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Pick-up locations and bus allocation for transit-based evacuation planning with demand uncertainty

Authors


SUMMARY

This paper develops a decision-support model for transit-based evacuation planning under demand uncertainty. Demand uncertainty refers to the uncertainty associated with the number of transit-dependent evacuees. A robust optimization model is proposed to determine the optimal pick-up points for evacuees to assemble, and allocate available buses to transport the assembled evacuees between the pick-up locations and different public shelters. The model is formulated as a mixed-integer linear program and is solved via a cutting plane scheme. The numerical example based on the Sioux Falls network demonstrates that the robust plan yields lower total evacuation time and is reliable in serving the realized evacuee demand. Copyright © 2012 John Wiley & Sons, Ltd.

1 INTRODUCTION

Natural and man-made disasters have always been a major concern to humanity, and recent trends illustrate an increase in vulnerability towards these disasters. Aiming to reduce adverse consequences of these disasters, the field of emergency management has evolved substantially over the years [1]. Evacuation is often the most viable response action for protecting affected people. Between 1990 and 2003, there are 230 evacuations that involved 1000 or more evacuees [2].

Evacuation planning has drawn a significant amount of attention over the past several decades. There are various previous studies that focus on different aspects of evacuation planning such as evacuees' behaviors (e.g., [3-5]), traffic control strategies (e.g., [6-8]), sheltering site selection (e.g., [9, 10]), demand management (e.g., [11, 12]), and real-time route guidance (e.g., [13-15]). However, most of these studies are focused on auto-based evacuation. Unfortunately, not all the people in risk areas will own or have access to personal vehicles during evacuation.

Recent experiences with hurricane Katrina and Rita in 2005 have highlighted the need of evacuation planning for transit-dependent peoples (e.g., [16-18]). More specifically, during hurricane Katrina, there was no effective plan to address the evacuation of those who were dependent on public transportation. The plans lacked important details such as how to use the buses, availability of drivers, bus routes, or the pick-up locations (staging areas) to collect evacuees [19]. A report published by the Victoria Transport Policy Institute describes New Orleans's public transportation evacuation as follows: “…bus deployment was ad hoc, implemented by officials during the emergency without a detailed action plan…Katrina's evacuation was relatively effective for people with automobiles but failed transit-dependent residents” [16]. State and local plans during hurricane Rita appeared to be better coordinated compared with hurricane Katrina, in providing public transportation during the evacuation. However, the plan still lacked many details concerning the evacuation of transit-dependent residents with special needs [19].

In the USA, New Orleans is not unique in terms of its high percentage of carless residents (27%). In fact, according to the U.S. Census [20], seven cities, including New York (56%), Washington DC (37%), Baltimore (36%), Philadelphia (36%), Boston (35%), Chicago (29%), and San Francisco (29%), have carless populations higher than New Orleans [17]. The extra risks faced by carless population during an evacuation are well recognized, and thus, transit evacuation planning has been recently viewed as an important component of evacuation planning and management. Despite the subsequent improvements, deficiencies still remain in the state and local emergency response plans to evacuate using public transportation (e.g., [6, 18, 21, 22]).

There are a limited number of studies on modeling of transit-based evacuation. Recently, Margulis et al. [23] developed a deterministic decision-support model for hurricane evacuation. The model determines the optimal bus allocation for specific pick-up points to shelters, while maximizing the number of people evacuated. The model does not consider any uncertainty during evacuation and also assumes that the locations of pick-up points are known. Sayyady [24] presented a mixed-integer linear program to optimize transit routing plans to minimize the total evacuation time during no-notice disasters. The locations of pick-up points and shelters were also assumed to be known, and transit vehicles were only allowed to perform one trip considering the limited evacuation time during a no-notice disaster. Song et al. [25] proposed a similar model to generate bus routing plan to minimize the total evacuation time while considering demand uncertainty. More recently, Abdelgawad et al. [26] developed an approach to optimally operate the available capacity of mass transit to evacuate transit-dependent people during no-notice evacuation of urban areas. An extended vehicle routing problem was proposed to determine the optimal scheduling and routing for the buses in order to minimize the total evacuation time. All these previous studies assume that the pick-up locations for transit evacuation are given and known. The locations of pick-up points are important decisions in transit-based evacuation planning and influence the efficiency of the evaluation plans. However, no analytical model has been proposed to help state and local governments with this decision making. This paper aims to fill this void.

On the other hand, evacuation planning is challenging because of the uncertainties involved. There can be numerous sources of uncertainties during an evacuation, and most of them are not easy to quantify. Uncertainties in the evacuation problem can be related to either the demand side, such as total number of evacuees, their locations, decisions to evacuate or not, and timing (e.g., [27-30]); the supply side, such as reduction in link capacities and limited network connectivity due to impacts of the disaster on infrastructure (e.g., [29-31]); or disaster characteristics, such as intensity and time (e.g., [32]). However, the majority of existing evacuation planning models assumes determinism. There can be significant consequences of ignoring the uncertainties during real-life mass evacuations, and the whole evacuation process could end up in a failure.

A large number of problems in various engineering applications require decisions to be made in the presence of uncertainty. The problem of interest in this paper closely relates to facility location models and logistics planning models for disaster preparedness. There exists considerable literature on facility location and logistic planning under uncertainty (e.g., [33-39]). Recent reviews can be found in, for example, [40].

This paper optimizes the use of public transportation for evacuating transit-dependent residents. More specifically, for evacuation planning, we determine optimal locations where evacuees will assemble, and allocate transit vehicles to assembly locations to pick up the evacuees and transport them to public shelters. We consider that the number of transit-dependent evacuees is uncertain and attempt to design a location–allocation plan that performs reasonably well in any realization of the uncertain demand on one hand, and is not overly conservative on the other hand.

To our best knowledge, this paper is the first study to develop an analytical approach to determine optimal pick-up locations for transit-based evacuation planning. A new solution algorithm is developed to solve the resulting model effectively. For the remainder, Section 2 elaborates our modeling considerations. Section 3 presents the model formulation, solved by a cutting plane scheme developed in Section 4. Results from a numerical example based on the Sioux Falls network are reported in Section 5. Finally, Section 6 concludes the paper.

2 MODELING CONSIDERATIONS

2.1 Basic assumption

Consider a situation when residents are required to evacuate in the event of a disaster. In response, many residents will use their own private vehicles to evacuate, whereas transit-dependent residents will rely on the public transportation system to take them to safe locations. When the evacuation orders are released, it is assumed that transit-dependent evacuees from different residential locations will assemble at designated pick-up locations, and allocated transit vehicles will pick up and transport them to public shelters. In this paper, we propose a model that determines the optimal locations of pick-up points and allocation of a given number of transit vehicles to pick up evacuees in order to minimize the total evacuation time.

In our modeling framework, it is assumed that evacuees will go to their nearest pick-up points. We further assume that evacuees will accumulate at the pick-up locations before the start of the evacuation process, and thus, the proposed model does not take into account delays or times for evacuees to go to the pick-up locations. The number of available transit vehicles and their capacities are assumed to be known in the model. For allocation of available vehicles, we assign each vehicle to a particular pick-up location and assume that the vehicle can make trips to different public shelters, but it only serves its assigned pick-up point without going to another one. In other words, buses are assigned a particular pick-up point to transport evacuees. The buses can make multiple round trips to evacuate them to different shelters but will always come back to the same assigned pick-up point. We consider that the locations of open public shelters and their capacities available for transit-dependent evacuees are known. We further assume that transit vehicles, in a limited number, have negligible impact on overall congestion, and thus, the travel times between pick-up points and public shelters are given and constant. In specifying these travel times, additional delays due to congestion during evacuation are considered.

2.2 Representing demand uncertainty

We consider that the numbers of transit-dependent evacuees at different residential locations are uncertain, and attempt to design an optimal location–allocation plan under against the uncertain demands.

Quite a few techniques have been developed in the literature to accommodate uncertainty in decision making. For example, the number of evacuees can be represented as a random variable, and a stochastic programming approach can be applied subsequently. However, this approach requires the knowledge of the probability distribution of the random number of evacuees, one type of information rarely available and difficult to obtain for evacuation planning.

Robust optimization is a more recent approach to optimize decisions under uncertainty. In this approach, the uncertain variable is not assumed to be random but varies within a bounded set. The approach then optimizes against the worst-case scenario realized in the set. To avoid being overly conservative, the uncertainty set needs to be carefully specified. Such a notion of modeling uncertainty offers computational advantages and has been used in many applications. For an overview on robust optimization, see, for example, [[41, 42]].

Considering that it is practically difficult to obtain the probability distribution of the uncertain demand, we adopt the robust optimization approach, that is, the uncertain demand is assumed to be confined to an uncertainty set, and the pick-up locations and bus allocation are optimized against the worst-case demand scenario realized from the set. More specifically, we assume that the evacuee demand at each residential location can be one of the several possible values whose probabilities of occurrence are unknown. These possible demand values are not difficult to obtain and can be identified, for example, with alternative forecasts of the magnitude and impact area of the disaster. Let math formula denote the possible demand value at the resident location or demand point i ∈ I, with math formula corresponding to the nominal demand value, that is, the most-likely forecast. To hedge against uncertainty, a location–allocation plan could be optimized against a worst-case scenario where maximum possible demand is realized at each demand point. However, this will lead to an overly conservative plan. More specifically, it is rare for all demand points to reach their maximum demand values simultaneously. Designing the location–allocation plan against such a rare case would be overly protective.

With a philosophy that the nature is neither our enemy nor friend, we assume in this paper that demands for at most Γ demand points can deviate from their nominal values simultaneously, whereas the others must remain at their nominal values. We call Γ as degree of pessimism, which is a parameter reflecting the decision makers' attitude toward risk. A larger Γ implies that decision makers are more risk averse (e.g., [43, 44]).

Mathematically, let math formula be a binary variable that indicates whether demand value s is realized at demand point i. Because only one demand value can be realized at each demand point, we have math formula Also, because at most Γ demand points are allowed to deviate from their nominal values and math formula represents that the nominal demand value is realized at demand point i, we have math formula. Thus, the uncertainty set for the transit-dependent evacuation demand at demand points can be characterized as follows:

display math

where d is the demand vector realized for the network and di is the demand realized at demand point i. Hereinafter, the superscript d denotes variables associated with a particular demand vector d ∈ D.

Note that the aforementioned uncertainty set does not necessarily include all the possible realizations of evacuation demand, which would be difficult to obtain because of limited information at the planning stage. The set is a purely mathematical construct to model demand uncertainty in the robust optimization approach, and choice of D affects the efficiency and robustness of the resulting robust plan [41]. Compared with other types of configuration, the aforementioned set is easier to obtain because planning agencies are often able to produce a few different demand forecasts as previously discussed. If a planning agency knows that substantial correlation among demands exists, the correlation information should be utilized to refine the uncertainty set accordingly.

Evacuation planning is multifaceted with a variety of objectives. In this paper, we optimize the location–allocation plan to minimize the maximum total evacuation time realized in the uncertainty set. Other objectives can be used, such as minimizing the longest evacuation time, maximizing the total number of people evacuated, minimizing the total number of vehicles needed. According to a survey by Zhang and He [45], the highest priority for transit evacuation planning is often to minimize the total evacuation time of all the transit-dependent evacuees.

3 MODEL FORMULATION

Let G(N,A) denote the network, where N and A are the sets of nodes and links in the network, respectively. Assume that the transit-dependent demand during evacuation is originated from a subset of the nodes, that is, the set of demand points I ⊂ N. Next, let j ⊂ N denote a set of shelter locations in the network. A binary variable yi represents the pick-up location decision, which is 1 if a pick-up point is located at i ∈ I and 0 otherwise.

We hereinafter assume that buses are the only public transportation available for evacuation. Let B denote the number of buses available for the evacuation and βb denote the capacity of bus b. A binary variable math formula represents the bus allocation decision, which is 1 if bus b is assigned to pick-up point i and 0 otherwise. We use an integer variable Xbij to represent the number of trips bus b needs to make between pick-up point i to shelter j. We further use a binary variable math formula to represent evacuees' choice of pick-up points. The variable is 1 if pick-up point p is nearest to the demand point i, and is thus selected. Otherwise, it is 0. Let Wi denote the distance of demand point i to its nearest pick-up location and qp the demand accumulated at the pick-up location p.

Using the aforementioned notations, a robust transit pick-up location and bus allocation (RTPL) model can be formulated as follows: RTPL

display math

s.t.

display math(1)
display math(2)
display math(3)
display math(4)
display math(5)
display math(6)
display math(7)
display math(8)
display math(9)
display math(10)
display math(11)
display math(12)
display math
display math

where Tij is the round-trip travel time from a pick-up point i to a shelter  j, Kj is the capacity of public shelter j available for transit-dependent evacuees, Cip is the distance between the node pair (i,p) in the network and is an input to the model, ω is the longest walking distance allowed from any demand point to its nearest pick-up point, Tmax is the maximum running time allowed for each bus during evacuation, and M is a sufficiently large number.

The objective function of RTPL is to minimize the total evacuation time. All buses are treated equally in the function. Because buses can be of different sizes, their capacities can be multiplied in the objective function to represent the total passenger evacuation time. Constraint (1) ensures all the evacuees under all demand scenarios realized from the uncertainty set can be transported by buses. Constraint (2) requires the total number of evacuees transported to a shelter to be less than the available capacity at that shelter. Constraint (3) ensures that each bus is assigned to only one particular pick-up point. Constraint (4) implies that the node a bus is assigned to is a pick-up point. Constraint (5) specifies that a bus can make trips to different shelters but it only goes to its assigned pick-up point. Constraints (6) and (7) calculate the walking distance of a demand point to its nearest pick-up point. Constraint (8) assumes that all the evacuees from a particular demand point go to the same nearest pick-up point. Constraint (9) ensures that the node where evacuees assemble is a pick-up point. Constraint (10) is to estimate the total number of evacuees assembling at each pick-up point. Constraint (11) requires that the walking distance from each demand point to its nearest pick-up point should be less than the maximum allowed. Finally, constraint (12) imposes an upper bound on the total running time of each bus during evacuation.

Note that constraints (6)(9) and (11) can be further simplified as the following equivalent set of constraints. We present as aforementioned to facilitate the understanding of the model structure.

display math
display math
display math

where I+ = {i|C(i,k) ≤ ω  ∀ i, k ∈ I }

In summary, the output of the aforementioned model includes the optimal locations of pick-up points (i.e., y), allocation of buses to pick-up points (i.e., μ), the number of trips made by each bus (i.e., X), evacuees' choice of pick-up points (i.e., δ), and the total demand accumulated at each pick-up point (i.e., q).

4 SOLUTION ALGORITHM

As formulated, RTPL is a mixed-integer linear program. However, constrains (1) and (10) are written for each demand scenario in the uncertainty set. The number of demand vectors in the uncertainty set D can be extremely large because of the combinatorial nature of the set. More importantly, it is difficult to enumerate them all. All these render RTPL difficult to solve. In this paper, we develop a cutting plane scheme (e.g., [46]) to solve RTPL. More specifically, we solve a sequence of restricted RTPL problems with respect to a subset of demand vectors math formula that contains a much smaller number of demand vectors and then solve another problem to generate demand vectors one at a time as needed to expand math formula.

The restricted RTPL problem (R-RTPL) can be written as follows: R-RTPL

display math

s.t.

display math(13)
display math(14)
display math
display math

(2)(9), (11), and (12)

The aforementioned restricted problem (R-RTPL) is same as the original RTPL problem, but written for a subset of known demand vectors rather than the whole uncertainty set. Compared with RTPL, the aforementioned formulation contains a less number of constraints, and is thus much easier to solve and can be solved directly. Let math formula solve the R-RTPL globally with the optimal objective value as math formula. Then, the solution math formula also solves the original RTPL problem if the bus allocation plan math formula is sufficient to transport the demands at pick-up points math formula under all demand scenarios in  D. Mathematically, this can be verified by solving the following worst-case demand problem (WCD): WCD

display math

s.t.

display math(15)
display math(16)
display math(17)
display math(18)
display math(19)
display math(20)
display math(21)
display math

where ei denote the excess demand at pick-up point i and M is a sufficiently large number.

The objective of WCD is to find a demand vector d that maximizes the total excess demand, that is, the number of evacuees not transported by the current bus allocation plan. Constraints (15)(17) are to calculate the excess demand at pick-up point i. Constraint (18) calculates the total number of evacuees at each pick-up point. Constraints (19)(21) ensure that the demand vector d belongs to the demand uncertainty set.

If the optimal objective value of WCD is no greater than zero, then the current bus allocation plan is sufficient; that is, math formula is the optimal solution to the original RTPL problem. Otherwise, an improved solution may be obtained by expanding the demand vector set math formula where math formula is the solution to the aforementioned WCD. What follows is the procedure of the cutting plane scheme outlined earlier.

Cutting Plane Scheme

  • Step 0: Choose an initial demand vector d1 ∈ D. Set n = 1 and  math formula.
  • Step 1: Solve the R-RTPL problem with a subset of demand vectors math formula and let math formula denote the resulting optimal solution.
  • Step 2: Solve the WCD problem associated with the current location of pick-up points math formula and the bus allocation math formula. Let dn denote the optimal solution and math formula denote the corresponding objective value.
  • Step 3: If math formula, stop and math formula is an optimal solution to the RTPL problem. Otherwise, set math formula and go to step 1.

Because the number of demand vectors in the uncertainty set D is finite, the aforementioned procedure terminates after a finite number of iterations.

5 NUMERICAL EXAMPLE

This section reports results from solving the proposed model on the Sioux Falls network (e.g., [47]). We implemented the cutting plane scheme in General Algebraic Modeling System (GAMS) [48] where the R-RTPL and WCD problems were solved using CPLEX [49]. The computation experiments were conducted on a 2.4 GHz Dell computer with 2 GB of RAM.

For the Sioux Falls network, the original data for each link consist of two parameters [47], and they are transformed into the link's free-flow travel time and capacity. Mass evacuation is considered for a hypothetical disaster in the region. As shown in Figure 1, we assume that nodes 1–12 and 16–18 are the demand points and public shelters are located at nodes 13, 20, 21, and 22. We further assume that the planning agency has estimated three possible demand values at each demand point, that is, nominal, low, and high. The nominal demand is reported in Table 1, and the other two demand values, low and high, are generated by multiplying the nominal demand by a uniformly distributed number between (0.5, 1) and (1.5, 2). The shelter location and capacity are reported in Table 2. It is further assumed that there are 10 buses available for evacuation and each has a capacity of 30. The maximum allowable running time of each bus is 180 minutes, and the maximum allowable walking time from a demand point to a pick-up point is 5 minutes.

Figure 1.

Sioux Falls network.

Table 1. Demand values at each demand point.
Demand pointsNominal demandLow demandHigh demand
Node 160.0035.15109.20
Node 242.0038.7066.35
Node 340.0031.0065.00
Node 446.0029.9284.38
Node 538.0024.5565.28
Node 650.0030.6084.00
Node 734.0022.9556.98
Node 844.0040.8468.90
Node 944.0023.4869.30
Node 1052.0039.0093.32
Node 1148.0047.9691.95
Node 1240.0031.5864.62
Node 1636.0035.8465.98
Node 1726.0022.9049.10
Node 1830.0016.9649.55
Total demand630.00471.431083.91
Table 2. Available capacity of public shelters.
Public sheltersAvailable capacity
Node 13240.00
Node 20333.00
Node 21360.00
Node 22300.00
Total capacity1230.00

Tables 3 and 4 present the optimal locations of pick-up points and allocation of bus trips for the nominal, robust, and worst-case plans. The robust plan is obtained using degree of pessimism with Γ = 3, whereas the nominal plan is obtained with Γ = 0 (deterministic demand with nominal value). The worst-case plan is the optimal plan against deterministic high demand at each demand point. It is equivalent, in this particular example, to solving the proposed model with degree of pessimism Γ as 15. As can be seen from the computational time reported in Table 3, the proposed solution algorithm was very efficient for solving the Sioux Falls network.

Table 3. Pick-up locations under nominal, robust and worst-case plans.
 Nominal planRobust plan (degree of pessimism = 3)Worst-case plan
 Node 3Node 3Node 3
 Node 6Node 6Node 6
 Node 10Node 10Node 10
 Node 18Node 17Node 12
  Node 18Node 17
   Node 18
Computational CPU time (second)1.568.762.40
Total evacuation time (minute)612.001128.801234.20
Probability of meeting demand with allocated bus trips2.13%97.94%100.00%
Table 4. Bus allocation under nominal, robust, and worst-case plans.
Nominal PlanRobust PlanWorst-Case Plan
Bus #TripsPick-up pointShelterBus #TripsPick-up pointShelterBus #TripsPick-up pointShelter
15Node 18Node 2012Node 6Node 2015Node 10Node 22
27Node 3Node 1324Node 10Node 2223Node 6Node 21
35Node 10Node 2221Node 10Node 2134Node 18Node 20
42Node 6Node 2034Node 10Node 2243Node 6Node 20
54Node 6Node 2047Node 3Node 1351Node 3Node 13
    54Node 6Node 2051Node 3Node 21
    61Node 17Node 2064Node 6Node 20
    62Node 17Node 2277Node 3Node 13
    74Node 18Node 2084Node 17Node 22
    83Node 6Node 2193Node 12Node 21
    91Node 3Node 13103Node 10Node 21
    92Node 3Node 21101Node 10Node 22
    101Node 17Node 21    

In order to evaluate the performance of these three different plans, we randomly generate 10 000 demand vectors by assuming that, for each demand point, there is an equal probability of occurrence for the nominal, low, and high demand values. For each sampled demand vector, we then evaluate whether the bus allocation is sufficient to transport all the evacuees at pick-up points. The probability of meeting the demand is calculated accordingly and reported in Table 3.

From Table 3, we can observe the difference in total evacuation time and the reliability of the nominal, robust, and worst-case plans. The worst-case plan is 100% reliable in meeting the demand realized under uncertainty, but the estimated total evacuation time (and the required bus running time and the associated resource) is very high. On the other hand, the robust plan requires less total evacuation time, saving approximately 9% of the evacuation time (and the associated resource) while achieving nearly the same level of reliability in meeting the realized demand. The nominal plan yields much saving in total evacuation time but meets the evacuation demand with a probability of 2%. Such an exceptionally low reliability underscores the importance of accommodating the demand uncertainty in evacuation planning.

Table 5 compares the nominal and worst-case plans against the robust plans with varying the degree of pessimism. In general, the probability of a robust plan in meeting the demand increases with the increase in the degree of pessimism Γ, but the total evacuation time estimate also increases. As previously mentioned, Γ controls the degree of conservatism in the solution. An appropriate value of degree of pessimism can be chosen depending on the decision makers' attitude towards risk.

Table 5. Reliability of plans with varying degree of pessimism.
Degree of pessimism (Γ)Total evacuation time (minute)Probability of meeting travel demand (%)
0 (nominal case)612.002.13
1782.0034.88
2965.6076.42
31128.8097.94
41181.0099.59
51227.40100.00
61227.40100.00
71227.40100.00
81234.20100.00
15 (worst case)1234.20100.00

6 CONCLUDING REMARKS

In this paper, we proposed a robust optimization approach to determine the optimal locations of transit pick-up points and trip allocations for buses in transit-based evacuation planning under demand uncertainty. Demand uncertainty refers to the uncertainty in the total number of transit-dependent people. The proposed model was formulated as a mixed-integer linear program and solved by a cutting plane scheme. Numerical results from the Sioux Falls network illustrate that the proposed robust model produces a less conservative but reliable plan, compared with the one optimized against the worst-case demand.

This research is useful to planners, transit providers, and emergency management officials for an effective and reliable transit evacuation planning. For emergency management agencies, how to efficiently utilize public transportation during evacuation while leaving no citizens behind is a key question. This paper offers an approach to provide answers to some of the critical issues for transit evacuation.

As part of future studies, we plan to implement the proposed model and solution algorithm for a large-scale realistic network to demonstrate the applicability of model to the real-world evacuation situations. In addition, although only demand uncertainty is considered in this paper, the proposed modeling framework is general enough to accommodate other types of uncertainty, for example, uncertain transit travel times between pick-up points and shelters. To this aim, another uncertainty set can be defined to represent the uncertainty associated with transit times, and then, an evacuation plan can then be designed to minimize the maximum total evacuation time. Our future work will also extend the current model to be more flexible by relaxing some of the assumptions made in this paper, for example, allowing transit vehicles to serve multiple pick-up points, considering more realistic behaviors of evacuees in selecting pick-up points and also capturing the time-dependent nature of evacuation.

7 LIST OF SYMBOLS AND ABBREVIATIONS

7.1 Symbols

i

index for demand points, i I

j

index for shelter locations, j J

N

set of nodes

A

set of links

B

total number of available buses

b

index for the buses

βb

capacity of bus b

Tij

round trip travel time from pick-up location i to shelter j

Cip

walking distance between the node pair (i,p) in the network

Tmax

maximum running time allowed for each bus during evacuation

ω

longest walking distance allowed from any demand point to its nearest pick-up location

Wi

distance of demand point i to its nearest pick-up locations

Kj

capacity of shelter j

M

sufficiently large number

Γ

degree of pessimism

math formula

demand for demand point i under scenario s

Si

number of demand scenarios for demand point i

di

demand realized for demand point i

qp

number of evacuees accumulated at the pick-up location p

math formula

binary variable indicating which scenario s is realized for demand point i

yYi

binary variable indicating pick-up location decision

math formula

binary variable indicating bus allocation decision

Xbij

integer variable representing number of trips bus b makes between pick-up location i and shelter j

math formula

binary variable indicating evacuees' choice of pick-up location

ei

excess demand at pick-up point i

7.2 Abbreviations

GAMS

General Algebraic Modeling System

RTPL

Robust Transit Pick-up Location problem

R-RTPL

Restricted RTPL problem

USDOT

U.S. Department of Transportation

USDHS

U.S. Department of Homeland Security

WCD

Worst Case Demand problem

ACKNOWLEDGEMENTS

The authors would like to thank three anonymous reviewers for their helpful comments. This research was partly funded by Center for Multimodal Solutions for Congestion Mitigation, University of Florida, and National Natural Science Foundation of China (71228101).

Ancillary