Optimal Scheduling of Evacuation Operations - Semantic Scholar

Optimal Scheduling of Evacuation Operations Hayssam Sbayti and Hani S. Mahmassani evacuate to, and which path to take so as to minimize the network clearance time (NCT). Furthermore, in this application, evacuees are assumed to adhere to the evacuation guidance information and not switch to different departure times, destinations, and paths. Although deviations from this assumption would be expected in actual evacuations, it is not totally unrealistic to expect the network users to follow the prescribed guidance information since they cannot possibly determine their optimal paths in the case of emergency evacuations. Nonetheless, the results obtained in this study are to be considered as a benchmark for assessing the value of better management and information systems.

Evacuations necessitated by extreme events are usually envisioned as taking place with all people evacuating simultaneously; this leads to premature congestion on the surface streets and excessive delays. With the evacuating load onto the network staggered, the onset of congestion may be delayed, and people can evacuate more quickly. In this study, the problem of scheduling evacuation trips between a selected set of origin nodes and (safety) destinations was considered, with the objective of minimizing network clearance time. A modified system-optimal dynamic traffic assignment formulation is proposed; in it the total system evacuation time, as opposed to the total system trip time, is minimized. An iterative heuristic procedure is used to solve this problem: the method of successive averages is used to find the flow assignments for the next iteration; a traffic simulator, DYNASMART-P, is used to propagate the vehicles on their prescribed paths and determine the state of the system. Therefore, the simulator serves as a tool to satisfy the dynamic traffic assignment constraints implicitly while evaluating the objective function. The output of this model will be the departure time, route, and destination choices for each evacuee. The output is then aggregated to produce a time-dependent staging policy for each selected origin.

EVACUATION MODELING A general approach to model evacuation processes has been to use network flow models, which have been applied primarily for evacuation of buildings, in which travel times are constant or possibly time-dependent. In vehicular traffic networks, travel times are flowdependent and increase nonlinearly with higher densities until traffic slows down to a crawl and queuing occurs. This phenomenon is particularly true under evacuation circumstances, in which the transportation system degrades quickly shortly after demand overwhelms supply. Although recognizing this flow dependence is essential to realistic traffic system evacuation modeling compared with assuming constant link travel times, it makes the problem considerably more difficult to solve. Traffic simulation models have been used in the past two decades to investigate emergency evacuation scenarios. Early evacuation models can be classified in two main categories: microscopic and macroscopic simulation-based models. The main disadvantage of microscopic simulators is the extensive data and computer resource requirements, whereas the macroscopic ones, such as NETVAC1 (1), do not have the capability of keeping track of individual driver decisions. Besides, applications at that time mainly focused on evacuation approaches around nuclear power plants and lacked the capability to address other types of disasters (2). Hobeika and Kim (3) compared different traffic assignment procedures in evacuation modeling that use a traffic simulator and concluded that a user equilibrium (UE) assignment showed better results than a shortest-path algorithm based on two performance measures, namely, evacuation time and the number of congested links. That study did not consider system-optimal (SO) assignment to minimize NCT. A model that uses an SO type of analysis was proposed by Sattayhatewa and Ran (4) and is specifically designed for nuclear power plant evacuations. Their model addresses two different evacuation applications: minimization of total evacuation travel time of the disaster zone without predefining a target evacuation time and minimization of travel times for each origin–destination (O-D) pair. The

Congestion in most major urban areas results in loss of productivity and environmental degradation, but in the case of a human-made or natural disaster, it can be catastrophic and life threatening because the shift in the demand pattern results in excessive loads on streets not designed to handle them. Therefore, better utilization of the available transportation network capacity during disasters is essential to manage traffic and lead people to safety. One way to do so is to address the demand side of the problem by scheduling and spreading the evacuation operation. Another is to address the supply side of the problem by the available capacity being reallocated and the direction on strategically selected roadway links being reversed in a process known as contraflow design. This design can lead to a temporary increase in the network’s operational capacity without major infrastructure changes. The current study focuses on the former approach, namely, managing the loading process to maximize evacuation throughput. This study uses a mesoscopic simulation tool to investigate the benefit of evacuating a given zone in stages rather than simultaneously. In simultaneous evacuation, vehicles are advised to evacuate immediately to their preferred destination, whereas in staged evacuation, vehicles are advised when to evacuate, which destination to Maryland Transportation Initiative, Department of Civil and Environmental Engineering, University of Maryland, 1173 Glenn L. Martin Hall, College Park, MD 20742. Transportation Research Record: Journal of the Transportation Research Board, No. 1964, Transportation Research Board of the National Academies, Washington, D.C., 2006, pp. 238–246.

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other notable aspect of their model is, unlike most of the previously used tools, the use of an analytical dynamic traffic assignment (DTA) procedure, though application of the proposed approach was only reported for a three-link network. DTA models, unlike static models, are able to model time-varying flows and hence provide the ideal platform to model system degradation, especially in an evacuation setting. Chiu et al. (5) applied DYNASMART-P (6), a simulation-based DTA model, to represent the evacuees’ response to UE- and SO-based evacuation advisory information and the effect on system degradation. In their experiments, the time-dependent evacuation demand is taken to be the day-to-day O-D demand without consideration of the fact that the evacuation demand has a different temporal pattern from the day-to-day demand. Recently, there has been a general interest among researchers and planners in adopting contraflow measures during emergency evacuations in an effort to increase the network capacity and accommodate the resulting excessive demand (7–9). Although most of these studies show a marked improvement over normal conditions, safety continues to be a major issue, especially when the driver population is not familiar with this type of operation. An alternative strategy to reduce and delay system degradation is to spread the demand. If some evacuees can be convinced that it is “faster to wait,” it would be possible to move closer to an SO state and therefore reduce average evacuation times. Chen and Zhan (10) did one of the few documented studies on the effectiveness of staging evacuation operations. The study concludes that staging is not effective when traffic is near free-flow conditions. The staging strategies considered in that study were defined a priori by the analyst and not generated to satisfy specific systemlevel objectives. The current study investigates the effect of intelligently spreading the evacuation demand for a given impacted traffic analysis zone (TAZ) by scheduling the evacuation operations so as to minimize NCT. The model described here represents the case in which a super user (central controller) will assign a departure time, a destination, and a corresponding path to that destination for all evacuees, who are assumed to follow these instructions at all times. The model therefore will estimate a lower bound for NCT. The results of this model will then be aggregated and provided as a staging policy in the form of how many vehicles to send to safety in each time period for a given zone.

PROBLEM STATEMENT The following notation and definitions are used: N= A= I= Iv = J= Jv = B(n) = C(n) = λ= λ*i =

set of nodes in network; set of links in network; set of origins in network; set of evacuation origins in network; set of destinations in network; set of evacuation (safety) destinations in network; set of outbound links from node n ∈ N; set of incident links to node n ∈ N; superscript denoting departure time period, λ = 1, . . . , T; superscript denoting time period when incident occurs in origin i, ∞ otherwise; τi = superscript denoting evacuation advisory departure time period for origin i, τi ≥ λ*i ; v = superscript denoting class of vehicles that receive route guidance information;

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o = superscript denoting class of vehicles that do not receive route guidance information; u = superscript denoting general class of vehicles; i = subscript denoting origin node, i ∈ I; j = subscript denoting destination node, j ∈ J; n = subscript denoting node in network, n ∈ N; t = superscript denoting time step index; k(v) = subscript denoting path followed by evacuee; Δ = length of time step t; T = total number of departure time periods considered; Ri = total population, expressed in terms of vehicle trips, in origin i at time of evacuation; Γ˜ i = total number of vehicles to be evacuated when an incident is detected in origin i; d ta = number of vehicles exiting link a during time step t; m ta = number of vehicles entering link a during time step t; x ta = number of vehicles on link a during time step t; Etn = number of vehicles generated in network at node n during time step t; Otn = number of vehicles exiting network at node n during time step t; Qi = capacity of safety destination j, j ∈ Jv; q λj = maximum service inflow rate that can be handled at safety destination j in departure time period λ; = number of class u vehicles going from origin i to destid λ,t,a,u i,j,k nation j assigned to path k during departure time period λ that will exit link a during time step t; = number of class u vehicles going from origin i to destim λ,t,a,u i,j,k nation j assigned to path k during departure time period λ that will enter link a during time step t; r i,jλ = total number of trips leaving origin i to destination j in departure time λ; λ,u = number of user class u vehicle trips leaving origin i to R i,j,k destination j on path k in departure time λ; λ,o = number of user class o (nonimpacted) vehicle trips leavR i,j,k ing origin i to destination j on path k in departure time λ; i,v = decision variable: number of user class v vehicles leaving Rˆ τi,j,k evacuation origin i to shelter destination j using path k in departure time τi ≥ λ*i ; Sλi 1,λ2 = number of vehicles departing during departure time period λ1 reaching origin i during departure time period λ2; T λ,u i,j,k = travel time for class u vehicle going from origin i to destination j using path k during departure time period λ; δ λ,t,a,u i,j,k = time-dependent link–path index, equal to 1 if class u vehicles going from origin i to destination j during departure time λ and assigned to path k are on link a during time step t, 0 otherwise; and f() = function of. A traffic network that is considered is represented by a directed graph G(N,A) with N nodes and A directed arcs. Also considered are multiple origins i ∈ I ⊆ N, destinations j ∈ J ⊆ N, and the timedependent O-D vehicle trip desires r λi,j, ∀ i ∈ I, ∀ j ∈ J, and λ = 1, . . . , T. It is assumed that the preevacuation incident path assignments R λ,u i,j,k are known: ∀ i ∈ I, ∀ j ∈ J, k ∈ K, and λ = 1, . . . , T. The problem may then be stated given the following: (a) occurrence of an evacuation incident in origin i at departure period λ*i , (b) preevacuation incident vehicle assignments R λ,u i,j,k , and (c) a set of possible shelter locations j ∈ Jv. The problem is to provide route guidance information only for those vehicles directly affected by the incident (referred to hereafter as impacted vehicles) so as to evacuate the designated origins (referred to hereafter as impacted origins) as early as possible. The

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route guidance instructions are expressed in terms of time-dependent i ,v path reassignments Rˆ τi,j,k , τi > λ*i, for user class v (evacuees). Existλ,o ing preincident path assignments R i,j,k for the background traffic are assumed to remain fixed except when they are en route to affected origins. Evacuees (user class v) will receive route guidance in the form of SO evacuation information advising them on their departure times, shelter destinations, and paths in order to evacuate the zone and reach safety as soon as possible. Evacuees are assumed to follow the evacuation guidance information at all times. Vehicles currently en route to impacted origins at the time of incident will receive information from variable message signs or en route guidance information advising them to make a U-turn back home if they were destined to danger zones or to find alternate routes to their destinations if their paths traversed impacted origins. Future planned trips to impacted origins after the evacuation incident occurs will be cancelled. In the current application, two user classes are considered: (a) class v users or evacuees, who receive SO evacuation guidance information provided by the central controller, and (b) class o users, who represent the remaining vehicles (background traffic) in the network and will retain their original preevacuation incident paths unless diverted back to their origin. The model assumes that the central traffic controller has full a priori information on the O-D desires (R λi,j ) of the i,u vehicles and their original path assignments (Rτi,j,k ) in the network. Upon the detection of an evacuation incident in an origin i, the evacuation route advisory period starts, at time period λ*i . Starting from this time period the controller will provide SO evacuation advisory information for class v vehicles. The decision variables in this model are the time-dependent path i ,v reassignments Rˆ τi,j,k for class v vehicles (evacuees). These reassignments will then form the basis for calculating the staging policy for each evacuation origin i.

T

∑ ∑ ∑ ∑ Rˆ v

v

τi = λ*i i ∈I v j ∈J v t b

t c

b

find ∀i ∈ I v , j ∈ J v , τ i ≥ λ*i , k

Rˆ iτ, ij,,vk

Objective function: T

minimize Z = ∑ ∑ ∑ ∑ ( τ i + Ti,τji,, kv ) Rˆ iτ, ij,,vk τi = λ*i j ∈Jv i ∈I v

(1)

k

subject to λ*i

λ*i

λ =1

λ =1

* Γ i = Ri + ∑ Siλ ,λi + ∑ ∑ ∑ ∑ Riλ, ,j,ok

∑ ∑ ∑ Rˆ

τi , v i, j, k

j

o

∀i ∈ I v , τ i ≥ λ ∗i

≥ ∑ Siλ, τi λ*i

T

* ∑ ∑ ∑ ∑ Rˆ τi,i,j,vk = Γ i + ∑ Siλ ,λi

v

∀i ∈ I v

(2)

k

(3))

λ

k

k

λ

(6)

∀τ i ≥ λ ∗i

τi , u i, j, k

(7)

k

∀t , n , b ∈ B ( n ) , c ∈ C ( n )

= Ont − Ent

x at = x at−1 + mat−1 − d at−1

(8)

∀t , a

(9)

x at = ∑ ∑ ∑ ∑ ∑ Riλ, j,o, k δ iλ, ,t,a,o j, k λ

i

j

o

k

T

+ ∑ ∑ ∑ ∑ ∑ Rˆ iτ, ij,,vk δ iτ,i j,t,a,v ,k τi = λ*i i ∈I v

j

v

λ ,o δ iλ, ,t,a,o j, k = f ( Ri, j, k )

(10)

∀t , a, v , k , τ i ≥ λ ∗i , i ∈ I v , j ∈ J v

(11)

∀t , k , λ , i , j , a

(12)

Ti,λj,,uk = ∑ ∑ δ iλ, ,jt,,ka, u Δ t

∀t, a

k

)

(

δ iτ,i,t,a,v = f Rˆ iτ, ij,v, k j, k

∀t , u , k , λ , i , j , a

(13)

a

mat = ∑ ∑ ∑ ∑ ∑ miλ, ,j,t,ka, u i

j

λ

u

j

λ

u

∀t , a

(14)

k

d at = ∑ ∑ ∑ ∑ ∑ diλ, j,,tk, a , u

λ ∗i ≤ τ i

∀i , j , λ , k

R

∀j ∈ J v , λ , τ i ≥ λ ∗i

c

i

∀i ∈ I v , j ∈ J v

λ ,o i, j, k

τi = λ*i j ∈Jv

u

(5)

t

k

T

a τ = λ* i i

δ iλ, ,t,a,u j, k = 0 or 1

I v , Jv

o

∀j ∈ J v , t

k

∀t , a

(15)

k

T

given

v

i

∑d − ∑m

o

+ ∑ ∑ ∑ Riλ, ,j,ok ≤ q λj

∑ ∑ ∑ ∑ ∑ Rˆ

Qj ≥

i

O tj = ∑ ∑ ∑ ∑ ∑ miτ,ij,,tk, a, u

The problem may be formulated as follows:

j ∈J v

τi , v i, j, k

k

i

PROBLEM FORMULATION

λ

k

∑ ∑ ∑ Rˆ i

+ ∑ ∑ ∑ ∑ Riλ, ,j,ok = ∑ O tj

τi , v i, j, k

τi = λ*i i ∈I v

∀i ∈ I v

(4)

u

∀t , j

(16)

k

∀i , j , k , λ , t , a , u

(17)

∀i

(18)

all variables ≥ 0

(19)

The decision variables in the foregoing formulation are the path i ,v reassignments Rˆ τi,j,k for class v vehicles, which form the basis for calculating the staging policy for each evacuation origin i. The staging policy consists of the optimal number of vehicles that need to evacuate from each danger zone in each time period to minimize NCT. The objective function (Equation 1) is a modified SO formulation and represents the minimization of the total system evacuation time. τi The term (τi + T i,j,k ) represents the number of time periods τi, measured i,v from the start of evacuation in affected origin i plus the trip time T τi,j,k for vehicle v. Therefore Equation 1 essentially measures the total system evacuation time and not the total system trip time as in classical SO formulations. It should be noted that in minimizing such an objective function, the triple optimization result holds true in that (a) the total evacuation trip time is minimized, (b) NCT is minimized, and (c) the number of evacuees reaching safety in each time period is maximized (11). Constraint 2 states that the total number of vehicles to be ultimately evacuated from an impacted origin i is the sum of the total zonal pop-


ulation in that origin, expressed in vehicle trips, and the number of vehicles that have reached the impacted origin when the evacuation incident occurs reduced by the number of vehicles that had already left the impacted origin before the start of evacuation. Constraint 3 ensures that no storage takes place in the impacted evacuation area. The path reassignments must be greater than or equal to the number of vehicles that reached the evacuation area. Constraint 4 ensures mass balance at the origin level. The path reasi ,v signments Rˆ τi,j,k (v) summed over all paths, destinations, and time periods for a given impacted origin i must be equal to the number of vehicles present in impacted origin i at the start of evacuation. Constraint 5 ensures mass balance at the destination level. Constraints 6 and 7 ensure that inflow and total capacity at safety destinations are respected. Constraint 8 represents the conservation of vehicles at any node in the network. Constraint 9 defines the continuity equation for a link. It states that the number of vehicles on a link a at time step t is the sum of the number of vehicles and the net inflow of vehicles in previous time step t − 1. Constraints 10 through 13 represent the time-dependent link–path incidence relations in a dynamic network. Constraints 14 through 16 are definitional constraints. Constraint 17 restricts the time-dependent incidence variable to binary values of 0 or 1. Constraint 18 restricts the evacuation advisory periods to start after the occurrence of an incident. Constraints 19 are the nonnegativity constraints.

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a departure time period, a destination, and the best route to that destination to minimize NCT. Therefore the vehicles are loaded with different rates along links simultaneously in each departure time period. The steps of the algorithm are described in detail next. Step 0. Initial Solution • Set iteration counter n = 0. • Load user class o vehicles onto their original path assignments R τ,o i,j,k . • Assign evacuation demand (user class v) to an initial set of feasible departure times, an initial set of feasible shelter destinations, τ,v,n . and an initial set of feasible paths to these destinations to obtain Rˆ i,j,k Step 1. Evaluate Objective Function • Under the given set of departure time and destination and path assignments, perform simulation of the traffic demand with DYNASMART-P traffic simulator to obtain vehicle trip times and link marginal travel times. • Scan all evacuees (user class v) and determine their total trip time (objective function) measured from the start of the evacuation procedure and the NCT for iteration n (NCTn). • Update the minimum NCT (MNCT). If NCT n is less than MNCT, then set MNCT to be equal to NCT n.

SOLUTION ALGORITHM Step 2. Find Descent Direction The solution procedure assumes that the traffic control center has perfect information about the O-D desires and the corresponding vehicles’ initial time-dependent historic assignment paths. Such paths are assumed to be the result of some sort of equilibrium among competing multiple user classes with pretrip, en route, UE, and SO information. Once an evacuation incident is detected, the affected zone and consequently the set of impacted origins Iv are delineated. All newly generated (after evacuation starts) vehicles from these impacted origins will be classified as user class v, provided with evacuation trip guidance including when to evacuate (departure time), where to go (destination choice), and which route to take (path choice) in such a way as to minimize their NCT. All other vehicles will be classified as user class o and will retain their original assignment paths. A modified multiple user class DTA algorithm is then used to determine the time-varying SO reassignment paths for class v vehicles. The solution algorithm represents a heuristic iterative procedure in which the objective function is evaluated and constraints are satisfied through a simulation model. The simulation model is used to move vehicles along their assigned departure times and paths until they reach their destinations, capturing the state of the system in the process. Traffic flow in the simulator is represented with a mesoscopic approach in which vehicles are tracked individually but moved in packets according to macroscopic traffic flow relations between average speed and concentration on roadway links. At each iteration, the heuristic defines a search (descent) direction along which the objective function is expected to improve. Assignments (step sizes) to the identified search direction are determined with the commonly adopted method of successive averages, originally introduced for static traffic assignment problems (12) and adapted to time-dependent DTA problems by Peeta and Mahmassani (13). The solution of this procedure is embedded in the vehicle loading strategy (scheduling strategy). Each vehicle is assumed to belong to a block within the zone and is assigned a generation link in that block. The model will then assign to each vehicle, given its generation link,

• Using the link marginal travel times (from Step 1), compute the minimum-cost time-dependent k-shortest-path tree between each origin and shelter destination for each feasible departure time. • For each origin, determine the departure time, path, and destination combination that yields the least marginal travel time cost k τ,n ij . This will be the descent direction. Step 3. Determine Auxiliary Solution • Perform an all-or-nothing loading for all evacuees R τ,v ij for a given origin i onto path k τ,v,n ij . This step will give the auxiliary number of vehicles on paths Y τ,v,n i,j,k . Step 4. Update Assignment • Update the assignments for the next iteration by using the method of successive averages, ∀ i, j, τ, and k as follows: Riτ, ,jv, ,kn +1 =

⎛ 1 1 ⎞ τ ,v ,n Yi,τj,,vk,n + ⎜ 1 − ⎟ Ri, j, k ⎝ n +1 n + 1⎠

Step 5. Check Convergence • Check the difference in the number of vehicles assigned to various departure times, destinations, and paths over two successive iterations as follows: Riτ, ,jv, k,n +1 − Riτ, ,jv, k,n ≤ • Denote the number of violations N() for which the above criterion has failed over all sets of ∀ i, j, τ, and k.

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Step 6. Terminate Procedure

Step 2. Start Evacuation Incident Scenario

• Specify a preset upper bound Ψ on the number of violations N() and terminate the algorithm if N() ≤ Ψ. • If the termination criterion is not met, increment iteration counter (n = n + 1) and go to Step 1 with new path assignments R τ,v,n+1 i,j,k .

• Use DYNASMART-P to simulate the vehicles’ trips following their original set of path assignments (from Step 1). • Start evacuation incident scenario at time Ts = 25 min.

Step 3. Detect Evacuation Incident EXPERIMENTAL DESIGN AND MODEL RESULTS Numerical experiments are presented to illustrate the application of the evacuation model framework and to evaluate the benefits of staging evacuation operations in an urban network. The test network, shown in Figure 1, is selected only to test the proposed procedure and not for being a high-risk area for major emergencies. The test network is extracted from the Fort Worth, Texas, network and corresponds to one sector of Interstate 35W between I-20 and I-30, with a surrounding network of signalized arterials and stop- or yield-controlled local streets on both sides of the freeway. The network consists of 168 nodes, 441 links, and 13 TAZs. The origins and destinations are regular nodes and are distributed over the whole network. Freeways, frontage roads, and ramps are modeled as directed arcs, and the remaining links are two-way directed arcs, one per direction. The free mean speed on freeway links is 70 mph, on major arterials is 50 mph, on ramps and local streets is 40 mph. The jam density is assumed to be 200 vehicles/mi for freeways and 120 vehicles/mi for arterials, ramps, and local streets. All intersections are controlled (either signalized or stop- or yield-controlled) except freeway nodes. The simulated evacuation incident involves a case in which a hypothetical bioterrorism attack was launched in Zone 7 (shaded area in Figure 1) and is set to start at Minute 25 of the simulation. The traffic controller will (a) detect this incident at Minute 30 into the simulation, (b) close all entrances to Zone 7, and (c) start the evacuation advisory procedure. Safety destinations (decontamination centers) with sufficient parking capacities are assumed to be located at Nodes 116 and 117. The advisory procedure will determine for the evacuee the departure time, destination, and path to this safety destination. The controller will also provide pretrip, variable message sign, and en route information for nonevacuees who are on their way to the impacted zone for necessary detours and trip changes. It should be noted that the selection of safety destinations is made externally to the procedure, though it could be determined through an extension of the current formulation. Also, one could select nodes just outside the hot zone as safety destinations with no loss of applicability of the procedure. Three sets of experiments were designed to investigate the convergence pattern of the solution algorithm. However, the network must first be prepared for the solution algorithm to work properly.

Network Preparation This phase is used to prepare the network for later use in the evacuation-scheduling algorithm.

Step 1. Initialization • Run DYNASMART-P with 100% UE assignment to determine the initial vehicle path assignments. • Classify all vehicles to be of user class o (nonimpacted).

• Keep simulating until an evacuation incident is detected at time Td = 30 min. • Determine evacuation incident or incidents and TAZs (Zone 7, Figure 1, shaded area). • Determine the set of impacted origin nodes Iv ⊆ I (Nodes 99, 106, 107, 108, 146, 147, 152, 153, 155, 158, 165, 177, and 178 in this application). • Identify possible shelter locations Jv ⊆ J (Nodes 116 and 117 in this application).

Step 4. User Class Specification • Determine nonimpacted vehicles and classify them as user class o. • Determine vehicles that need to stop, cancel their trip (have not yet departed) to impacted origins or make a detour, classify them as user class o, and assign them new destinations. • Determine vehicles that need to be evacuated, classify them as user class v, assign them to (a) the first departure time after the evacuation incident, (b) a shelter destination at random, and (c) the best path to that destination. This will be the initial solution to be later used in Step 0 of the solution algorithm. • Keep simulating until all vehicles are out of the network. • Denote the resulting path assignments as R τ,u,0 ijk .

Convergence Pattern Three scenario runs are considered in investigating the convergence pattern of the solution algorithm. Three levels of evacuation demand are considered, namely, 1,794, 3,558, and 5,692 vehicles. The total demand (evacuees and nonevacuees) loaded on the network is fixed at 47,300 vehicles. In all these experiments, the evacuation incident is located in Zone 7 and starts at Minute 25 of the simulation. The central traffic controller detects the incident 5 min later and starts providing SO evacuation advisory information 30 min into the simulation. The objective function to be minimized is the total system evacuation trip time as measured from the start of evacuation advisory period (Minute 30 of the simulation). Minimizing such an objective is equivalent to minimizing the NCT, the primary objective for any evacuation procedure, as well as maximizing the number of evacuees reaching safety in each time period. The objective function is computed as follows: Z = total system evacuation trip time = ∑ [arrival time ( j ) − 30 min ]

j ∈ user class v

j

Any algorithm using the method of successive averages is expected to converge rather slowly; however, the convergence rate may be increased considerably by intelligent selection of the number of


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Safety Destination 115 117

FIGURE 1

Test network.

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244


Total Trip Time (min)

130,000 105,000 80,000 55,000 30,000 0

FIGURE 2

10

20

30

40 50 60 Iteration Number

70

80

90

Convergence pattern for solution algorithm (demand level 1,794 vehicles).

vehicles to update their path. This selection is done by the restriction of the algorithm to update only those vehicles that have longer trip times rather than by random selection of vehicles. The convergence results reported here are based on such an updating technique. Figures 2 through 4 depict the convergence patterns for the three scenarios. As expected, the solution algorithm converges to an optimal point, albeit somewhat slowly. As can be seen, gains can be significant in the first few iterations but can be rather small as the number of iterations grows. It took 40 iterations for the algorithm to stabilize (within 5% of optimality) for the first scenario, 20 iterations for the second scenario, and 70 iterations for the remaining scenario. On the bright side, for such a network, the model execution time has been remarkably fast and requires less than 30 s per evacuation origin per iteration on a 2.8-GHz Intel processor Dell 4700 system. However, because of the time-dependent minimum-cost path algorithm embedded in DYNASMART-P and the tracking of individual vehicles, the memory requirements are relatively large.

Analysis of Results To analyze the benefit of the proposed evacuation-scheduling model, three sets of experiments are considered, each corresponding to the previously selected demand levels of 1,794, 3,558, and 5,692 evacuees. Each set consists of two scenario runs, one in which simultaneous evacuation is simulated and the other in which scheduled evacuation is simulated. In the first set of experiments, under the simultaneous evacuation procedure, evacuees required 90 min to clear the network with a total evacuation trip time of around 54,650 veh-min and an average trip time of 22 min. With the evacuation-scheduling algorithm, the total evacuation trip time is reduced by 31%, to around 37,639 veh-min; NCT is also reduced by 20%, to 71 min, and the average trip time is reduced by 61%, to 9 min. In the second set of convergence experiments, under simultaneous evacuation rules, the evacuees required 202 min to clear the network with a total evacuation trip time of


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TABLE 1

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Summary of Scenario Results

Demand Scenario Number

Evacuation Type

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Evacuation Demand (veh)

Network Clearance Time (min)

Total Trip Time (veh-min)

Average Trip Time (min)

1,794 1,794 3,558 3,558 5,692 5,692

90 71 202 84 278 108

54,650 37,639 315,324 91,221 532,427 204,130

22 9 69 13 80 28

around 204,130 veh-min; NCT is also reduced by 61%, to 108 min; and the average trip time is reduced by 65%, to 28 min. Table 1 summarizes the results for these three sets of convergence experiments. As expected, under all demand levels, intelligently spreading the evacuation demand over the possible set of departure times results in significant benefits over the simultaneous evacuation procedure. Moreover, the model outputs the fraction of evacuees to depart in each time period along with the path and destination. Figures 5, 6, and 7 present the optimal staging policy for evacuating the demand for Scenarios 1, 2, and 3, respectively. For example, in Figure 5, 6%,

Percentage of Evacuation Demand

around 315,324 veh-min and an average trip time of 69 min. With the evacuation-scheduling algorithm, the total evacuation trip time is reduced by 71%, to around 91,221 veh-min; NCT is also reduced by 59%, to 84 min; and the average trip time is reduced by 81%, to 13 min. In the last set of convergence experiments, under simultaneous evacuation rules, the evacuees required 278 min to clear the network with a total evacuation trip time of around 532,427 veh-min and an average trip time of 80 min. With the evacuation-scheduling algorithm, the total evacuation trip time is reduced by 62%, to

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Staging policy for Scenario 1 (demand level 1,794 vehicles).

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10%, and 5% of the total evacuation demand, expressed in vehicle trips, are advised to evacuate in the first 3 min. The procedure effectively is front-loading the roadway links in the first few time periods (30% to 45% of total evacuation demand) just enough to prevent system breakdown. After that, the procedure manages the network’s residual capacity, as manifested by the sending of as little as 0% to 2% of total evacuation demand in subsequent time periods. Therefore, it can be concluded that by intelligently staggering the evacuees’ departure process and advising them to wait until their scheduled departure time, the NCT can be significantly reduced for the benefit of all evacuees.

CONCLUSIONS An evacuation-scheduling model based on SO assignment is presented. The model represents the case in which a super user (central traffic controller) has a priori information about the path assignments for two types of users (evacuees and nonevacuees) in the network and seeks to minimize the NCT for evacuees while trying to minimize the disruption to the nonimpacted vehicles. Evacuees are provided with trip information regarding their departure times, destination choice, and the path to that destination. In addition, this model assumes that evacuees follow their route guidance information and do not switch paths or departure times on their own. Such a model will result in estimating a lower bound for NCT to be later used in subsequent emergency planning applications. Model results indicate that the solution algorithm converges under three evacuation demand levels ranging from low to high to an SO state, whereby the NCT cannot be further reduced by vehicles’ unilaterally switching departure times, paths, or destinations, or all three. Furthermore, the results show that scheduling evacuation operations yields significant improvement over the simultaneous case in three network performance measures, namely, NCT, total trip times (measured from the start of the evacuation advisory period), and average trip time.

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