Robust stochastic vehicle routing and scheduling for bushfire emergency evacuation: An Australian case study

Transportation Research Part A 104 (2017) 32–49

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Robust stochastic vehicle routing and scheduling for bushfire emergency evacuation: An Australian case study Shahrooz Shahparvari ⇑, Babak Abbasi School of Business IT & Logistics, College of Business, RMIT University, Melbourne, Australia

a r t i c l e

i n f o

Article history: Received 11 March 2016 Received in revised form 30 January 2017 Accepted 28 April 2017 Available online 2 August 2017 Keywords: Evacuation decision support system Stay or go policy Stochastic modeling Bushfire Emergency situation Vehicle routing problem Uncertainty Black Saturday

a b s t r a c t This study proposes a stochastic modeling approach as an evacuation decision support system to determine the required vehicles, scheduling and routes under uncertainties in evacuee population, time windows and bushfire propagation. The proposed model also considers road availability and disruptions. A greedy solution method is developed to cope with the complex nature of vehicle routing problem. Furthermore, the effectiveness of the proposed solution is evaluated by comparison with a designed genetic algorithm on sets of various numerical examples. The model is then applied on the real case study of the 2009 Black Saturday bushfires in Victoria, Australia. Several plausible evacuation scenarios are generated, utilizing the historical data of Black Saturday. The results are analyzed using the frequency approach to determine the optimal evacuation plan. The results show that it would have been possible to evacuate the late evacuees on Black Saturday, even within hard time windows and a maximum population. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction A wildfire, known as a bushfire in Australia, is a freely burning, uncontrolled and unplanned fire often occurring in regional areas, but which can also consume houses or agricultural resources (CFA Definitions, 2012). Uncontrolled bushfires can lead to loss of human lives as well as widespread damage to property, buildings, livestock, crops and road networks. Bushfires tend to be more prevalent in countries with harsh weather conditions like Australia, the United States, Canada and Russia. Factors such as global warming and climate change will potentially create more conducive conditions for bushfires in the future. The spread of a bushfire is influenced by the strength and direction of wind, and the surrounding landscape. The propagation rate can double for example with only a slight increase in wind speed and with a ten degree rise in land slope. Temperatures during a bushfire can exceed 800 °C, with flames potentially reaching heights of 30 m or more (Whittaker et al., 2009). Bushfires in Australia caused A$2.5 billion worth of damage between 1967 and 1999. 223 bushfire deaths were recorded during the same period, accounting to 39% of all natural disaster-related deaths in Australia. A$4.4 billion worth of damage was caused by the 2009 Black Saturday In addition, 173 people lost their lives and a further A $645 million was incurred by survivors as injury-related expenses (Victorian Bushfires Royal Commission Report, 2009). Evacuation is defined as the procedure for moving people from a target location (an arranged meeting point) to a nearby safer location (Southworth, 1991). This can be a complex process with multiple considerations. ‘‘Evacuation strategies” are the most common method for controlling natural disasters and attempting to ensure the safety of affected people

⇑ Corresponding author. E-mail address: [email protected] (S. Shahparvari). http://dx.doi.org/10.1016/j.tra.2017.04.036 0965-8564/Ó 2017 Elsevier Ltd. All rights reserved.

S. Shahparvari, B. Abbasi / Transportation Research Part A 104 (2017) 32–49

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(Southworth, 1991). Bushfire evacuations can be mandatory, suggested, or optional. Most nations, including Australia, only require mandatory evacuation in extreme bushfire circumstances (Arnol, 2007). Instead, affected people are usually left to decide for themselves whether they want to evacuate or stay to try and protect their properties. Th decision is therefore a crucial in terms of community safety in a bushfire scenario. Affected people may decide to evacuate, shelter-in-home or take shelter-in-refuge nearby (Lindell and Perry, 1992). In terms of protecting human life, early evacuation is obviously the safest option. Early evacuation is particularly important for the disabled, the elderly, and people without transport. Speed of evacuation is a critical consideration in evacuation crisis management. High passenger-capacity public vehicles such as buses should therefore be available at assembly points as part of an evacuation strategy (Vuchic, 2005). However, the desire among affected people to stay and protect properties in a bushfire scenario is often very strong. As a bushfire’s intensity increases, so too does the potential for factors such as road network blockages, radiant heat and toxic smoke to impact evacuation efforts. Time is always a crucial consideration. Evacuation strategies should attempt to ensure the maximum potential number of evacuees using a finite number of vehicles within the available clearance times (time windows). They should also consider landscape patterns and a range of bushfire scenarios contingencies such as sudden changes in wind direction. Such factors potentially affect evacuation cost, transport network links and associated travel times. There is also the potential that these factors may disrupt the evacuation operation to the extent that human lives may be lost). Contingencies must be considered in emergency evacuation planning because multiple factors are uncertain in an emergency evacuation situation. In bushfire scenarios, insufficient historical data and the multiple potential patterns of propagation often make it difficult to frame the problem parameters as exact values. In addition, time windows for affected locations tend to be difficult to deterministically model. Accordingly, interval ranges are a more appropriate means of representing time windows for a bushfire evacuation scenario. To minimize the potential loss of human life, an integrated optimization model evacuation routing approach needs to be developed that can consolidate the dynamic components of an emergency bushfire evacuation scenarios. This paper presents a stochastic logistics mathematical modeling approach to enable effective planning for short-notice emergency evacuation operations. The model enables planning to be conducted for transporting the maximum number of late evacuees to safe shelters via using available resources i.e. vehicles and shelters, within a short time window. The model’s objective function is to maximize the total number of evacuees within very tight pre-determined time windows, assuming road block conditions. The main contributions of this study is to consider uncertainty (stochastic) and robustness of evacuees demands and time windows during an emergency situation. In order to address the inherent uncertainty in an emergency evacuation scenario, the evacuation plans for 1000 plausible scenarios from the 2009 Black Saturday bushfire are analyzed and solved by the proposed greedy algorithm. The remainder of this paper is organized as follows: Section 2 surveys existing research literature on emergency evacuation and VRP problems; Section 3 introduces the proposed mathematical formulation; Section 4 introduces the 2009 Black Saturday case study; Section 5 provides computational results using the 2009 Black Saturday bushfire data and plausible evacuations scenarios; and in Section 6 conclusions will be drawn.

2. Literature review The vehicle routing problem (VRP) tries to determine the optimal routes for a vehicle fleet to service sets of customers (Dantzig and Ramser, 1959). The majority of existing research literature has investigated VRP variations, providing insights such as introducing exact solution approaches and greedy solution algorithms (Eksioglu et al., 2009). In this paper, several variants of classic VRP are utilized and consolidated: (1) Capacitated VRP (CVRP), where vehicles have a limited, pre-defined capacity. (2) Multi-destination VRP (MDVRP), where vehicles can visit a destination more than once. (3) VRP time-window (VRPTW), where a time window limitation is introduced within which vehicles must service customers. (4) Dynamic VRP (DVRP), where problem elements such as the availability of a road network may vary over time. (5) Stochastic VRP (SVRP), where random uncertainties are involved. Stochastic programming (SP) is an appropriate planning tool for emergency evacuation because it can handle uncertainty via probabilistic disaster and outcome scenarios. SP has been successful applied to many disaster management problems (Barbaroso and gcaron, 2004; Beraldi et al., 2004; Chang et al., 2007; Lamiri et al., 2008). Stochastic vehicle-routing problems (SVRP) are divided into two types: ‘wait and see’ and ‘here and now’ (Dror et al., 1989). In ‘wait and see’ problems, routes are determined after demand is known. They tend to become deterministic vehicle-routing problems (DTVRP). In ‘here and now’ problems, routes are set based on expected demand, and this type of problem is the focus of our study, using a chance-constrained model (Stewart and Golden, as cited in Dror et al. (1989)). Under the chance-constrained model, the selection of routes is strictly based on probable demand and time windows. ‘Here and now’ problems are also referred to as probabilistic optimization (Joshi, 1997) or priori optimization (Bertsimas et al., 1990). ‘Here and now’ SVRP problems can be further embellished as stochastic location-routing related problems (SLRP). Integrating other VRP variants to SVRP increases the complexity of the problem. Satisfactory solution procedures for these problems are still evolving as a result (Bertsimas et al., 1995). It is far easier to solve the deterministic version of a problem than every instance of its probabilistic formulation. From other perspective, few studies have investigated the use of operations research techniques in natural disaster evacuation routing problems (Cova and Johnson, 2003). There is also limited published research on the applications of VRP mod-

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els in bushfire events. Most studies have focused on using public transport such as buses for the evacuation of those without a car, including the elderly and disabled. In one toxic smoke case study a model optimizing travel time based on a static routing strategy was applied, without noting the number of evacuees at the assembly points (Margulis et al., 2006). The impact of bushfire on the routing strategies of utilizing a fleet of buses was the focus of another study (Mastrogiannidou et al., 2009). In this research, buses were assigned to evacuee assembly points via a greedy solution algorithm. The model was applied to small transit-based emergency evacuation case study. The focus in another study was the development of a nonlinear formulation to measure the effectiveness of transit-based evacuation in a highly populated area (Chen and Chou, 2009). Various criterion such as average traffic density, start time, congestion time, time windows and total process time were evaluated. This study found that the total time required for the evacuation process could be improved significantly if more evacuees used the transit system. A scenario-based approach to disaster management has both advantages and disadvantages. One realistic advantage is the ability to allow parameters to be statistically dependent allowing specific future events to be considered (Snyder, 2006). A disadvantage is that the number of scenarios necessarily must be limited. Snyder (2006) concludes that ‘‘the scenario based approaches generally result in more tractable models.” Barbaroso and gcaron (2004) presented optimal transportation plans for an earthquake response, utilizing a scenario-based two-stage SP model. They defined two response phase stages; the first is the early response phase in various earthquake scenarios, and the second covers later responses under various detailed earthquake impact scenarios. The location and assignment of emergency vehicles has been variously studied (Beraldi and Bruni, 2009; Beraldi et al., 2004). Beraldi et al. (2004) used a mixed integer formulation with probabilistic constraints, assuming independent random vehicle requests. Beraldi and Bruni (2009) made no such assumption in their SP model, allowing for spatial dependence of vehicle requests. Chang et al. (2007) developed a scenario-based two-stage stochastic programming model for locating and distributing rescue resources during a flood emergency. Alsalloum and Rand (2006) and Rajagopalan et al. (2008) studied emergency management in terms of the minimum number of rescue vehicles and hospital locations. Alsalloum and Rand (2006) used goal programming to minimize the number of vehicles required to cover expected demand. Rajagopalan et al. (2008) offered a model to achieve the dynamic redeployment of ambulances based on fluctuating demand over time. Özdamar et al. (2004) provided a vehicle routing problem formulation for dispatching commodities such as medical supplies and personnel to distribution centers in emergency logistics planning situations. In their model, they used multi-period future demand forecasts instead of probabilistic demand. Yi and Özdamar (2007) developed a location routing and personnel allocation network-flow model to maximize the support coverage area during evacuation operations. Iakovou et al. (1997) researched a location problem with equipment capacity decisions in an emergency oil spill cleanup. Table 1 summarizes the public transit emergency evacuation optimization studies. The development of VRP to include high levels of uncertainty in input parameters should be explored to provide a robust solution to complex multi-criteria emergency evacuation problems. As it is shown in Table 1, there are only a few studies that have investigated the application of VRP on specific disasters whereby various operations research (OR) techniques to applied to solve evacuation-routing problems as an integrated system. The VRP were applied on predictable (e.g., hurricane, bushfire; see (Margulis et al., 2006; Shahparvari et al., 2015)) and unpredictable disasters (e.g., earthquake, terrorist attack; Sayyady (2007), Pourrahmani et al. (2015)). Furthermore, most studies have assumed a single time window constraint for the entire network, which may not necessarily mimic the real case phenomenon. There are however a few studies that have considered different time windows for each of the road segments to model the network risk in a bushfire situation in Australia (Shahparvari, 2016a, 2016b). Uncertainty in input parameters representing the situated context of a disaster, however, is often overlooked in evacuation studies (Kulshrestha et al. (2014), He et al. (2009)), The incorporation of uncertainty associated with bushfire propagation, available time-windows within which evacuees need to be transported to safe shelters and travel times should be considered as a vital constraint in emergency evacuation modeling. The deterministic approach to public transit evacuation modeling is relatively less effective in solving the dynamic problem such as the case with bushfire emergency evacuation. Despite the existence of several studies that have focused on developing decision support system for emergency situations (Amailef and Lu, 2013; Yoon et al., 2008), the use of public transit systems for disaster evacuation during, public evacuation planning for a wildfire disaster scenario is still broadly missing from the majority of the emergency evacuation research literature. In the most closely related study, Shahparvari et al. (2015, 2017a, 2017b) addressed the short-notice bushfire evacuation problem by aiming to maximize the number of evacuees while simultaneously minimizing resource allocation. Shahparvari et al., 2016a formulated the problem as a multi-objective integer problem within short time windows. The epsilon-constraint method was utilized to solve the model. They applied their model to a set of wildfire propagation scenarios. However, their model did not consider multi-modal road connections or the road disruption risk in generating emergency evacuation plans. Shahparvari et al. (2017a, 2017b) have developed novel model addressing the multi-modal road connections and road disruption risk in emergency evacuation. However, their model still did take into account the robustness and stochastic uncertainty in the evacuation demand. The specific focus of this paper’s research therefore is on robust stochastic capacitated resources evacuation during short-notice bushfire that has not been well studied. Given the uncertainty of bushfire gradual propagation, the potential impact of this natural disaster on infrastructures, and the significance of clearance time in an emergency response scenario, these factors need to be incorporated into an effective decision support tool. Although travel time has been considered as an uncertain variable in many evacuation studies,

Table 1 Public transit (bus) emergency evacuation optimization literature review (MILP – Mix integer linear programming, NLP – Non-linear programming, ILP, PMILP – Possibilistic MILP, Integer linear programming, ⁄ – multi-objective). Authorns

Modeling MILP

Perkins et al. (2001) Margulis et al. (2006)

Yi and Özdamar (2007) Mastrogiannidou et al. (2009) He et al. (2009) Rui et al. (2009) Chen and Chou (2009) Sayyady and Eksioglu (2010) Chan (2010) Zhang et al. (2010) Abdelgawad et al. (2011) Bish (2011) Chen et al. (2011) Kaisar et al. (2012) Goerigk et al. (2013) An et al. (2013) Kulshrestha et al. (2014) Goerigk and Grün (2014) Zheng (2014) Zhang and Chang (2014) Shahparvari et al. (2015) Pourrahmani et al. (2015) Shahparvari et al. (2016) Qazi et al. (2016) Shahparvari et al. (2016)

NLP

Mode Other

Deterministic

Objective\s Stochastic

Time (distance)

People

Costs

Solution approach Resources

Exact

Heuristics

MetaHs

Disaster type Simulation

General

Huricane

Bushfire

Comments Earthquake

Others

Buses only carry out one single trip Small size case study implemented No real case stufy was implemented Fix rate of evacuees transferred assumed No time window was considered No large-scale evacuation scenario used Small case study multi-stage problem

Two routing paths is considered Two step location-routing models is used Small case by two stage modeling Disruption risks was not considered Small size numerical example is appleid Small size case study was implemented The network was not calibrated Small size numerical example was used Routing plans is not well considered Small size case study is implemented No probabilities for scenarios considered Small numerical example is appleid Small case in two separated modules Small case study is used

S. Shahparvari, B. Abbasi / Transportation Research Part A 104 (2017) 32–49

Sbayti and Mahmassani (2006) Pages et al. (2006)

ILP

Buses only carry out one single trip Fetch evacuation mode is assumed Small case study is implemented Possibilistic approach is implemented

35

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other uncertain variables such as bushfire propagation rates, time windows, shelter capacities and evacuee populations have not received sufficient consideration. 3. Mathematical formulation 3.1. Model description The based multi-destination capacitated dynamic vehicle routing problem with time window (MDCDVRP-TW) developed by Shahparvari (2016c) has the following characteristics: – The distribution of late evacuees population is assumed to be randomly generated from Poisson distribution with mean of actual recorded data. – The distribution of time windows is assumed to be randomly generated by utilization of uniform distribution within a range of ±25% of the actual recorded time windows for each town. – The number and locations the capacitated shelters are known. – Each route, township and shelter can be served by several vehicles. – The vehicles have limited passenger capacity. Considering the above characteristics, the proposed Multi-Destination Capacitated Dynamic Vehicle Routing Problem with Time Window (MDCDVRP-TW) model is formulated as below. This objective function aims to maximize total number of transferred evacuees from the townships with lowest risk.

Max f ¼

XXX X

X mn ijr Vð1  lijr Þ

ð1Þ

n2N m2I i2I ðj;rÞ2Sm i:

This objective is subject to the following constraints: k XX X

e X mn ijr V < C j þ V

8j 2 J

ð2Þ

e X mn ijr V < P i þ V

8i 2 I

ð3Þ

n2N m¼1ðj;rÞ2Sm :j

k XX X n2N m¼1ði;rÞ2Sm

X

i:

amn ¼1 i

8n 2 N;

m2I

ð4Þ

8n 2 N;

m 2 f1g

ð5Þ

8n 2 N;

m2I

ð6Þ

8n 2 N;

m 2 fjIjg

ð7Þ

8n 2 N;

m 2 f1; . . . ; jIj  1g

ð8Þ

8n 2 N;

i 2 ðI [ JÞ;

ð9Þ

8n 2 N;

k2I

i2I

X

amn ¼0 j

j2J

X

bmn ¼1 i

i2ðI[JÞ

X mn bi ¼ 0 i2I

mþ1;n bmn i2ðI[JÞ ¼ ai2I X X mn mn bmn X mn X ijr i2ðI[JÞ þ ijr ¼ ai2I þ ði;rÞ2Sm :i k X X

X mn ijr t ijr þ

m¼1ði;j;rÞ2Sm

X

X mn ijr þ

r2Sm

X

ð10Þ

mn X mn jir 6 Z ijr  BM

X mn ijr þ

X

8i 2 I;

j 2 J;

m 2 I;

n2N

ð11Þ

mn X mn jir P Z ijr

r2S

8i 2 I;

j 2 J;

m 2 I;

n2N

ð12Þ

8ði; j; rÞ 2 Sm ;

m 2 I;

n2N

m

6 Yj þ

Z mn jir

P2

i2I

X

ði;j;j

e X mn jir t jir 6 T k

m¼1ði;j;rÞ2Sm

X mn Pijj0r  1

0

k X X

r2Sm

m

r2S Z mn ijr Z mn ijr

X

m2I

ði;rÞ2Sm i:

Pmn ijj0 r P

ðj;rÞ2Sm :j

8ði; j; j0 ; rÞ 2 Sm ; 8j;

X

;rÞ2Sm :j

X

Pmn ijj0r

mn Z mn ijr P ai

Yj  1

0

j 2 J;

8m 2 I;

0

j–j; 0

ð13Þ

n2N m 2 I;

ð14Þ n2N

ð15Þ

j–j;

n2N

ð16Þ

m 2 I;

n2N

ð17Þ

j

8i 2 I;

S. Shahparvari, B. Abbasi / Transportation Research Part A 104 (2017) 32–49

37

þ X mn ijr 2 Z

8ði; j; rÞ 2 Sm ;

a

ð19Þ

2 f0; 1g

8i 2 I; m 2 I; n 2 N 8i 2 ðI [ JÞ; m 2 I; n 2 N 8i 2 ðI [ JÞ; j 2 J; n 2 N; m 2 I

2 f0; 1g

8ði; j; j0 ; rÞ 2 Sm ;

ð22Þ

mn i mn bi Z mn ijr Pmn ijj0 r

2 f0; 1g 2 f0; 1g

n 2 N;

m2I

0

j–j;

n 2 N;

ð18Þ ð20Þ ð21Þ

m2I

Constraint (2) is the shelter absorption capacity. Constraint (3) is the rescue vehicle passenger capacity. Constraint (4) requires all vehicles to start from the assembly points in each time window. Constraints (5)–(7) restrict the start and finish point of each vehicle at each time window. Rescue vehicles may start in any town and they can proceed from a shelter to their next assembly point in the next time window (m > 1). Vehicles may finish service in either a shelter or an assembly point in each time window except for the last service where they should finish in one of the assigned shelters (m = {I}). Constraint (8) guarantees a continuous flow with vehicles starting their next service from their previous finish point. Constraint (9) ensures flow conservation of the network within each time window. The quantity of vehicles that arrive to a node must equal the quantity that departs. Constraint (10) is time windows and guarantees that total vehicle travel time does not exceed the total available evacuation time window. Constraints (11)–(17) are sub-tour elimination constraints. Constraints (18)–(22) define the domain of variables. 3.1.1. Robust model with uncertain right-hand side Robust optimization (RO) is one of the most common approaches to solve linear optimization problems with uncertain data. The RO approach focuses on data uncertainty related with hard constraints: those that are guaranteed for data within an appropriate prescribed set. Soyster (1973) pioneered this approach by proposing a linear optimization model to construct a feasible solution for all data that belongs to a convex set. Recently, robust optimization has witnessed a significant growth (Ben-Tal et al., 2005; Ben-Tal and Nemirovski, 2002; Bertsimas and Sim, 2003, 2004). For a summary of the state of art in RO, please refer to Ben-Tal et al. (2009) and references therein. RO has been proposed to apply in network and transportation systems (Atamtürk and Zhang, 2007; Erera et al., 2009; Mudchanatongsuk et al., 2008; Ordóñez and Zhao, 2007). Bertsimas and Sim (2004) proposed a solution approach for a linear mathematical model with an uncertain coefficient matrix at the left-hand side. Their approach provides a robust solution whose level of conservatism can be flexibly adjusted in terms of probabilistic bounds for constraint violation. Since the proposed model of this study has a deterministic zero-one coefficient matrix like most of the logistics models, their approach is not capable to provide its robust counterpart. Najafi et al. (2013) customized the Bertsimas and Sim (2004) approach for a robust optimization of the linear programming models with uncertain right-hand side. They considered a LP model with uncertain right-hand side as below:

Min z ¼ s:t:

X

X c j xj j

aij xj 6 b~i 8i;

xj P 0

8j;

ð23Þ

8i 8j

In which bi represents uncertain parameter that takes a value according to a symmetric distribution with mean equals to ~i . aij ; cj are deterministic parameters. They showed that the robust counterpart of the nominal value (bi ) in the interval ½bi  b model can be re-written as follows:

Min z ¼

X X cj xj or Max w ¼  c j xj j

j

X X s:t: aij xj þ ai Ci þ bis 6 bi

8i

s2si

j

ð24Þ

ai þ bis P b~is 8i; s 2 si xj P 0 8 j

bis P 0 8i; s 2 si

ai P 0 8i

In which Ci adjusts the level of conservatism of the solution acquired by the proposed method. Ci is not necessarily integer and takes a value in the interval ½0; si ; si being the set of uncertain parameters in the ith constraint). ai and bi are dual variables of the linear equivalent of protective function of constraint. In this research we have applied the mentioned approach to deal with the uncertainty related to capacity of shelters. Therefore, constraint 2 can be rewritten as following two constraints:

XX X

m2I n2N ði;rÞ2Sm :j

X nm ijr V þ

X s B j þ Aj C j < C j þ V

X s B j þ Aj Cj > C j þ V s2si

8j 2 J;

s2si

8j 2 J;

s 2 si

8j 2 J;

s 2 si s 2 si

ð25Þ

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4. Solution approach The proposed model is NP-hard. In the simple case of considering just one bushfire scenario, one shelter, and not including road blocks and capacities, it reduces to the NP-hard vehicle routing problem with time windows (VRP-TW). When several shelters are to be served, the multi-destination capacitated dynamic vehicle routing problem with time window (MDCDVRP-TW) reassembles to run several VRP-TWs (one per scenario). These are composed by prioritizing the time windows. 4.1. Greedy algorithm method Because the Capacitated Dynamic EVRP-TW model is a form of an NP-hard problem, solutions of large scale experiments are intricate. To assure the applicability of the developed model in realistic case studies, this section develops a greedy solution algorithm method to solve the model as follows: Algorithm 1. Greedy solution algorithm (Shahparvari et al., 2017b).

n ¼ 1; P While i2I Di > 0, Run the model P P P Pi P i  m2I i2I ðj;rÞ2Sm X mn ijr V i: P P P Cj C j  m2I i2I ðj;rÞ2Sm X mn ijr V :j n¼nþ1 End while ‘‘The number of required vehicles is” = n;

As Algorithm 1 demonstrates, a constructive greedy algorithm is implemented to generate routing plans by individually assigning the required number of vehicles. In other words, at the first step the algorithm assigns only one vehicle ðn ¼ 1Þ to be served. The model prioritizes the assembly points based on pre-defined time windows and travel times and generates the first vehicle routing plan to maximize the total number of transferred people. Then after considering the total number of transferred people by vehicle number one, the algorithm updates the remaining population values and shelter capacities in the model before assigning the next vehicle. In the same way, the second vehicle ðn ¼ 2Þ is assigned to travel among assembly points and shelters and transfer the maximum number of remaining population within the available time window. The proposed greedy algorithm continues the assignment of vehicles until all of the population has been evacuated. Therefore, the algorithm assigns vehicles and continuously amends the parameters until the optimal number of required vehicles is determined. In order to validate the effectiveness of the proposed greedy algorithm, sets of various randomly generated deterministic problems with different sizes are considered. The results are compared to results of application of a designed genetic algorithm as another solution approach. The performance of the greedy algorithm is validated by the application of a genetic algorithm as an alternative solution approach (Shahparvari et al., 2017b). Taking into account their results, we can apply the greedy technique on a real sized case study with stochastic parameters.

5. Case study – Black Saturday Murrindindi Mill fire Murrindindi is a small regional town situated approximately 100 km north-east of Melbourne, the capital city of the State of Victoria in Australia. On 7 February 2009, Black Saturday, a series of severe bushfires engulfed the region. The first fire commenced at approximately 3.00 pm in Wilhelmina Falls Road. The bushfire subsequently propagated rapidly. Within an hour and a half it had spread more than 50 km to reach the town of Narbethong. A change in wind direction two hours later caused the fire to sweep through the neighboring towns of Marysville, Buxton and Taggerty. Ultimately, the bushfire spread through approximately 168,000 hectares of land causing massive damage. Melbourne’s water reservoirs were disrupted. Final statistics reveal that the Black Saturday Murrindindi bushfire resulted in 40 deaths, 71 casualties, and 500 household dislocations, mainly in Narbethong, Marysville and Buxton (see Fig. 1). The focus of this study is the six towns most affected by Black Saturday. Five safer locations adjacent to these towns are considered as capacitated shelter for emergency late evacuees. According to Australian Bureau of Statistics data, the total population of the six towns is approximately 2160. Based on Black Saturday bushfire research data compiled by the Country Fire Authority (CFA), 51% of residents are considered as the emergency late evacuee population (Teague et al. (2009)) (see Table 2). Table 2 also outlines the finite evacuee capacity of each destination shelter.

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Fig. 1. An overlay of the transportation network on the disruption risk map of the 2009 Black Saturday bushfire-affected area in Murrindindi Shire, Victoria, Australia. Each road between the towns contains several segments that are represented by the index of ‘‘l” and indicated by dashed lines. Disruption risks of the road routes are marked in three colors. Green indicates low road- disruption risk, while the orange and red colors respectively denote medium and high road-disruption risk. The gray and red arcs show the wind direction on Black Saturday. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 2 The case study data. Townships i

Population Di

Time windows m

i1 i2 i3 i4 i5 i6

Narbethong Marysville Taggerty Buxton Cambarville Rubicon

240 260 170 130 110 190

130 190 240 300 360 400

j1 j2 j3 j4 j5

Shelters j Alexandra Thornton Eildon Yea Yarra Glen

Capacity ðCapj Þ 1500 500 500 1000 1000

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Fig. 2. Vehicles routing pattern for the baseline. The start points are marked by circle while the finish point (shelters) are indicated by triangle. The solid lines indicate evacuee boarded vehicle trips while dashed lines indicate returning vehicle (empty) trips.

The network of routes in the case study area contains several segments (see Shahparvari et al., 2017b, Appendix Table A1). In this research, VicRoads1 bushfire risk assessments have been utilized as the source information to determine the risk of disruption for each route (see Shahparvari et al., 2017b, Appendix Tables A2 and A3). A VicRoads disruption risk measure has been allocated to each specific road segment in each route. A standard deviation approach has been utilized for the risk classification levels. Fig. 1 illustrates the three disruption risk group classifications, as follows: high-risk roads (red), moderate-risk roads (orange), and low-risk roads (green). Each vehicle (bus) has a maximum capacity of 45 passengers. Travel time between any two points on the proposed network is calculated based on geographic data and travel speed zone maps (VicRoads, 2014). The time windows, bushfire spread directions and disruption data have all been sourced from the 2009 Black Saturday Murrindindi bushfire records (Victorian Bushfires Royal Commission Report, 2009). 6. Dynamic visualization of results This section aims to demonstrate how the proposed model can be utilized to enhance emergency evacuation planning and performance, in order to save more lives in natural disaster situations. The 2009 Murrindindi bushfire case study was comprehensively examined and plausible scenarios analyzed in order to demonstrate the problem methodology and the model’s analytical solutions in order to enable insights to be drawn on how such changes affect the emergency evacuation process. The model was implemented using the CPLEX solver 12.6, on a PC with 8 GB RAM and 3.40 GHZ CPU. 6.1. Evacuation planning in the baseline (deterministic solution) The evacuation plan is successfully implemented via utilizing seven rescue buses to transport evacuees to safe shelters. Fig. 2 illustrates the evacuation routing pattern of assigned vehicles for the baseline. The route of each rescue vehicle is 1 VicRoads (or Roads Corporation of Victoria) is a statutory corporation which is the road and traffic authority in the state of Victoria, Australia. VicRoads plans, develops and manages the arterial road network and delivers road safety initiatives and customer registration and licensing services.

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Fig. 3. Total left over population histograms.

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Fig. 4. Average estimation error with 95% confidence.

Fig. 5. Total average uncovered population.

indicated via different colors. Furthermore, it outlines the routing schedule and designated rescue vehicles for the evacuation on the network. As it is shown, Bus 1 starts its evacuee pick-up service from Marysville, transporting its first passengers to the closest available shelter at Thornton, via the safest route. The model determined that Bus 1 could utilize the most reliable routes in order to safely transport all late evacuees from three bushfire-impacted towns (440 people). the evacuation plan for the other towns are plotted in Fig. 2.

6.2. Probabilistic scenarios This section demonstrates the application of the proposed model to the Murrindindi bushfire for 1000 randomly generated bushfire propagation and demand distribution scenarios. Time windows for each township are considered to be randomly generated with a range of ±25% of the actual recorded uniform distribution time window. The township populations are also assumed to be randomly generated by utilization of a Poisson distribution. Actual reported data from Table 4 is used as the mean value for parameter k. Fig. 3 presents histograms of total left over population by number of vehicles, derived from solving model by utilization of the proposed greedy solution approach. This figure demonstrates that utilizing nine vehicles provides coverage for all evacuee demands under any bushfire scenarios in the networks. Descriptive statistics also are provided in the histogram. The average left over population by utilizing one vehicle is 616 with standard deviations of 41. Q90%, Q95% and Q99% factors for vehicle one are 661, 679, and 709 respectively. Fig. 4 indicated the average estimation error with 95% confidence based on the number of available buses. It also highlights the convergence of results. Fig. 5 illustrates that the model’s total left over population decreases as the number of available buses increases. Fig. 5 also includes the descriptive statistics for the various scenario outputs.

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Fig. 6. Optimistic and pessimistic scenarios for time window and population values.

Fig. 7. Average left over population in each township.

Furthermore, in order to investigate the model’s ability to generate effective evacuation plans, it was implemented in four optimistic and pessimistic settings by considering maximum and minimum values for population and times windows of each town. The results are illustrated in Fig. 6 and show that the value of n (the required number of buses for evacuation) varies between 4 and 9. This means at least four buses can cover the entire evacuation demand for a minimum population and maximum time window scenario, while nine buses are required to evacuate all the people when time is tight and each town has a maximum late evacuee population. Fig. 7 depicts the average left over population with different numbers of buses in each township. It illustrates that the Rubicon and Taggerty townships could have been entirely evacuated under any bushfire scenario if more than 3 buses were utilized in the evacuation process. It also demonstrates that townships with higher populations and less time windows require more vehicles to completely evacuate. For example, nine buses are required to cover the Narbethong population. As it is shown, Narbethong and Marysville have the highest percentage of evacuee population left over in all scenarios.

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Fig. 8. The ratio of the number of uncovered people in the towns by the total uncovered people.

Fig. 9. Average left over evacuees at Naberthong by 10 min intervals.

Fig. 8 illustrates detailed presentations of ratio of uncovered population in the townships over the total uncovered population when different numbers of buses are used. It also demonstrates that Narbethong is more sensitive to bushfire scenarios than other townships. However, the left over population of townships which are located closer to shelters (e.g., Rubicon and Taggerty) do not have any sensitivity to bushfire scenarios. This is shown in Fig. 8 by the green color bars decreasing from 60% (when only 3 buses are used) to 0% (when 8 buses are used). Furthermore, an analysis was conducted on the Narbethong results to investigate the effect of the time window on the left over population. Fig. 9 illustrates the variation of left over population to the number of utilized vehicles. The time window range (100–160 min) is considered in ten minute intervals. Interestingly, the results prove the direct impact of time windows on the left over population in Narbethong as more of the population is left within the tight window intervals and utilizing fewer vehicles. However, the left over population of Narbethong is not more sensitive when the number of available vehicles is increased. Fig. 9 illustrates the average number of transferred evacuees to shelters under the proposed model in 1000 bushfire scenarios, as well as the overall cumulative covered population. The transportation of evacuees to the shelters varies and as expected the total cumulative covered population is increased by utilizing more vehicles. Interestingly, results show that most evacuees are transferred to shelters that are located close to each other in the northern central area. In most scenarios, the majority of evacuees are evacuated to the Thornton shelter which is indicated in green. This shows the significant role of this town in sheltering late evacuees. Interestingly, the results indicate that in the absence of Thornton, the Alexandra shelter is assigned more than Eildon. This is most likely due to its higher capacity, less risky road route and its proximity to

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Fig. 10. Average transferred evacuees to shelters.

Fig. 11. Average route usage to transfer evacuees to shelters.

Thornton. Put differently, the Alexandra and Eildon shelters are respectively assigned as back-up shelters when Thornton shelter has reached full capacity. It indicates that emergency evacuation authorities should pay more attention to back– up shelters close to the central shelters in comparison to the more remote shelters (Fig. 10). Furthermore, Fig. 11 uses the outputs to display the recommended shelters in terms of the routes used for each township. The results indicate that in most scenarios the model attempts to route evacuees to shelters via the safest and shortest routes. Finally, Fig. 12 depicts the total average objective function values by number of assigned rescue vehicles over all scenarios. As mentioned earlier, the objective function of the proposed model maximizes the total transferred population via the most reliable routes. Interestingly, Fig. 12 shows that total objective values are increased by increasing the number of assigned vehicles. 6.3. Worst plausible scenario: Disruption in both the arterial road and the most used shelter This section aims to investigate the model’s ability to generate routing plans in a scenario of both route and shelter disruption. The analyses of the results of previous section indicate that the Maroondah Highway plays a critical role in all rout-

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Fig. 12. Total average objective function values by number of assigned rescue vehicles.

Fig. 13. Evacuation routing pattern in a central shelter disruption scenario.

ing plans. This is because it is the major road route between Buxton and Taggerty. The majority of late evacuees are transferred to the northern shelters via this Highway. On Black Saturday in 2009, police reported doubts over the accessibility of the Maroondah Highway. In this scenario it is assumed that the Maroondah Highway is no longer available due to a collision immediately after bushfire ignition on the right side of the network. This scenario assumes that Thornton is not functioning as an available shelter due to unforeseen operational failures. The results show that Bus 1 departs from Rubicon to Alexandra. Bus 1 is then scheduled to return to Taggerty to evacuate its entire late evacuee population via four round trips to Alexandra. Route 2 between Taggerty and Alexandra has a higher reliability against bushfire disruption and has therefore been assigned as the route from Taggerty. Fig. 13 illustrates the evacuation routing plans for other townships. The model is

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able to provide routing arrangements to maximize evacuation even in the event of a severe disruption to the most highly used network infrastructure (the most used shelter and the major road route). Seven buses could safely transfer the entire late evacuee population within the time windows. 7. Conclusion This paper proposed an innovative formulation of emergency evacuation response decision support system. A stochasticbased vehicle routing problem (VRP) model was developed to maximize the number of late evacuees transported from assembly points to designated shelters via the safest routes. The proposed model considered several realistic constraints and integrated several VRP variants. The constraints were formulated as a probabilistic multi-destination capacitated dynamic vehicle routing problem with time windows (MDCDVRP-TW). The proposed model was tested using case study information from a 2009 major bushfire event that occurred in Victoria, Australia. The model was retrospectively applied to 1000 plausible scenarios from the Victorian Black Saturday bushfire disaster. This was done to assess the model’s ability to compile actual large scale case study data. We proposed a frequency approach solved by the application of a greedy algorithm to obtain a reliable evacuation plan from among the recommended options. The model provided suitable emergency evacuation operational plans that maximized efficiency in a variety of contexts. An analysis of the results provided by the model indicated that it would have enabled the rescue of the entire late evacuee population in the 2009 Black Saturday Murrindindi bushfire under uncertain clearance times and evacuation demands. This would have been achieved by utilizing only a limited number of rescue buses via the safest routes. As expected, the results proved that towns with higher populations and less time windows require more rescue vehicle to ensure evacuee coverage in all scenarios. Thornton was shown to have a significant role to play in sheltering evacuees, emerging as the most suitable destination in 33% of the 1000 bushfire scenarios investigated by the model. Alexandra was the next best option, being most suitable in 29% scenarios. The proposed model was generated effective evacuate plans even in an evacuation scenario where there was disruption to the most used shelter and road network. The model has therefore proven its ability to generate safe and effective routing plans for emergency evacuation in a wide variety of realistic scenarios. However, it is important to understand that evacuation procedures and plans are affected by other factors in addition to tight time windows, safety and potential infrastructure failures. Further research into additional factors is therefore necessary. A realistic and versatile emergency response decision tool may be able to be developed by the relaxation of other restraint assumptions in this study. In addition, it would be useful to evaluate how the proposed model may be able to be adapted for potential use in other propagating natural disasters, such as hurricanes and floods. Appendix A A.1. Notations Symbol I J R N Sm Sm :i Sm i: i j; j0 m k n r Pei Cej fk T tijr

lijr V BM

Description Sets Sets set of townships I 2 f1; 2; . . . ; jIjg; Set of all the shelterJ 2 fjIj þ 1; jIj þ 2; . . . ; jIj þ jJ jg;   Set of routes R 2 r 1 ; r 2 ; . . . ; rjRj ; Set of vehicle N 2 fn1 ; n2 ; . . . ; jN jg; Set of the available transportation network at time window m; m 2 f1; 2; . . . ; jIjg; Set of all the arrival routes to node i at time period m,i 2 ðI [ J Þ; Set of all the egress routes from node i at time period m; i 2 ðI [ J Þ; Indices Index for townships; Index for shelters; Index for time periods (m 2 f1; 2; . . . ; jIjg, e.g. m ¼ 3 indicates time period between T 2 and T 3 ); Index for time windows (k 2 I, e.g., k ¼ 3 is time from 0 to T 3 ); Index for vehicle n; Index of routes; Parameters Population of evacuees at town i, ðiIÞ; Capacity of shelter j; ðjJÞ; Available evacuation time window for town i (kI, e.g. township i is available from 0 to T k¼i ); Travel time between town i and shelter j via route r; Risk of disruption route r between towns i and shelter j; Maximum boarding capacity of the rescue vehicle; Big number;

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A.2. Decision variables The objective function of the model to be maximized is given by: Integer variable X mn ijr

amn i

bmn i Z mn ijr Yj Pmn ijj0 r

Number of times that the vehicle n travels from node i to node j via route r at time window m by vehicle n; Binary variables 1 if vehicle n starts evacuation process from node i at time window m, 0 otherwise, ði 2 I [ JÞ; 1 if vehicle n finishes evacuation process at node i at time window m, 0 otherwise, ði 2 I [ JÞ; 1 if vehicle n travels from town i to shelter j via route r at time window m, 0 otherwise, ði 2 I [ JÞ ; 1 if node j is assigned as a shelter, 0 otherwise s, ðj 2 JÞ; Auxiliary variable for sub-tour elimination ði 2 I [ JÞ;

References Abdelgawad, H., Abdulhai, B., 2011. Large-scale evacuation using subway and bus transit: approach and application in city of Toronto. J. Transport. Eng. 138, 1215–1232. Alsalloum., O.I., Rand, G.K., 2006. Extensions to emergency vehicle location models. Comput. Oper. Res. 33 (9), 2725–2743. Amailef, K., Lu, J., 2013. Ontology-supported case-based reasoning approach for intelligent m-Government emergency response services. Decis. Support Syst. 55 (1), 79–97. An, S., Cui, N., Li, X., Ouyang, Y., 2013. Location planning for transit-based evacuation under the risk of service disruptions. Transport. Res. Part B: Methodol. 54, 1–16. Arnol, C., 2007. Firefighting Operations on the Urban Interface. Fellowship Report 20081, The Winston Churchill Memorial Trust of Australia. Atamtürk, A., Zhang, M., 2007. Two-stage robust network flow and design under demand uncertainty. Oper. Res. 55 (4), 662–673. Barbaroso, gcaron, G., 2004. A two-stage stochastic programming framework for transportation planning in disaster response. J. Oper. Res. Soc. 55 (1), 43– 53. Ben-Tal, A., El Ghaoui, L., Nemirovski, A., 2009. Robust Optimization. Princeton University Press. Ben-Tal, A., Golany, B., Nemirovski, A., Vial, J.-P., 2005. Retailer-supplier flexible commitments contracts: a robust optimization approach. Manuf. Serv. Oper. Manage. 7 (3), 248–271. Ben-Tal, A., Nemirovski, A., 2002. Robust optimization–methodology and applications. Math. Program. 92 (3), 453–480. Beraldi, P., Bruni, M.E., 2009. A probabilistic model applied to emergency service vehicle location. Eur. J. Oper. Res. 196 (1), 323–331. Beraldi, P., Bruni, M.E., Conforti, D., 2004. Designing robust emergency medical service via stochastic programming. Eur. J. Oper. Res. 158 (1), 183–193. Bertsimas, D., Chervi, P., Peterson, M., 1995. Computational approaches to stochastic vehicle routing problems. Transport. Sci. 29 (4), 342–352. Bertsimas, D., Sim, M., 2003. Robust discrete optimization and network flows. Math. Program. 98 (1–3), 49–71. Bertsimas, D., Sim, M., 2004. The price of robustness. Oper. Res. 52 (1), 35–53. Bertsimas, D.J., Jaillet, P., Odoni, A.R., 1990. A priori optimization. Oper. Res. 38 (6), 1019–1033. Bish, D.R., 2011. Planning for a bus-based evacuation. OR Spectrum 33, 629–654. CFA Definitions 2012, Definitions, 2013. . Chan, C.P., 2010. Large scale evacuation of careless people during short-and long-notice emergency. Chang, M.-S., Tseng, Y.-L., Chen, J.-W., 2007. A scenario planning approach for the flood emergency logistics preparation problem under uncertainty. Transport. Res. Part E: Logist. Transport. Rev. 43 (6), 737–754. Chen, C.-C., Chou, C.-S., 2009. ’Modeling and performance assessment of a transit-based evacuation plan within a contraflow simulation environment. Transport. Res. Rec.: J. Transport. Res. Board, No. 2091, 40–50. Chen, D.A., Xu, W.T., Zhang, W., 2011. Optimize the transit vehicle routing for the emergency evacuation. In: Advanced Materials Research. Trans Tech Publ, pp. 1075–1081. Cova, T.J., Johnson, J.P., 2003. A network flow model for lane-based evacuation routing. Transport. Res. Part A: Policy Pract. 37 (7), 579–604. Dantzig, G.B., Ramser, J.H., 1959. ’The truck dispatching problem. Manage. Sci. 6 (1), 80–91. Dror, M., Laporte, G., Trudeau, P., 1989. Vehicle routing with stochastic demands: properties and solution frameworks. Transport. Sci. 23 (3), 166–176. Eksioglu, B., Vural, A.V., Reisman, A., 2009. The vehicle routing problem: a taxonomic review. Comput. Ind. Eng. 57 (4), 1472–1483. Erera, A.L., Morales, J.C., Savelsbergh, M., 2009. Robust optimization for empty repositioning problems. Oper. Res. 57 (2), 468–483. Goerigk, M., Grün, B., 2014. A robust bus evacuation model with delayed scenario information. Or Spectrum 36, 923–948. Goerigk, M., Grün, B., Heßler, P., 2013. Branch and bound algorithms for the bus evacuation problem. Comput. Oper. Res. 40, 3010–3020. Iakovou, E., Ip, C.M., Douligeris, C., Korde, A., 1997. Optimal location and capacity of emergency cleanup equipment for oil spill response. Eur. J. Oper. Res. 96 (1), 72–80. Joshi, R.R., 1997. A new heuristic algorithm for probabilistic optimization. Comput. Oper. Res. 24 (7), 687–697. He, S., Zhang, L., Song, R., Wen, Y., Wu, D., 2009. Optimal transit routing problem for emergency evacuations. In: Transportation Research Board 88th Annual Meeting. Kaisar, E.I., Hess, L., Portal Palomo, A.B., 2012. An emergency evacuation planning model for special needs populations using public transit systems. J. Pub. Transport. 15, 3. Kulshrestha, A., Lou, Y., Yin, Y., 2014. Pick-up locations and bus allocation for transit-based evacuation planning with demand uncertainty. J. Adv. Transport. 48, 721–733. Lamiri, M., Xie, X., Dolgui, A., Grimaud, F., 2008. A stochastic model for operating room planning with elective and emergency demand for surgery. Eur. J. Oper. Res. 185 (3), 1026–1037. Lindell, M.K., Perry, R.W., 1992. Behavioral Foundations of Community Emergency Planning. Hemisphere Publishing Corp. Margulis, L., Charosky, P., Fernandez, J., Centeno, M.A., 2006. Hurricane evacuation decision-support model for bus dispatch. Paper presented to Fourth LACCEI International Latin American and Caribbean Conference for Engineering and Technology (LACCET ‘2006), Breaking Frontiers and Barriers in Engineering: Education, Research, and Practice. Mastrogiannidou, C., Boile, M., Golias, M., Theofanis, S., Ziliaskopoulos, A., 2009. ’Using transit to evacuate facilities in urban areas: a micro-simulation based integrated tool. Transport. Res. Rec.: J. Transport. Res. Board 3439, 1–15. Mudchanatongsuk, S., Ordóñez, F., Liu, J., 2008. Robust solutions for network design under transportation cost and demand uncertainty. J. Oper. Res. Soc., 652–662

S. Shahparvari, B. Abbasi / Transportation Research Part A 104 (2017) 32–49

49

Najafi, M., Eshghi, K., Dullaert, W., 2013. A multi-objective robust optimization model for logistics planning in the earthquake response phase. Transport. Res. Part E: Logist. Transport. Rev. 49 (1), 217–249. Ordóñez, F., Zhao, J., 2007. Robust capacity expansion of network flows. Networks 50 (2), 136–145. Özdamar, L., Ekinci, E., Küçükyazici, B., 2004. Emergency logistics planning in natural disasters. Ann. Oper. Res. 129 (1–4), 217–245. Pages, L., Jayakrishnan, R., Cortés, C., 2006. Real-time mass passenger transport network optimization problems. Transport. Res. Rec.: J. Transport. Res. Board, 229–237. Perkins, J.A., Dabipi, I.K., Han, L.D., 2001. Modeling Transit Issues Unique To Hurricane Evacuations: North Carolina’s Small Urban And Rural Areas. Pourrahmani, E., Delavar, M.R., Pahlavani, P., Mostafavi, M.A., 2015. Dynamic evacuation routing plan after an earthquake. Nat. Hazards Rev. 16 (4), 04015006. Qazi, A.N., Okubo, K., Kubota, H., 2016. Short-notice bus-based evacuation under dynamic demand conditions. Asian Transp. Stud. 4, 228–244. Rajagopalan, H.K., Saydam, C., Xiao, J., 2008. A multiperiod set covering location model for dynamic redeployment of ambulances. Comput. Oper. Res. 35 (3), 814–826. Rui, S., Shiwei, H., Zhang, L., 2009. Optimum transit operations during the emergency evacuations. J. Transport. Syst. Eng. Inform. Technol. 9, 154–160. Sayyady, F., Eksioglu, S.D., 2010. Optimizing the use of public transit system during no-notice evacuation of urban areas. Comput. Ind. Eng. 59, 488–495. Sayyady, F., 2007. Optimizing the Use of Public Transit System in No-notice Evacuations in Urban Areas (Doctoral dissertation, Mississippi State University). Sbayti, H., Mahmassani, H., 2006. Optimal scheduling of evacuation operations. Transport. Res. Rec.: J. Transport. Res. Board, 238–246. Shahparvari, S., Abbasi, B., Chhetri, P., Abareshi, A., 2015. Vehicle routing and scheduling for bushfire emergency evacuation. In: 2015 IEEE International Conference on Industrial Engineering and Engineering Management (IEEM). IEEE, pp. 696–700. Shahparvari, S., Chhetri, P., Abbasi, B., Abareshi, A., 2016a. Enhancing emergency evacuation response of late evacuees: revisiting the case of Australian Black Saturday bushfire. Transport. Res. Part E: Logist. Transport. Rev. 93, 148–176. Shahparvari, S., Abbasi, B., Chhetri, P., 2017a. Possibilistic scheduling routing for short-notice bushfire emergency evacuation under uncertainties: an Australian case study. Omega 72, 96–117. http://dx.doi.org/10.1016/j.omega.2016.11.007. ISSN 0305-0483. Shahparvari, S., Abbasi, B., Chhetri, P., Abareshi, A., 2017b. Fleet routing and scheduling in bushfire emergency evacuation: a regional case study of the Black Saturday bushfires in Australia. Transport. Res. Part D: Transp. Environ. in press. Snyder, L.V., 2006. ’Facility location under uncertainty: a review. IIE Trans. 38 (7), 547–564. Southworth, F., 1991. Regional Evacuation Modeling: A State-of-the-art Review (No. ORNL/TM-11740). Oak Ridge National Lab., TN (USA). Soyster, A.L., 1973. Technical note—convex programming with set-inclusive constraints and applications to inexact linear programming. Oper. Res. 21 (5), 1154–1157. Teague, B., Mcleod, R., Pascoe, S., 2009. Victorian Bushfires Royal Commission Interim Report. Parliament of Victoria, Melbourne. VicRoads, 2014. VicRoads Speed Zones Maps. In: V. U. (Ed.). http://www.data.vic.edu.au/. Victorian Bushfires Royal Commission Report 2009, July 2009. Interim Report, Section, vol. 7. Vuchic, V.R., 2005. Urban Transit: Operations, Planning, and Economics. John Wiley & Sons, New Jersey. Whittaker, J., McLennan, J., Elliott, G., Gilbert, J., Handmer, J., Haynes, K., Cowlishaw, S., 2009. Victorian 2009 Bushfire Research Response: Final Report. Bushfire CRC Post-fire Research Program in Human Behaviour.(Bushfire CRC: Melbourne) Available at . Yi, W., Özdamar, L., 2007. A dynamic logistics coordination model for evacuation and support in disaster response activities. Eur. J. Oper. Res. 179 (3), 1177– 1193. Yoon, S.W., Velasquez, J.D., Partridge, B., Nof, S.Y., 2008. Transportation security decision support system for emergency response: a training prototype. Decis. Support Syst. 46 (1), 139–148. Zhang, H., Liu, H., Zhang, H., Wang, J., 2010. Modeling of evacuations to no-notice event by public transit system. In: 2010 13th International Ieee Conference On Intelligent Transportation Systems (ITSC). IEEE, pp. 80–484. Zhang, X., Chang, G.-l., 2014. A transit-based evacuation model for metropolitan areas. J. Pub. Transport. 17, 9. Zheng, H., 2014. Optimization of bus routing strategies for evacuation. J. Adv. Transport. 48, 734–749.