SUPPORTING MOBILITY-IMPAIRED POPULATIONS IN …docs.trb.org/prp/15-2436.pdf · SUPPORTING MOBILITY-IMPAIRED POPULATIONS IN EMERGENCY EVACUATION Lei Feng ... [email protected]

SUPPORTING MOBILITY-IMPAIRED POPULATIONS IN

EMERGENCY EVACUATION

Lei Feng

Operations Analysis, FedEx Corporation

1000 Ridgeway Loop Rd, Suite 500, Memphis, TN, 38120

Tel: (901) 355-0098

Email: [email protected]

Neža Vodopivec

Applied Mathematics & Statistics and Scientific Computation

University of Maryland

Mathematics Building, College Park, MD 20742

Tel: (301) 405-7768


Elise Miller-Hooks

Department of Civil & Environmental Engineering

University of Maryland

1173 Glenn L. Martin Hall, College Park, MD 20742

Tel: (301) 405-2046


Word Count: 5540 + (6 Figures + 2 tables)*250 = 7,540 words

Revised Paper

October 16, 2014

Submitted for presentation at the 2015 Transportation Research Board 94th Annual Meeting and for

publication in the Transportation Research Record

mailto:[email protected]



Feng, Vodopivec and Miller-Hooks

ABSTRACT 1

One of the populations’ most vulnerable in emergencies is persons with impaired mobility. During an 2

evacuation, the mobility-impaired need special arrangements while being transported to shelters and special 3

accommodations during their shelter stays. With limited resources, not all shelters can be specially equipped 4

for low-mobility evacuees. In this paper, we address the problem of simultaneously choosing a particular 5

subset of shelters to equip, matching persons with impaired mobility to these shelters, and routing 6

paratransit vehicles to bring passengers to shelters accordingly. We formulate the problem as a mixed-7

integer program and propose a heuristic appropriate for solving real-world, large-scale instances. These 8

solution strategies can form the basis of a practical tool to be used for evacuation planning. We construct 9

an evacuation scenario based on real-world evacuation zones, shelter locations, and population statistics for 10

New York City, and we use this scenario to evaluate our heuristic. 11

Keywords: mobility-impaired population evacuation, location-routing problem, tabu search. 12

Feng, Vodopivec, Miller-Hooks 1

INTRODUCTION AND MOTIVATION 13

Urban populations are especially vulnerable when a disaster strikes. Whether such a disaster is caused by a 14

natural event, industrial accident, or terrorist attack, how well a city prepares and responds can be critical 15

in determining the outcome. Large-scale evacuation, though often the most viable response, requires 16

effective communication with the public, well-coordinated management of efforts, and carefully planned 17

traffic-control strategies. Urban evacuations are complicated by the fact that, in many large metropolitan 18

areas such as New York City (NYC) and Chicago, over 30% of households are without cars (1). A 19

mismanaged evacuation of the carless population during Hurricane Katrina has subsequently led emergency 20

planners and researchers to devote attention to bus-based evacuations (2-4). 21

In 2006, the United Nations (5) adopted a convention urging member states to undertake “all 22

necessary measures to ensure the protection and safety of persons with disabilities in situations of risk, 23

including situations of armed conflict, humanitarian emergencies and the occurrence of natural disasters.” 24

Nevertheless, NYC’s evacuation during Tropical Storm Irene in 2011, while successfully incorporating 25

buses, failed so completely at meeting the needs of disabled residents that the city was sued by disability-26

rights organizations. The city’s further failure to evacuate the disabled during Hurricane Sandy in 2012 led 27

the presiding judge to grant an expansion of the case to a class-action lawsuit (6). 28

While all persons with disabilities are disproportionately affected in disasters, it is the mobility-29

impaired who are particularly likely to be left behind or neglected during evacuation. For many persons 30

with low mobility, it is difficult or impossible to access the general public transit stops. Those who manage 31

are often forced to part with their wheelchairs, walkers, or other medical equipment at bus loading areas. 32

Upon their arrival at shelters, the mobility-impaired are frequently unable to obtain needed assistance and 33

medical care (7). 34

In 2010, 2.8% of United States households had at least one mobility-impaired member (1) and, as 35

the population ages, this percentage is expected to increase. Any successful emergency-response plan must 36

account for the fact that mobility-impaired populations need special accommodations both while being 37

transported to shelters and during their stay there. 38

The best, and perhaps only, way that emergency responders will be able to reach these populations 39

is by dispatching lift-equipped vehicles to provide them with door-to-door transportation. The most efficient 40

approach would be to use local paratransit providers who possess vehicles capable of transporting large 41

medical equipment and are familiar with the needs of the low-mobility population. Many local governments 42

already maintain contracts with such providers to support the daily activities of this population in 43


compliance with the Americans with Disabilities Act (ADA). The ADA mandates that all residents have 44

access to public transit (8). 45

All shelters housing mobility-impaired populations should staff medical personnel, maintain 46

necessary equipment, and incorporate wheelchair-accessible amenities. It is impractical to provide such 47

accommodations at the full range of shelters serving the general population. A more effective approach 48

would concentrate the necessary resources at a few shelters and assign mobility-impaired evacuees to only 49

these designated shelters. 50

This paper proposes optimization-based techniques to assist in planning large-scale evacuations of 51

a mobility-impaired population. Specifically, the problem of effectively: (1) selecting the subset of special 52

shelters to house the mobility-impaired population, (2) assigning each mobility-impaired evacuee to a 53

special shelter not too far from his or her home, and (3) designing a set of paratransit routes that will 54

transport evacuees to their assigned shelters is formulated as a mixed integer program. We refer to this 55

tripartite problem as the Mobility-Impaired Population Evacuation Problem (MIPEP). 56

Our model describes how emergency planners can most effectively allocate a city’s resources in an 57

evacuation of the mobility-impaired. More specifically, its solution identifies the most cost-effective way 58

to transport and house such populations while maintaining a high level of care. The model explicitly 59

accounts for evacuee welfare by removing solutions that are bad for them from consideration entirely. The 60

constraints achieving this removal account for several major medical and psychological considerations. In 61

particular, for medical reasons, there is an absolute maximum on passenger ride time. Each vehicle is 62

modeled as having some finite capacity, and each passenger with large medical equipment as taking up 63

more of this capacity. No equipment is left behind. Each shelter is modeled as having some finite capacity. 64

Consequently, no overcrowding or shortage of medical support occurs. Furthermore, for psychological 65

reasons, an evacuee must not be assigned to a shelter more than some constant times as far from his or her 66

home as the closest shelter, special or not. An evacuee’s route to his or her assigned shelter must not take 67

more than a constant times as long as the direct route. Finally, a vehicle must unload fully whenever it 68

arrives at a shelter. 69

The MIPEP is combinatorially difficult because all three of its components—location, allocation, 70

and routing—are interdependent. Our model and solution treat them as such. A more simple-minded 71

solution might proceed by: (1) making an educated guess at the right number of shelters S to open to the 72

mobility-impaired (by balancing an estimated marginal cost of an additional shelter against an estimated 73

marginal savings in transportation); (2) locating the S shelters as centrally as possible; (3) allocating each 74

evacuee to his or her nearest shelter; and then (4) solving the resulting fixed routing problem. Figure 1 75


shows an example of how such stage-wise solutions can perform badly. Specifically, the figure illustrates 76

that the choice of a best set of routes can vary significantly over different client assignment decisions. In 77

Figure 1(a), clients a - f are assigned to shelter s1 and g - i are assigned to shelter s2. In the route depicted 78

in Figure 1(b), client c is reassigned to shelter s2. The corresponding best routes can vary significantly. That 79

is, this seemingly minor modification in client assignment can lead to very different routing solutions with 80

significantly different performance. The MIPEP avoids these pitfalls by making location, allocation and 81

routing decisions simultaneously. 82

83

FIGURE 1. Illustration of One Vehicle Route with Different Assignments 84

Direct solution of the mathematical program on large-scale, real-world problem instances is 85

formidable, and thus, we develop a heuristic based on concepts of tabu search to solve the proposed 86

mathematical program. The solution methodology is applied on a case study of a hypothetical evacuation 87

of NYC’s mobility-impaired population using real data obtained on location from various government 88

databases. These aspects of the paper are described following a review of the related literature. 89

RELATED LITERATURE 90

Several works have addressed the use of public transit in evacuation. Margulis et al. (9) developed a 91

deterministic decision-support model for bus dispatching to maximize the number to egress in hurricane 92

evacuation. This model assumes that evacuees are assembled at their closest pickup points and the locations 93

of shelters are known. With a similar assumption of predetermined pickup and shelter locations, Sayyady 94

and Eksioglu (10) proposed a mixed integer linear program to optimize transit routing plans with the 95

objective of minimizing total evacuation time for no-notice evacuation. In their work, transit vehicles are 96

only allowed to perform one trip. Abdelgawad and Abdulhai (11) and Bish (4) formulated this transit-based 97

evacuation problem as types of vehicle routing problems. The objectives of both models are to transport 98

evacuees from preset pickup locations to known shelter locations in the minimal amount of time by 99

a

bc

d

e

f

g

hi

S1

S2

Holding Lot

Shelter

Client

a

bc

d

e

f

g

hi

S1

S2

(a)(b)

Trip Distance

Deadhead

Transit Distance


efficiently routing and scheduling a fleet of buses from a set of bus yards. Both works propose heuristic 100

solution concepts. 101

More recently, Kulshretha et al. (12) proposed a mixed integer linear program to determine the 102

optimal pickup locations for evacuees to assemble. Their model simultaneously considers the allocation of 103

available buses to transport the assembled evacuees between the pickup locations and specified shelters. 104

While these works are related in that they assign pickup locations to shelters, they do not capture many of 105

the elements of the MIPEP, such as the need for simultaneous routing, assignment and shelter location 106

decisions. They also focus on aspects of an evacuation that are not relevant in this context, such as selection 107

of pickup locations at which evacuees will assemble. 108

We are aware of only two prior studies in the literature that have proposed tools to aid in decisions 109

regarding the location of shelters for transit-based evacuation. Song et al. (13) formulated this transit-based 110

shelter location and evacuation problem in the context of hurricanes as a location-routing problem (LRP) 111

with uncertain demand. The LRP is used to determine optimal shelter locations and transit routes with the 112

objective of minimizing total evacuation time. Shelter locations are chosen from a pool of potential 113

locations based on their distances from the pickup locations assuming all pickup locations are assigned to 114

their nearest open shelter. Only the transportation cost is considered in the objective and each bus is 115

restricted to a single route per shelter. The authors compared three heuristics based on concepts of genetic 116

algorithms, artificial neural networks, and hill climbing. 117

Goerigk et al. (14) considered a location-routing bus evacuation problem in which both a subset of 118

shelters and a subset of pickup locations are chosen optimally from a set of candidates. They proposed a 119

branch-and-bound technique that takes an integrated rather than sequential approach to the location-120

allocation and routing components of the problem. The objective is to choose shelters and create bus 121

schedules that minimize the time required to transport all evacuees. In the model, a fixed number of buses 122

is available and the number of shelters used in limited. Buses must transport evacuees directly from a pickup 123

location to a shelter without visiting intermediate pickup locations. A case study is performed on a very 124

small instance of such a problem in which the population of a 500-meter radius must be evacuated from a 125

choice of 14 possible pickup locations. The computational complexity of generating an exact solution limits 126

real-world application of the algorithm proposed. The MIPEP generalizes the problem of Goerigk et al. by 127

assuming a more complex routing component in which multiple evacuees are picked up at their individual 128

locations on a single route to a shelter. The MIPEP is solved with a heuristic that can be applied to real-129

world, large-scale instances as we demonstrate with a case study for the evacuation of a mobility-impaired 130

population in New York City. 131


The general LRP has received significant attention over past decades. Applications primarily relate 132

to logistics. See (15) for a review of both models and solution methods for LRPs. Traditional LRP models 133

determine where to locate facilities and how to distribute or collect goods to or from customers through 134

simultaneously solving a joint location and routing problem. The MIPEP differs from the LRP in several 135

important ways. The MIPEP involves two types of facilities: the holding lots for paratransit vehicles 136

(depots) and shelters. The LRP has only depots from which vehicles start out and to which they return once 137

the goods are distributed or collected. The MIPEP has several additional constraints, such as that evacuees 138

should not spend an unreasonable amount of time onboard while additional pickups are made. The operating 139

patterns of these two classes of problems also differ. Specifically, in the LRP, each vehicle is restricted to 140

serve only one depot, while in the MIPEP each vehicle is allowed to perform multiple tours for multiple 141

shelters. 142

The MIPEP also has commonality with other ridesharing problems, including the Dial-A-Ride 143

Problem (DARP) (see (16) for a review of DARP). Like the MIPEP, the DARP is characterized by pairing 144

and precedence constraints, such that for each request the origin must precede the destination and both 145

locations must be visited by the same vehicle, and user inconvenience constraints, including a maximum 146

ride time limit. A primary concern in DARP, however, is the time windows that limit pickup and delivery 147

for customers. The routing aspects of MIPEP are similar, with some exceptions: (1) pickups assigned to the 148

same shelter share an identical destination; (2) customers do not choose their time windows and instead are 149

expected to be ready for the vehicle when it arrives; and (3) customers with different shelter destinations 150

must be transported on different vehicles. Additionally, shelter destinations are chosen for the evacuees in 151

the MIPEP in coordination with routing decisions; whereas, customer destinations in DARP are set by the 152

customers. Finally, the MIPEP is a multi-depot type of ridesharing problem, since resources from multiple 153

companies’ fleets will be drawn upon. 154

Thus, it appears that no previous work has proposed optimization tools that can directly support 155

sheltering and routing decisions for the mobility-impaired population in an evacuation. This paper seeks to 156

fill this gap. 157

MATHEMATICAL FORMULATION 158

The MIPEP is formulated next. Before proceeding to the formulation, notation is introduced. 159

Notation 160

w Number of paratransit holding lots

{1,..., }H w Set of paratransit holding lots


hV Number of vehicles at holding lot h H

{1,..., }h

h H

K V

Set of all vehicles in all holding lots

n Number of pickup nodes

1,...,P w w n Set of pickup nodes

( , )a bi iq q Number of persons, a, and wheelchairs, b, at pickup node i P

i Service time at pickup node i P

dim Distance of pickup node i P to its nearest shelter

tm Maximum onboard time for all passengers

Deviation parameter for the assigned distance to dim

m The number of potential shelters

{ 1,..., }S w n w n m Set of potential shelters

N Set of all nodes {H }P S

sF Fixed opening cost of shelter s S

sQ Capacity of shelter s S

C Capacity of vehicle

D Maximum driving distance for each vehicle

ijd Distance from node i to j, ,i j N

ijt Travel time from node i to j, ,i j N

dC Unit cost of driving distance for all vehicles

M Arbitrary large number

Decision Variables 161

1, if shelter is open

0, otherwise s

s Sy

1, if pickup is assigned to shelter and is transported by vehicle

0, otherwise

kis

i P s S k Kz

1, if arc ( , ) is traversed by vehicle

0, otherwise

kij

i j k Kx


kiL Load of vehicle k K after visiting node i N

Auxiliary decision variables associated with routing: 162

1, if vehicle comes from holding lot

0, otherwise kh

k K h Hu

kijT Trip duration of vehicle k K starting from node i N and ending at node j N

Formulation 163

Given the above notation, the MIPEP can be defined on a digraph ),,( ANG where N is the set of nodes 164

representing holding lots, pickup locations and shelters, and A is the set of directed arcs 165

{( , ) : , }A i j i j N connecting these locations. 166

min kMIPEP s s d ij ij

s S k K i N j N

C F y C d x

(1)

Subject to 1 ,kis

k K s S

z i P

(2)

( 2 ) ,a b ki i is s s

k K i P

q q z Q y s S

(3)

,k dis is i

k K s S

d z m i P

(4)

1 ,kij

k K i N

x j P

(5)

,,0 KkNixxNj

kij

Nj

kji

(6)

1 ,kh

h H

u k K

(7)

, ,khj kh

j P

x u k K h H

(8)

,kh h

k K

u V h H

(9)

(1 ) (1 ) , , , ,k k k kij is js ijM x z z M x i P j P k K s S (10)

1 (1 ) , , ,k kis isz M x i P s S k K (11)

0 , ,kiL i H S k K (12)


(1 ) 2 (1 ) , , ,k k a b kij j j j ijM x L q q M x j P i N k K (13)

, ,k kj ij

i N

L C x j P k K

(14)

,kij ij

i P S j P S

d x D k K

(15)

,,,0 KkNjSHiT kij (16)

(1 ) (1 ) , , ,k k k k kij ij ij ij j jq ij

q N

M x T t x T M x i P j P k K

(17)

, , ,k kij ijT M x i N j N k K (18)

, , ,k tijT m i P j N k K (19)

\

1 , 2 | |,kij

k K i P j N P

x P P P

(20)

, , , {0,1}, , , , , , .k k k ks ps ij kh i ijy z x u L T R h H p P i N j N s S (21)

Total costs include the fixed cost of operating shelters that can support this population and 167

operational costs of transporting these individuals to their assigned shelters. Objective function (1) 168

minimizes the sum of fixed and operational costs. It is reasonable to assume that the second term 169

(transportation cost) will be roughly proportional to the total distance driven by all paratransit vehicles. 170

Similarly, we assume that the first term (cost of providing special services at shelters) will be proportional 171

to the number of shelters at which special services are provided. 172

Constraints (2) ensure that each pickup node is assigned to exactly one shelter. By Constraints (3) 173

pickup nodes are assigned only to those shelters that are open, and capacity limits of open shelters are met. 174

Constraints (4) force that, for each pickup node, the assigned shelter is within times the distance to its 175

nearest shelter. Each customer is served exactly once through Constraints (5). Flow conservation is 176

expressed in Constraints (6). Constraints (7) and (8) ensure that each vehicle is used at most once, while 177

Constraints (9) force the number of vehicles that come from each holding lot do not exceed the number of 178

available vehicles in it. Constraints (10) and (11) ensure that clients travel toward their assigned shelters 179

without stopping at other intermediate shelters. Constraints (12) ensure vehicles are empty when leaving a 180

holding lot and after each stop at a shelter. Constraints (13) express that vehicle load when leaving a pickup 181

node increases by the number of passengers and wheelchairs that are loaded. It is assumed that one person 182

occupies one seat and one wheelchair two seats. Constraints (14) guarantee that vehicle load restrictions 183

are met. Constraints (15) define that the distance that each vehicle travels from its first pickup to its last 184


dropoff location is restricted to a maximum limit. Constraints (16) reset the incurred trip duration of each 185

vehicle to zero every time the vehicle leaves a holding lot or shelter. Constraints (17) and (18) express that 186

the trip duration of each vehicle increases when it traverses the nodes. Constraints (19) ensure that trip 187

duration is lower than the passenger maximum onboard time. Subtours are eliminated in Constraints (20). 188

Finally, integrality of the decision variables is stated in Constraints (21). 189

The formulation involves O(|K|∙|N|2) binary decision variables, O(|K|∙|N|2) integer decision 190

variables, and O(2|N|-2|P|) constraints. Only very small-scale instances can be solved exactly. The most recent 191

exact solution method is proposed by Akca et al. (17). The authors formulated a traditional location routing 192

and scheduling problem, which is comparable in complexity to the MIPEP, as a set-partitioning problem 193

and proposed a column generation framework with two-phase pricing in the subproblem. To deal with large 194

instances, they also proposed two heuristic pricing algorithms to solve the subproblem. The largest 195

instances that were tackled involved 5 facilities and 40 customers requiring 8 CPU hours of computing 196

time. Thus, the practical MIPEP instance is solved heuristically, as described next. 197

SOLVING THE MIPEP 198

Heuristics proposed for traditional LRPs with a size seen in real-world operations provide inspiration for 199

the proposed solution strategies. Three solution strategies for LRPs were described in (18): sequential, 200

interactive, and nested. Sequential methods are often ‘locate first and route second’ type heuristics, where 201

the location problem is solved first and the routing problem second. Without consideration for the 202

interrelations between the two problems, they usually obtain low-quality solutions (19). Interactive methods 203

treat the location and routing problems equally and iterate between the two problems until a stopping 204

criterion is met. For example, Tuzun and Burke (20) and Wu et al., (21) proposed tabu search and simulated 205

annealing solution methods, respectively, to solve traditional LRPs. Although these methods can provide 206

better solutions than the sequential methods, these interactive methods cannot explore the neighborhood 207

space extensively due to the equal treatment of the two problems. Nagy and Slahi (18) proposed a 208

hierarchical structure for the LRP, where the location problem is solved in the main problem and the routing 209

problem as a ‘subproblem’ to it. Based on this hierarchical structure, Gündüz (22) proposed a tabu search 210

algorithm to solve a combined location problem and multi-depot vehicle routing problem with time 211

windows. Results show that the nested methods outperformed the other two solution methods. 212

We present such a nested tabu search solution strategy for the MIPEP. Our strategy applies a 213

hierarchical structure similar to that proposed in (18). It decomposes the MIPEP into subproblems: (1) a 214

Capacitated Shelter Location-Allocation Problem (CSLAP) and (2) a Multi-depot Dial-A-Ride Problem 215

(MDARP). Thus, interactions between the CSLAP and the MDARP subproblems are explicitly considered. 216


Capacitated Shelter Location-Allocation Problem (CSLAP) 217

Letting , ,kis is

k K

z z i P s S

, CSLAP can be formulated as follows. 218

min CSLAP s s d is is

s S i P s S

C F y C d z

(22)

subject to 1 ,is

s S

z i P

(23)

( 2 ) a bi i is s s

i P

q q z Q y s S

(24)

,dis is i

s S

d z m i P

(25)

, {0,1} , ,s isy z i p s S k K (26)

Objective function (22) minimizes the sum of fixed and assignment costs. The assignment cost is 219

calculated through the direct distance between the pickup location and the associated evacuees’ assigned 220

shelter. This direct distance is an approximation to the actual transportation distance. Constraints (23 - 25) 221

play the same roles as constraints (1-3) in the MIPEP formulation. Binary integrality is required through 222

Constraints (26). 223

The CSLAP can be solved exactly through a commercial solver. Two decision sets are identified 224

from its solution: 0 ,y which indicates whether or not each shelter s S is open, and 0 ,z which indicates 225

if a passenger i P has been assigned to shelter s S . 226

Multi-Depot Dial-A-Ride Problem (MDARP) 227

With 0z identified in CSLAP, the MDARP is formulated as follows. 228

min kMDARP d ij ij

k K i N j N

C C d x

(27)

subject to Constraints (5)-(20)

, {0,1}, , , Z , , , , , .k k kij kh i ijx u L T h H p P i N j N s S (28)

The MDARP is a difficult combinatorial problem. Its simpler single-depot problem even is known 229

to be NP-hard. Solutions of problem instances of a size seen in the real world are unobtainable with typical 230

computing power. Thus, a cheapest insertion heuristic adapted from (23) is proposed for its solution. This 231


algorithm first builds optimal routes for all pickup-shelter location pairs and then assigns the built routes to 232

vehicles in holding lots. The objective of this assignment is to minimize total route costs subject to 233

limitations on the number of available vehicles in each holding lot. The main steps of the cheapest insertion 234

heuristic are summarized next. 235

Cheapest Insertion Heuristic for the MDARP

Step a: Copy each pickup-shelter pair obtained from the CSLAP into the unrouted request list, URL.

Step b: Create an empty route. Set r =1.

Step c: Select the first unrouted pair (p, s) from URL. Find all feasible insertions within all existing routes,

1 to r.

(i) If a feasible insertion is found, insert (p, s) to the route r* with minimum insertion cost,

and update route r*.

(ii) If no feasible insertion exists, create a new empty route and insert (p, s) in it. Set r = r+1.

Step d: Repeat step c until URL is empty.

Step e: Add each depot to the starting and ending point of each route; calculate the updated route cost

matrix.

Step f: Assign routes to holding lots according to the updated route cost matrix.

236

Potential Feasible Insertions 237

Due to Constraints (10) and (11) that prevent vehicles from stopping at intermediate shelters while en route 238

to another destination shelter, the potential feasible insertions of pickup-shelter pair (p, s) on route r in step 239

c take one of the following forms: 240

(1) Insert (p, s) at the beginning of route r. 241

(2) Insert (p, s) immediately after each shelter, if any shelter already exists on route r. 242

(3) Insert p immediately before s and each of the pickup nodes before s until reaching another shelter 243

or the beginning of the route, if shelter s already exists on route r. 244

Feasibility Checking 245

An insertion of a pickup-shelter pair must ensure that vehicle capacity constraints (14), maximum driving 246

distance limit for drivers (15) and passenger maximum onboard time (19) are satisfied and do not create 247

violations of these constraints for other nodes already included in the route. 248

249

Nested Tabu Search Strategy 250

Figure 2 provides the overall framework of this strategy. 251


252

FIGURE 2. Flowchart of Nested Tabu Search Algorithm 253

The procedure begins with generation of an initial solution through a sequential solution strategy, 254

and setting the best solution to the obtained initial solution. Candidate solutions around the current best 255

solution are generated in a diversification step. A tabu list is maintained to prevent the search from getting 256

stuck at a local solution. An identification and intensification procedure is employed to filter out infeasible 257

candidate solutions that are generated in this procedure and calculate total costs. The best generated feasible 258

solution is assessed to see if it meets the aspiration criterion. Details of the aspiration, tabu list management, 259

and termination criteria are summarized next. This is followed by detailed descriptions of diversification 260

No

Step 2- Diversification: generate candidate solutions,

Ycand, within the exploration region of ybest.

Step 4- sort Ynew according to

corresponding cost in non-decreasing

order, select the first point ynewbest.

Step 5- aspiration

satisfied?

Step 6- update best known

solution, ybest, and total cost, Cbest

Yes

Step 7- tabu?No

Step 11- expand tabu

list

Step 12-

termination criteria

sastified?

Step 13- output optimal

solution ybest with Cbest

Yes

Step 8- is Ycand

fully scanned?

Yes

Step 9- select next element in Ycand

No

Step 10- set best known solution

to first element in tabu list

No

End

Step 1-Initialization: solve the MIPEP through solving

CSLAP first and MDARP second to generate an initial

solution y0 and calculate total cost CMIPEP(y0). Set the best

solution ybest = y0 and the best cost Cbest =CMIPEP(y0).

Step 3- Identification and Intensification: for each

generated candidate solution, Ycand, if it is feasible

and not tabu, calcuate the updated cost CMIPEP(Ycand).

Yes


and intensification steps. 261

Aspiration criterion Any feasible candidate solution that has the best total cost of all discovered solutions 262

will become the best solution regardless of its tabu status. 263

Tabu list Two tabu lists are applied within the overall procedure. Both employ a complete memory 264

approach. One maintains a list of solutions, considered in the last certain number of iterations (the tabu 265

tenure) of the search procedure. This list prevents revisiting of solutions within the iterations associated 266

with the tabu tenure. A solution may be removed from the list prematurely if no neighboring candidate 267

solution of the best solution outperforms this best solution. The second tabu list includes infeasible 268

candidate solutions generated in the diversification step. These solutions are maintained as tabu 269

permanently during the entire search procedure. This list is used to filter out infeasible candidate solutions 270

in steps 2 and 3. 271

Termination criteria The procedure terminates when either a predefined maximum number of iterations or 272

a predefined maximum number of iterations without improvement is reached. 273

Step 2-Diversification 274

In this step, a diversification strategy is applied to generate a set of candidate solutions Ycand, a set of 275

solutions vectors, within the neighborhood space of the best solution, ybest. The candidate solutions are 276

generated through adapted exploration moves ‘drop’, ‘add’ and ‘switch’, originally introduced in (24). 277

Before introducing the adapted exploration moves, a neighborhood relation between two shelters is defined 278

based on the definition proposed in (18): 279

Given constraints (4) or (25), which enforce that each client cannot be sent to shelters that are 280

beyond times the distance to the nearest shelter, a concept of neighboring shelters is defined. 281

Neighboring Shelters (depicted in Figure 3): Two shelters 1s and 2s are neighbors if and only if at least 282

one pickup node p exists such that 1s and 2s are the first and second nearest shelters to p, respectively, 283

and if 2 1d d , then 1 2( )s Nb s and 2 1( )s Nb s , where ( )Nb s denotes a set of neighboring shelters 284

of shelter s. 285

286

s1

s2

p

d2d1

2 1d d


FIGURE 3. Illustration of Neighboring Shelters 287

288

289

FIGURE 4. Flowchart of Identification and Intensification Processes 290

291

Based on the definition of neighboring shelters, Ycand can be generated through the following 292

actions. 293

294

select the 1st element, y, in Ycand

is y tabu or in the

infeasible list ?

solve the CSLAP(y)

is Ycand fully scanned ?

Step 4- sort Ynew according to

corresponding cost set CMIPEP(Ynew)

Yes

select next element

No

Yes

No

is y feasible ?

solve the MDARP with trip pairs from CSLAP(y), calculate

the cost, CMIPEP(y), add y to the new solution set, Ynew, and

CMIPEP(y) to the corresponding cost set CMIPEP(Ynew)

Yes

Step 2- generate candidate solution set,

Ycand

put y into the

infeasible listNo


For each open shelter ,s S and each closed shelter ( )s Nb s : 295

Drop - setting best ( ) 0s y 296

Add - setting best ( ) 1s y 297

Switch - setting best ( ) 0s y and best ( ) 1s y 298

In each case, if the resulting solution is not tabu, add it into Ycand. 299

Step 3-Identification and Intensification 300

Candidate shelter location decisions, Ycand, generated in the diversification step are not guaranteed to adhere 301

to Constraints (4). CSLAP is therefore re-solved to determine a feasible assignment given each candidate 302

solutions in Ycand. 303

Figure 4 gives details of the identification and intensification processes. In these processes, all 304

feasible generated candidate solutions y in Ycand are identified, the MDARP is solved with the assignment 305

results from CSLAP(y), and the total cost CMIPEP(y) is calculated. 306

Numerical Experiments 307

Experimental design 308

Concepts developed herein are illustrated on a real-world based case study. The case study involves a 309

hypothetical hurricane evacuation in NYC, one of the many hurricane vulnerable areas along the coastline 310

of the U.S. The evacuation scenario is shown in Figure 5 and involves evacuating mobility-impaired 311

individuals from 588 pickup locations (round nodes) within the hurricane evacuation zone (shaded area) to 312

238 potential shelters (triangles) using paratransit vehicles affiliated with 39 paratransit depots (squares). 313

The data on potential shelters, including location and capacity, as well as the hurricane evacuation zones, 314

were obtained from a technical report from the U.S. Army Corps of Engineers (25). 315

The information on paratransit vehicles comes from the NYC data website (26), which includes 316

information from 161 companies, including the depot locations and number of affiliated vehicles. Only 39 317

companies with more than 15 paratransit vehicles are considered. Each vehicle is assumed to have a 318

capacity of seven spaces with each person occupying one space and each wheelchair two spaces. To 319

preserve privacy issues, data on real pickup locations were not available. The 588 pickup locations were 320

thus chosen as the centroids of census tracts in U.S. Census 2010. In reality, the needed pickup information 321


for the mobility-impaired can be obtained information gathered through registration for paratransit services 322

during ordinary circumstances. Random numbers were generated to determine with equal odds whether 323

each location contained one or two evacuees awaiting assitance from each pickup location. Similarly, each 324

passenger was assigned a wheelchair with probability 0.5. The fixed opening cost of each shelter is assumed 325

to be proportional to the shelter’s capacity. For simplicity, the proportionality constant is assumed to be 326

$10/unit of capacity unit. 327

328

FIGURE 5. NYC Hurricane Evacuation 329

Parameters used in the proposed model and solution algorithm are presented in Table 1. The tabu 330

tenure was tuned from 10 to 80 iterations in increments of 5. A setting of 15 iterations was found to have 331

the best performance in terms of convergence and best solution found. Other parameters of the model and 332

algorithm were set as follows: unit cost of travel distance of vehicle Cd = 15.20 $/mile; identical vehicle 333

capacity C = 7 spaces; identical service time σi = 3 minutes; maximum driving distance (excluding deadhead 334

distance) D = 320 miles; maximum passenger onboard time mt = 2 hours; tabu tenure = 15; maximum 335

iterations = 500; and maximum non-improvement iterations = 30. 336

The proposed solution strategies were implemented in Visual C++ 2010 and run on a personal 337

computer with Intel(R) CPU 3.10GHz and 4.0GB RAM. The C++ Concert Technology of CPLEX in the 338

IBM-ILOG CPLEX 12.51 was applied to solve the CSLAP and CSAP problems. 339


RESULTS ANALYSIS 340

TABLE 1. Final Shelter Locations 341

Final Shelter Locations

# Shelter Index Shelter Design Capacity

(1 unit per person and 2 units per wheelchair) Fixed Opening Cost ($)

1 629 220 2,200

2 630 65 650

3 632 195 1,950

4 639 340 3,400

5 640 50 500

6 651 61 610

7 659 63 630

8 673 146 1,460

9 677 82 820

10 698 50 500

11 700 58 580

12 704 74 740

13 727 71 710

14 742 96 960

15 752 94 940

16 765 94 940

17 766 87 870

18 767 78 780

19 769 59 590

20 772 52 520

21 774 50 500

22 775 77 770

23 778 98 980

24 832 62 620

25 839 40 400

26 842 90 900

27 848 481 4,810

28 852 83 830

29 856 52 520

Opening Cost ($) 30,680

The convergence process of the proposed nested tabu search strategy with 5 was also studied. 342

The procedure terminates after 70 iterations; no improvement in solution value was obtained after 30 343

iterations. The resulting solution has a total cost of $56,880 compared with an initial total cost of $59,740 344

obtained in the initial step. 345


29 of 238 shelters are identified for providing service to the mobility impaired at a total fixed 346

opening cost of $30,680. These are listed in Table 1. 347

Fixed and operational costs are further explored in Table 2. Table 2 shows the final cost results for 348

different ƞ values. As expected, total cost decreases with increasing value of ƞ, because a larger ƞ relaxes 349

the constraints. Further, with increasing ƞ, the ratio of fixed to total cost decreases, while the ratio of 350

operational to total cost increases. This is reasonable. Consider that a lower value of ƞ means more shelters 351

should be opened, which infers that shorter distances will need to be traveled. On the other hand, a higher 352

value of ƞ means fewer shelters will be open or farther but cheaper shelters can be opened. This likely infers 353

that longer travel distances will be incurred, and thus higher operational (transportation) costs will exist. 354

TABLE 2. Cost Results with Different ƞ Values 355

Final Results Statistics

Fixed Cost ($) Transportation

Cost ($)

Total Cost

($) Iteration #

CPU Time

(hours:minutes:seconds)

ƞ=2 58,260 23,900 82,160 38 0:58:20

ƞ=3 45,820 23,600 69,420 52 1:06:35

ƞ=5 30,680 26,200 56,880 70 1:26:46

ƞ=7 20,190 27,700 47,890 87 1:57:22

ƞ=10 19,280 27,040 46,320 108 2:33:49

Figure 6 shows the final route set generated with ƞ=5. This solution contains a total of 15 routes 356

that are to be performed by 15 vehicles originating from 11 holding lots (identified in the figure). It may be 357

possible to reduce the number of vehicles serving this population as is noted by the existence of several 358

rather short vehicle routes. This would be achieved, for example, if higher costs for introducing additional 359

vehicles to perform the routes are assumed. Work balance constraints might also be enforced. 360

The number of pickup points visited, number of demand units (people and wheelchairs) transported, 361

number of different shelters providing service to the mobility impaired, and number of trips performed by 362

each vehicle are also indicated in the figure. A trip herein is defined as a circuit that starts from a holding 363

lot or shelter, collects passengers from pickup points en route, and ends at a shelter (or returns to the holding 364

lot). The number of trips performed, therefore, is equivalent to the number of shelters to serve the mobility 365

impaired. 366


367 Figure 6 Final Route Set with ƞ=5 368

TABLE 4. Characteristics of Routes in Final Set of 15 Routes 369

Max.

Onboard

Distance

(miles)

Ave. Onboard

Distance

(miles)

Max. Trip

Distance

(miles)

Ave. Trip

Distance

(miles)

Ave. Trip

Load

9.68 1.95 21.64 3.98 5.63

7.08 1.75 19.25 4.07 5.84

6.84 1.93 22.78 4.74 5.26

8.17 1.82 17.40 4.43 5.60

8.32 1.51 19.84 3.77 5.26

8.95 1.87 14.94 4.17 5.50

6.86 1.67 12.69 3.88 5.91

7.61 2.06 14.40 4.70 5.79

2.79 1.48 5.83 3.15 4.75

8.08 1.48 17.03 3.71 5.93

7.21 2.62 16.02 7.17 5.42

3.71 1.68 7.70 3.61 5.71

8.38 3.15 16.86 5.95 5.33

8.35 3.40 16.90 7.80 5.50

7.70 3.27 15.74 7.38 5.86

Details of maximum and average onboard distance, maximum and average trip distance, as well as 370

average trip load (passengers and wheelchairs) for each route are listed in Table 4. These results show that 371

the longest distance a passenger traveled is 9.68 miles and the longest trip distance is 22.78 miles. The last 372


column indicates that almost all routes have average passenger loads of more than 5.5, inferring high vehicle 373

utilization rates. 374

Conclusions and Extensions 375

This paper addresses an important humanitarian problem and provides a practical, efficient optimization-376

based approach that can be used in large-scale emergency-response planning. A primary outcome of the 377

developed model and solution methodology is the selection of shelters that can, once identified, be 378

adequately prepared to support an evacuation. These developments have additional tactical and operational 379

utilities. Pre-disaster, forecasts specific to a given impending hazard may be available, affecting the 380

potential utility of shelter locations. Thus, in this tactical phase, a subset of the equipped shelters may be 381

open, and the CSLAP model can be applied to re-allocate and route the evacuees using these services for 382

this smaller set of destinations. Similarly, in a no-notice evacuation, this routing and allocation component 383

of the decision problem can be re-solved once knowledge of the viability of the shelter locations and/or 384

roadways are determined. If the original locations are to be operated, the allocation of evacuees to the 385

shelters may be maintained and only the routing decisions may be reassessed in response to information 386

about roadway closures or other conditions affecting the predetermined routes. This latter problem can be 387

addressed by solving an MDARP. 388

This study addresses emergency evacuation with notice as is the case in situations involving 389

hurricanes; however, for no-notice emergency evacuations, an alternative objective function in which the 390

number of pickups that can be completed in a fixed amount of time is maximized or the time required to 391

complete the pickups is minimized given a fixed fleet might be appropriate. Additionally, emergency 392

situations are inherently uncertain. For example, travel times, shelter capacities, demand for assistance and 393

even available resources may be affected by the disaster and a priori knowledge of quantities may be 394

precluded. Thus, it could be beneficial to account for such uncertainties within the proposed model. Also, 395

as some disasters evolve over time, it may become necessary to evacuate the shelters, sending the evacuees 396

even further from the affected area. That is, a two-stage evacuation might be required. One could explore 397

the possibility of applying the proposed model and solution methodologies to each stage. 398

Acknowledgments. The authors are thankful to Sergeant Ira Promisel of New York for sharing his expertise on NYC 399

evacuation concerns. This work was partially funded by the National Science Foundation, United States Department 400

of Transportation through the Mid-Atlantic University Transportation Center, the Maryland Industrial Partnerships 401

(MIPS) and ITCurves. This support is gratefully acknowledged, but implies no endorsement of the findings. 402


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SUPPORTING MOBILITY-IMPAIRED POPULATIONS IN …docs.trb.org/prp/15-2436.pdf · SUPPORTING MOBILITY-IMPAIRED POPULATIONS IN EMERGENCY EVACUATION Lei Feng ... [email protected]