoorn & Bovy, 2004) is that the preference of individuals for paths can be expressed in terms of their utility. This utility depends on a number of factors such as time, distance, safety and physical effort. Among these, walking time and walking distance are the most influencing attributes for an individual‟s route choice (Fisk, 1980). Hence, in this paper, the utility of a path is assumed to be a function of the path length and the travel time on that path. Individuals select the path that provides the highest utility to them. It is assumed that individuals choose a path based on their perceived knowledge of route performance rather than perfect knowledge. The lower level problem finds the path flow based on stochastic user equilibrium assignment (Fisk, 1980).
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ACCEPTED MANUSCRIPT The rest of this paper is structured as follows: section 2 explains how individuals choose a path from given alternatives. The mathematical formulation and solution methodology for the bi-level model is explained in Section 3. The model is non-linear in nature, therefore a heuristic for finding the initial feasible solution has been proposed in section 4. The solution methodology has been applied on a hypothetical network and the results obtained are reported in section 5. Section 6 gives the discussion and insights of developed models. Conclusions drawn and proposed future work have been presented in section 7.
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2. Pedestrian Assignment Model
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Pedestrian assignment problem attempts to determine the route that pedestrians take from the given options. The physical structure offers the set of paths from which individuals select their preferences. This physical structure can be converted to a network graph of nodes and edges by applying graph theory. In the network, a path can be defined as a sequence of joined edges from the origin to the destination. The characteristics of a path used in the model are its length, travel time and utility. The network is allowed to have multiple origins and destinations. Let O be the set of all origins, and D be the set of all destinations. Let K be the set of all paths, and E the set of all edges. Then the length Lk of path k from origin „o‟ to destination „d‟ is the summation of the length of all edges in that particular path, i.e.,
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Lk leodk ,e ,
o O; d D; k K ,
(1)
eE
where le is the length of edge e and odk ,e is a binary parameter, which is 1 if path k contains
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the edge e, and 0 otherwise.
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Similarly, travel time on a path can be computed as the sum of travel times on all the edges in k that particular path. Let Todk (bod ) be the travel time on path k, between origin „o‟ and
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destination „d‟, and bodk be the corresponding flow. Then,
Todk (bodk ) teodk ,e ,
o O; d D; k K
(2)
eE
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where te is the travel time on edge e. The edge travel time te can be computed using the „BPR based travel time function‟ (Branston, 1976). It is modelled as a function of the flow and capacity of an edge. The function considers congestion effect through the intensity factor (Ie) and is given by: f (3) e E te ( f e ) te0 (1 I e ( ce )2 ), e
where te0 ,fe and ce are the free flow travel time, flow and capacity of an edge e, respectively. As stated earlier, the preference of individuals for a particular path can be expressed by the path utility. Individuals are assumed to be utility maximizers, i.e., they choose a path which helps them in maximizing their utility. Let and be the sensitivity parameters (Daamen, Bovy, & Hoogendoorn, 2005) for individual preference over distance and time respectively. k Then the utility uod for a path k between origin „o‟ and destination „d‟ is modelled as:
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ACCEPTED MANUSCRIPT k k uod Lk Todk (bod ) ln( psk ),
o O; d D; k K ,
(4)
where psk is the path size factor of path k and is the corresponding sensitivity parameter. The path size factor is a correction term added to incorporate the correlation between overlapping routes. It is computed using equation 5 (Frejinger, Bierlaire, & Ben-Akiva, 2009): l 1 (5) psk t , tk Lk N t
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where Nt is the number of paths that pass through edge t. Fisk (1980) reports that Kuhn-tucker conditions of the utility maximization model generate equilibrium flows on each path that follow a logit-based assignment. The procedure for computing equilibrium flows on the edges, i.e., the lower-level problem, is presented next.
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During the evacuation, individuals may choose any exit they want. Therefore, it is a multi-destination problem which is computationally complex. An approach to convert this into a single destination problem (Han, Yuan, Chin, & Hwang, 2006) is as follows: All exits are connected to a single destination called super-sink (S) through dummy edges which have infinite capacity and zero travel time. The problem of evacuation from a given source to multiple exits is equivalent to evacuation from the source to the super-sink.
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For the resultant single destination problem, the equilibrium flow on paths can be computed quickly. The procedure for the same is shown in figure 1. The first step is to find out the set of paths from all the origins to the super-sink (S). Individuals will choose a path that will maximize their utility. Successive average with logit assignment (Feng et al., 2014) is used for finding the equilibrium flows. The logit assignment is performed based on the initial utility and then the corresponding path flows are calculated.
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k Let uoS be the utility of path k, between the origin „o‟ and super-sink S; the scaling
k exp( uoS ) p , exp( uoSr ) k oS
o O; k K
(6)
rRoS
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parameter of the utility function; RoS the set of path from origin „o‟ to super-sink S. Then the k probability ( poS ) of selecting path k from the origin „o‟ to „S‟ is given by:
Pedestrians are assigned to paths according to the computed probabilities. Let moS be the number of individuals to be evacuated from the origin „o‟ to S. Then the flow on path k ( bodk ) is given by:
k k boS mo S poS ,
o O; k K
(7)
Edge flow fe over edge e can be calculated after computing path flows using equation 8:
fe
b
oO kKoS
k oS
.oSk ,e ,
(8)
where oSk ,e is a binary parameter which is 1 if path k contains edge e, and 0 otherwise.
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Figure 1. Steps for computing equilibrium flows on paths
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Based on the current edge flows, utility values need to be updated. The logit assignment is applied again to find the new edge flow based on the current utility. This iterative process is repeated until convergence is reached.
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3. Mathematical Modelling and Formulation To assess the process of evacuation, evacuation time is used as a measure. Evacuation time is defined as the time by which the entire crowd in the hazardous zone is evacuated to safe destinations. Network redesign is used as a method to control the evacuation time. As indicated in section 1, a bi-level model is developed which takes into consideration the goal of both the crowd manager and individuals. The individual goal is achieved through their utility maximization as described in section 2. Crowd managers fulfil their goal by influencing the demand through capacity change. Individuals then try to maximize their utility for that particular capacity change. The outcome of the model is the optimal change in the network capacity required to achieve a given evacuation time. The model incorporates
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The following assumptions have been made in developing the models: All the origins and their expected demand are assumed to be known and constant. Individuals have a flexibility to choose an exit of their choice. Individuals are assumed to be utility maximizers. Monetary cost for all edges are known and assumed to be constant. There is an upper limit to the increase in the capacity of an edge. The herding behavior of individuals is not considered while calculating their route choice. Let ce be the capacity of an edge e, ceup the upper limit on capacity of an edge e, ge the
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monetary cost for unit capacity change, i.e., cost incurred in changing the capacity of an edge by one unit and we the change in capacity of an edge e. Then the total budget (B) of network redesign is given by:
B g e we
(9)
eE
The bi-level model can then be formulated as mathematical program P1. P1:
min B
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f ,w
subject to
eE
e
0
ToSk Tgiven
k K ; o O
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w
e E
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0 ce we ceup
(10)
(11) (12) (13)
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The objective function (equation 10) minimizes the budget for capacity change. The budget incurred is a function of the edge flow vector (f) and capacity change vector (w). The constraints for the model are given in equations 11-13. Equation 11 puts an upper limit on the increase in the capacity of an edge. This is done to make the problem more realistic. The value of the parameter ceup comes from the relevant stakeholders such as network planners and users. Equation 12 ensures that the total space before and after redesigning is the same so that the context of the problem does not change. The objective function, i.e., the budget for the network change makes sense only when it is used in conjunction with a target evacuation time. This feature of the model is incorporated in equation 13, which shows that the travel time ToSk on the chosen path k is less than the target time Tgiven. Travel time on any path can be computed using equation 3. In calculating ToSk , the flow on the edges (fe) is required which is determined in the lower level problem explained in section 2.
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It can be clearly seen that the program P1 is a non-linear constrained optimization problem. Obtaining global minima for this program is extremely challenging. As an alternative strategy, an IPA (Byrd, Gilbert, & Nocedal, 2000) with global search is adopted to get a solution. Figure 2 describes the detailed procedure for finding a solution to the program using this methodology. The global search algorithm works by generating trial points on which an IPA is applied. All the solutions are stored in a global search object. The solution having minimum value of the objective function in the global search object is chosen as the solution of P1.
Figure 2. Solution methodology of bi-level model
Apart from calculating the capacity change vector required to bring the evacuation time less than or equal to the available time (Tgiven), the crowd manager may also be interested to know the least possible evacuation time achievable by network redesign. Evacuation time ( T ) is the maximum of travel time across all paths, i.e. T = max( ToSk ) k K ; o O . The least possible evacuation time can be obtained by solving the mathematical program P2.
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ACCEPTED MANUSCRIPT P2: min T
(14)
b, w
subject to 0 ce we ceup
0
b
moS
e
eE
kK
k oS
(15) (16)
o O
(17)
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w
e E
The objective function given by (14) seeks to find the flow vector and capacity change vector that minimize the evacuation time ( T ). The constraints 15 and 16 for program P2 are the same as constraints 11 and 12 of the program P1. The constraint 17 is for conservation of flow over different paths. Heuristic for Starting Point
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4.
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While solving a non-linear program, a good starting point often increases computational efficiency significantly. To leverage this behavior, a heuristic is developed to find a good starting point for IPA. The steps involved in the heuristic are summarized in figure 3. The initial step is to perform a logit-based assignment on the network. In the logit-based assignment, the path having the maximum travel time is recorded as Pmax. The edge having the maximum travel time among all edges on Pmax is selected. The capacity of the selected edge needs to be increased to reduce T . As stated earlier, it is ensured that the total space
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before and after redesigning is the same. Therefore, if we increase the capacity of the selected edge by , there must be an edge where the capacity needs to be reduced by the same amount .
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In order to find the edge on which the capacity needs to be reduced, the path having the least travel time, recorded as Pleast, is chosen. The capacity of the edge having the least travel time among all edges on Pleast is reduced. After this initial redesigning of the network, T is
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computed again and compared with the target evacuation time Tgiven. If T is less than Tgiven, the current capacity change vector and flow vector serve as a starting point for IPA. If it is more than Tgiven, the process is repeated until T becomes less than Tgiven. 5. Illustrative Example 5.1. Test Network To test the ability of the developed model, the solution methodology has been applied to a hypothetical network as shown in figure 4. The test network consists of 20 nodes and 30 0 up edges. Each edge of the network is characterized by le, Ie, ce, te , ce and ge. Table 3 shows
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ACCEPTED MANUSCRIPT the values of these parameters for the test network. The values of Ie and ceup are chosen
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according to other studies in the literature (Proulx, 2002).
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Figure 3. Heuristic proposed for starting point solution
Figure 4. Illustrative example network
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ACCEPTED MANUSCRIPT Various evacuation scenarios are considered for analyzing the impact of network redesign on the evacuation time. The evacuation time depends on the number of people evacuating, i.e., the evacuation demand. Therefore, to begin with, an initial scenario is considered in which the effect of evacuation demand on evacuation time has been analyzed. For this purpose, the evacuation time is calculated for different values of evacuation demand. Table 3. Parameters value of test network ce
ge
ceup
Edge
le
te0 (s)
Ie
ce
ge
ceup
49.30
8E-04
30
5
50
16
100
70.42
8E-04
30
5
50
50
35.21
8E-04
10
5
50
17
100
3
100
70.42
8E-04
30
5
50
18
200
4
100
70.42
8E-04
30
5
50
19
200
5
100
70.42
8E-04
30
5
50
20
200
6
100
70.42
8E-04
30
5
50
21
200
7
100
70.42
8E-04
30
5
50
22
225
8
100
70.42
8E-04
30
5
50
23
144
9
300
211.27
8E-04
30
5
50
24
100
10
260
183.10
8E-04
30
5
50
25
11
260
183.10
8E-04
30
5
50
12
260
183.10
8E-04
30
5
50
13
100
70.42
8E-04
20
5
14
100
70.42
8E-04
30
15
100
70.42
8E-04
30
70
2
70.42
8E-04
30
5
50
140.85
8E-04
20
5
50
140.85
8E-04
30
5
50
140.85
8E-04
30
5
50
140.85
8E-04
30
5
50
158.45
8E-04
10
5
50
101.41
8E-04
10
5
50
70.42
8E-04
20
5
50
100
70.42
8E-04
20
5
50
26
100
70.42
8E-04
20
5
50
27
100
70.42
8E-04
20
5
50
50
28
100
70.42
8E-04
20
5
50
5
50
29
100
70.42
8E-04
20
5
50
5
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t
50
30
100
70.42
8E-04
30
5
50
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Ie
Edge
0 e (s)
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For further analysis and better clarity of exposition, a specific value of evacuation demand is chosen (3,000). For this demand, the evacuation time ( T ) is calculated using the pedestrian assignment model detailed in section 2. To control this process of evacuation, the impact of
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network redesign on T is further analyzed. The bi-level model, i.e., program P1, is applied
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on the test network to achieve the target evacuation time. To increase the computational efficiency of the model, the heuristic developed in section 4 is applied to provide a good starting point for the IPA. The model gives the minimum budget required for different values of the desired state. 5.2. Results
The evacuation scenario is as follows: due to some emergency, all the people need to be evacuated to a safe destination. It is assumed that people shall evacuate spontaneously (i.e., without any guidance mechanism, and as soon as signaled) attempting to maximize their utility. This self-guided evacuation behavior is key aspect to model real life behavior. In all scenarios, nodes 1 and 6 are considered as origins while nodes 16 and 20 are the safedestinations. This is a multi-destination problem, which needs to be converted to a single destination problem by connecting all the destinations to S as shown in figure 5.
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Figure 5. Super-sink added to the test network
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All paths from the origins to S are recorded. Table 4 shows the list of these paths. During the evacuation, free flow speed on all the links should be known in advance. Then the parameter
te0 is calculated by taking the ratio of the length of an edge to its free speed. As the free flow
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speed increases, link travel time decreases (by equation 3) which in-turn decreases the path travel time. Evacuation time is calculated as the maximum travel time over all the paths. The path travel time is given by equation 2. To find the effect of free speed on evacuation time, a series of experiments is performed. The experiments reveal that as the free flow speed increases, evacuation time reduces and varies almost linearly with the free flow speed. For the purpose of illustration, a free flow speed of 1.42 m/s (Thalmann, Musse, & Braun, 2007) is assumed for all the links. The results for this entire analysis are presented next in sections 5.2.1-5.2.2.
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5.2.1. Evacuation Time without Network Redesign
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The evacuation time can be calculated once flows on the edges are known. The method to calculate the equilibrium edge flows is presented in section 2. As stated earlier, the evacuation time depends on the evacuation demand. In the test network, various values of the total demand, distributed equally over the origins are considered. Figure 6 summarizes these results. It can be seen that T increases with an increase in the total evacuation demand, as expected.
Next, the total demand is considered to be 3,000, and it is distributed equally across the origins. For this scenario, T is 1016.3 seconds. Table 5 shows the equilibrium edge flows in the column „original network flow‟. This evacuation time is without any network redesign. The crowd manager wants this time to be reduced to a specific value, i.e., the target evacuation time, Tgiven. In effect, the crowd manager wants to control the process of evacuation, i.e., directing the evacuation towards a desired state by redesigning the network configuration constrained through equation 11 and 12 of program P1. The rest of this section
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ACCEPTED MANUSCRIPT describes the results of applying this model to a variety of scenarios developed using different values of Tgiven. The details and results are presented in section 5.2.2. Table 4. List of paths from origins (1 and 6) to super-sink (S) S No.
S No.
1-S
1
1 - 2 - 3 - 4 - 5 - 10 - 15 - 16 – S
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1 - 7 - 8 - 9 - 10 - 15 - 16 - S
2
1 - 2 - 3 - 4 - 5 - 10 - 15 - 20 – S
15
1 - 7 - 8 - 9 - 10 - 15 - 20 - S
3
1 - 2 - 3 - 4 - 5 - 10 - 16 - S
16
1 - 7 - 8 - 9 - 10 - 16 - S
4
1 - 2 - 3 - 4 - 9 - 10 - 15 - 16 - S
17
1 - 7 - 8 - 13 - 14 - 15 - 16 - S
5
1 - 2 - 3 - 4 - 9 - 10 - 15 - 20 - S
18
1 - 7 - 8 - 13 - 14 - 15 - 20 - S
6
1 - 2 - 3 - 4 - 9 - 10 - 16 - S
19
1 - 7 - 8 - 13 - 14 - 19 - 20 - S
7
1 - 2 - 3 - 8 - 9 - 10 - 15 - 16 - S
20
1 - 7 - 8 - 13 - 18 - 19 - 20 - S
8
1 - 2 - 3 - 8 - 9 - 10 - 15 - 20 - S
21
1 - 7 - 12 - 13 - 14 - 15 - 16 - S
9
1 - 2 - 3 - 8 - 9 - 10 - 16 - S
22
1 - 7 - 12 - 13 - 14 - 15 - 20 - S
10
1 - 2 - 3 - 8 - 13 - 14 - 15 - 16 - S
23
1 - 7 - 12 - 13 - 14 - 19 - 20 - S
11
1 - 2 - 3 - 8 - 13 - 14 - 15 - 20 - S
24
1 - 7 - 12 - 13 - 18 - 19 - 20 - S
12
1 - 2 - 3 - 8 - 13 - 14 - 19 - 20 - S
25
1 - 7 - 12 - 17 - 18 - 19 - 20 - S
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1 - 2 - 3 - 8 - 13 - 18 - 19 - 20 - S Path
S No.
6-S
1
6 - 7 - 8 - 9 - 10 - 15 - 16 – S
10
6 - 7 - 12 - 13 - 14 - 19 - 20 - S
2
6 - 7 - 8 - 9 - 10 - 15 - 20 – S
11
6 - 7 - 12 - 13 - 18 - 19 - 20 - S
3
6 - 7 - 8 - 9 - 10 - 16 – S
12
6 - 7 - 12 - 17 - 18 - 19 - 20 - S
4
6 - 7 - 8 - 13 - 14 - 15 - 16 - S
13
6 - 11 - 12 - 13 - 14 - 15 - 16 - S
5
6 - 7 - 8 - 13 - 14 - 15 - 20 - S
14
6 - 11 - 12 - 13 - 14 - 15 - 20 - S
6
6 - 7 - 8 - 13 - 14 - 19 - 20 - S
15
6 - 11 - 12 - 13 - 14 - 19 - 20 - S
7
6 - 7 - 8 - 13 - 18 - 19 - 20 - S
16
6 - 11 - 12 - 13 - 18 - 19 - 20 - S
8
6 - 7 - 12 - 13 - 14 - 15 - 16 - S
17
6 - 11 - 12 - 17 - 18 - 19 - 20 - S
9
6 - 7 - 12 - 13 - 14 - 15 - 20 - S
18
6 - 11 - 17 - 18 - 19 - 20 - S
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5.2.2. Evacuation Time with Network Redesign Recall that network redesign can be used as a method to control the evacuation time, i.e., to achieve the target evacuation time (Tgiven). Redesigning the network helps in influencing the demand by enhancing or limiting the use of certain routes without the use of force. The bilevel model of program P1 has been applied on the test network to achieve the desired evacuation time. For various values of the target evacuation time, minimum budget required is calculated. The solution method used is IPA with global search and the starting point heuristic presented in section 4. More details are available in section 3. Figure 7 shows the budget required for various values of Tgiven and a total evacuation demand of 3,000. From the figure, the crowd manager can determine the budget required to achieve a particular evacuation time. It can also be observed from the figure that the minimum evacuation time Tmin that can be achieved by network redesigning is 774.86 seconds Tmin is obtained by solving Program P2 explained in section 4. The corresponding budget for Tmin is
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2,489.176. Table 5 shows the optimal capacity change vector (we) and the corresponding edge flow vector (fe) to achieve Tmin (in the column „min. achievable time‟ of the table).
Figure 6. Variation of evacuation time with total evacuation demand
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It is also observed from table 5 that the system moves towards a uniform exit usage in the redesigned network as Tgiven is decreased. For the original network, 1,721 individuals select destination 20, while destination 16 is selected by 1,279 individuals and the total evacuation time is 1,016.3 seconds. Consider the scenario where Tgiven is chosen to be 895 seconds. It is seen that the flows at the destinations become 1,529 and 1,471 (as indicated in the column „particular time‟ of table 5), i.e., more uniform than the original flow of 1,721 and 1,279, respectively. We further reduce Tgiven and observe the trend of destination flows getting more uniform. Lastly, as indicated above, for Tgiven equal to Tmin , the destination flows of 1,490
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and 1,510 are much more uniform than the original network. It can be interpreted from these results that the crowd manager is able to achieve the system goals of evacuees changing their exit route preferences to make destination flows more uniform, albeit without using any force.
Figure 7. Budget verses evacuation state required
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fe 1055 1055 792 630 8 406 611 772 445 263 161 630 469 502 479 381 205 1492 48 58 97 1305 1023 1038 1118 1217 15 80 99 273 1490 1510
we 9.46 34.06 7.52 13.55 -29.85 -9.33 -2.08 6.26 3.38 -9.49 -17.94 9.33 -4.37 -5.91 -7.26 -13.96 -18.65 26.19 -23.78 -23.75 -21.27 39.90 20.75 20.56 28.11 29.85 -17.68 -14.55 -14.39 -14.67 ---
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we 0.01 12.02 0 -0.51 -17.04 0.05 0 0 0.58 -1.64 -1.3 0 -0.04 -0.07 0 -0.01 0.02 0.01 -0.01 0 0 11.38 -1.36 0 0.01 0.87 -0.19 -1.61 -1.15 -0.03 ---
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fe 968 968 704 490 728 805 662 876 532 263 214 490 495 747 987 851 747 772 455 407 642 724 277 480 647 783 203 167 136 746 1529 1471
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fe 776 776 619 469 919 977 630 780 724 158 150 469 306 757 1060 911 882 581 665 504 852 397 275 490 691 840 215 201 149 882 1721 1279
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Edges 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 20-S 16-S
To increase the computational efficiency, the developed heuristic is applied to generate a good starting point for IPA. The steps involved in the heuristic are shown in figure 3. The details, applied to a specific instance, are as follows. Consider the scenario of achieving a target evacuation time of 956 seconds, i.e. Tgiven=956 seconds. The heuristic is then applied to find a good starting point solution for IPA. Initially a logit-based assignment is performed on the test network and it is seen that path 9 has the maximum travel time. The edges on path 9 are 1, 2, 10, 7, 8, and 22 as seen in table 4. The
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Following the steps shown in the figure 3, it is determined that the capacity of edge 22 needs to be increased by 30. To keep the total space fixed, the capacity of edges 27, 28 and 29 should be reduced by 10 each. The resultant evacuation time is 950.36 seconds, which is less than the required 956.15 seconds. This satisfies the termination condition of the heuristic in figure 3. Hence, this point is selected as a starting point for the IPA. Table 6 compares the computational time (cT) required to find the solution from the model using global search, i.e., the standard library of MATLAB, to that with using our heuristic for the IPA corresponding to various values of target evacuation time (from the values already considered for budget calculations i.e. figure 7). It can be seen from the table that the budget computed is comparable for both the methods. However, the cT is significantly less when the heuristic is used. Thus, the heuristic increases the computational efficiency of the solution.
Tgiven =956.15
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Table 6. Budget and computational time using global search and using heuristic for IPA Budget using Global Search 121.28
Tgiven =895.72
(cT): 11203 sec
(cT): 135.41 sec
251.715 640.64
249.63 (cT):372 sec 643.9
(cT): 18602 sec
(cT): 377.02 sec
(cT): 9990 sec
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Tgiven =835.29
Budget using Heuristic 124.7
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5.2.3. Algorithm Scale with Number of Edges With an increase in the number of edges and nodes, the number of alternate paths (from which individuals have to make a decision) increases as well. Due to increase in the number of paths, the computational time increases. Simulations are performed for the hypothetical network by running scenarios to find minimum achievable time for different network configurations. Each time four nodes are increased at the end of the network, seven edges corresponding to these four nodes are added. Figure 8a shows the part of the network increased during the different simulations. In each simulation, a part shown in the figure is added to the network.
Figure 8a. Additional part to be added to test network
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The evacuation demand of 3,000 is considered for all these simulations. Further, to find the computational time for each simulation of different network configurations, an initial starting point is given as the upper bound of the capacity for all the links. The results show that with an increase in number of edges and nodes, computation time increases non-linearly. Figure 8b shows the computation times for different simulations. Further, a polynomial trendline of order 2 is fitted to these results and its R-squared value comes out to be 0.9524.
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Figure 8b. Computational time for different number of nodes in the test network
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5.3. Robustness Analysis
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We further note that the evacuation demand may vary with time. Therefore, it is important for the redesigned network to be robust with respect to the evacuation demand. To check the robustness of our proposed solution, the network redesigned for a particular demand should be used for other values of the demand. If the evacuation time is still less than the original evacuation time (i.e., the evacuation time without any network redesign), it indicates the robustness of the proposed approach. Consider the scenario with a total demand of 3,000 and a target evacuation time of 895 seconds. The solution of the model gives the location and the extent of the capacity change required to achieve Tgiven. Network is redesigned according to the solution. Now consider the redesigned network with a total demand of 3,500. The evacuation time for this demand turns out to be 1020.8 seconds, which is less than the original evacuation time of 1,156.8 seconds. Figure 9 summarizes the evacuation time for different values of the total demand and Tgiven corresponding to the network optimally redesigned for a demand of 3,000. The results indicate that the model solution is robust for upto 15% variation in total demand (500 out of 3,000).
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Figure 9. Effect of change in demand on evacuation time 5.4. Alternate Measures
5.4.1. Location of Additional Emergency Exit
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The evacuation time can also be reduced if an emergency exit, a new edge etc. is added to the physical network. In the test network, there are four locations where an emergency exit can be built as shown in table 7. As computed in section 5.2.1, the evacuation time without adding an emergency exit is 1016. 3 seconds. The evacuation time at different locations after adding an emergency exit is reported in table 7.
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It can be seen from the table that adding an emergency exit at location 3 is the most useful. In addition, it can be observed from table 7 that the evacuation time could increase if an emergency exit is added to location 2. That happens because more congestion builds up near exits. Therefore, it is important to know the location(s) where an emergency exit should be added and the methodology proposed in the paper can determine the same.
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Further, a scenario has been considered where evacuation from only one exit is possible. Specifically, the exit at node 16 is not accessible. The two aims of considering this scenario are: i) to find the evacuation time when only one destination exists, and ii) to check the robustness in the identification of the emergency exit location. The evacuation time in this scenario comes out to be 1172.9 seconds which is greater, as expected, than that of the original test network as crowding and congestion increase due to a single exit. In addition, the effect of adding an additional exit is also studied in this scenario. We check all four alternatives for the emergency exit locations. It is observed that in this scenario, similar to the scenario with only two exits, adding an emergency exit at location 3 is the most useful (table 7, column „1D Evac time‟).
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5.4.2. Additional capacity
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While developing the bi-level model, it has been assumed that the total space before and after the redesign is the same. However, in some situations it might be possible to increase the total capacity. The method proposed in the paper can be used to estimate the benefit in those situations as well. To do so, the right hand side of equation 12 is increased from zero. As an illustration, table 8 presents a comparison when the right hand side is made two instead of zero. As expected, the required budget in the former case is lesser and the exact values are reported in the table. Table 8. Effect on budget of additional capacity Tgiven Budget (sumwe=0) Budget (sumwe= <=2) 835.2901 640.64 634.02 895.7201 251.715 240.47 956.1501 121.28 115.3 1016.3 0 -
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6. Discussion
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To make the evacuation process efficient, it is required to keep a sound balance between increased demand and the available network supply. To do so, different ways can be used including controlling the flow of the people, and determining the optimal velocity for different sections.
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However, implementing these controls to make evacuation efficient may be a challenging task. Therefore, an alternative is to manage the supply itself, i.e., redesign the network configuration, if possible. For this purpose, mathematical models have been developed. The results of these models help the crowd manager in the following ways:
Planning for a crisis: This model can find ways which control the process of an evacuation. Increasing safety: The models determine the modification required in the venue design for enhanced safety. Venue suitability: The model can help in finding the suitability of the venue for a given number of attendees of an event. Specifically, even after modifications, if the
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evacuation time cannot be reduced to an acceptable level, then the venue is not suitable for the event. Uniform exit usage: It has been demonstrated by the results that due to redesign, the system moves towards a more uniform usage of alternative exits, as desired by the crowd manager. Location of emergency exits: Different locations can be compared based on the evacuation time and the best ones can be chosen. Emergency staff training: The model can also help in finding the areas that will be most congested. This can help the crowd manager in training the emergency staff appropriately. Thus, we can conclude that the models can help the emergency manager make several key decisions for emergency management. 7. Conclusions and Way Forward
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Appropriate crowd management strategies are essential at places of mass gatherings for safe and smooth movement of pedestrians. These strategies must include planning for the evacuation in emergency situations that may occur at these places. A proactive measure that can be adopted for evacuation strategies is to make a plan to utilize the available capacity by an efficient network redesign. Redesigning the network helps in influencing the demand by enhancing/limiting the use of certain routes without the use of force or an explicit guiding mechanism. To accomplish the redesigning, a bi-level model is presented which determines the location and extent of change in the capacities required to achieve a desired evacuation state. The Interior Point Algorithm (IPA) with global search has been applied to determine the optimal capacity change vector to achieve the target evacuation time. Practical ways to redesign are changing the capacity of routes, closing or opening new routes, use of temporary barriers, and changing illumination of routes. A heuristic to compute a good starting point for better computational efficiency is also presented. The robustness of the redesigned network with respect to the evacuation demand is reported. Additional alternatives are proposed to further decrease the evacuation time, and the effects of implementing them using the proposed model are presented.
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Apart from developing crowd management strategies, this model may also help in determining the suitability of a venue. If the minimum evacuation time cannot meet the strategic evacuation time even by redesigning using program P2, then that venue may not be suitable. The developed model is tested on a hypothetical network and the results are encouraging. The results demonstrate that appropriate strategies can be formulated (made more efficient using the proposed heuristic), and the network redesign is indeed robust with respect to the evacuation demand. Further, the effect of alternate strategies is quantified to assist better decision making. It is observed that through network redesign, the evacuation process evolves towards a uniform use of all exits as desired from a management perspective. Thus, the system goals are achieved by influencing individual decisions and not by forcing them.
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ACCEPTED MANUSCRIPT Future extensions of this model can include incorporating group behavior in the utility function. Further, models for dynamic redesigning that incorporate the time dependence of the evacuation demand and edge capacity can be developed.
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