Enhancing emergency evacuation response of late evacuees: Revisiting the case of Australian Black Saturday bushfire

Transportation Research Part E 93 (2016) 148–176

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Transportation Research Part E journal homepage: www.elsevier.com/locate/tre

Enhancing emergency evacuation response of late evacuees: Revisiting the case of Australian Black Saturday bushfire Shahrooz Shahparvari ⇑, Prem Chhetri, Babak Abbasi, Ahmad Abareshi School of Business IT & Logistics, RMIT University, Melbourne, VIC 3000, Australia

a r t i c l e

i n f o

Article history: Received 21 September 2015 Received in revised form 17 April 2016 Accepted 21 May 2016

Keywords: Bushfire Multi-objective optimisation Emergency management Late evacuees Disruption risk Black Saturday

a b s t r a c t This paper develops a multi-objective integer programming model to support tactical planning decision-making during a short-notice evacuation using the situated context of the 2009 Black Saturday bushfires in Victoria. Various bushfire scenarios and sensitivity analysis considering short time windows, availability of resources and road disruptions were implemented to demonstrate the robustness and reliability of the model. The e-constraint technique was applied to solve the problem. Results showed that it would be possible to evacuate all late evacuees during the Black Saturday bushfire events, even if one or two resources are disrupted within the hard time window constraint. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Bushfires (wildfire) are a natural hazard that constantly threatens human lives and properties in Australia. In the past 150 years, bushfires have claimed hundreds of lives and resulted in billions of dollar worth of damage in Australia. During this period, 300 people have lost their lives and a further 4185 have suffered injuries in Victoria alone. These figures account for approximately 39 per cent of total deaths and 57 per cent of injuries from all major recorded natural disasters in Australia’s history (Haynes et al., 2010). The Black Saturday bushfires, which occurred on the 7th of February 2009 in Victoria, are the worst on record. 173 people lost their lives, more than 7500 were displaced, and an estimated $ 4.5 billion worth of financial losses were incurred (Teague et al., 2009). Evacuation is defined by the Emergency Management Australia (2005) as ‘‘a risk management strategy which may be used as a means of mitigating the effects of an emergency or disaster on a community. This process involves the movement of people to a safer location and to be effective it must be correctly planned and executed”. Bushfire evacuation can be mandatory, recommended or voluntary. In Australia, it is voluntary as people are permitted to stay and protect their houses and possessions during an emergency (Teague et al., 2009). There are three categories of people in an event of emergency: (i) those who leave early; (ii) those who decide to shelter in refuge and (iii) those who stay at their properties (i.e. Shelter-inPlace). They are also referred as ‘protective actions’ (Cova et al. (2011). This research focuses on the evacuation of the third group of people as late evacuees in a short-notice evacuation situation. The situated knowledge uncertainty of a short-notice evacuation may put evacuees’ lives at a greater risk. 32 per cent of all the bushfire fatalities in Australia (176 of 552 deaths) during the last century were related to short-notice evacuation. Data shows that over 50 per cent of those who were

⇑ Corresponding author. E-mail address: [email protected] (S. Shahparvari). http://dx.doi.org/10.1016/j.tre.2016.05.010 1366-5545/Ó 2016 Elsevier Ltd. All rights reserved.

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evacuated on Black Saturday were late evacuees (Haynes et al., 2010). The Victorian Bushfires Royal Commission (VBRC) highlighted the need for amendments by providing 67 recommendations in their final report to the Victorian Government in July 2010. The VBRC emphasised the importance of urgently developing emergency bushfire plans and improving the emergency evacuation components within these plans, such as the identification of candidate shelters and bushfire disruption risks. These recommendations were given serious consideration when developing the national bushfire policy: Prepare, Act, Survive (Australasian Fire Authorities Council, 2010). A short-notice evacuation during emergency is a costly, complex, disruptive and politically sensitive issue. Short-notice evacuation of late evacuees is often challenging as it is dependent on numerous uncertainties. It is also an unavoidable as there are people those who choose to stay and defend their properties instead of leaving early; those with disability, those with younger children, and the elderly and those with no access to a personal vehicle. They are heavily dependent on fire agencies for last-minute evacuation. While early evacuation often occurs by car, the short-notice evacuation to transfer people from assembly points to the nearest safe shelters typically is carried out using high-capacity vehicles such as buses (Vuchic, 2005). Compulsory evacuation warnings are broadcasted to inform late evacuees to assemble at a pre-defined location and prepare for evacuation to a safer shelter via organised rescue vehicles. Lessons from the 2009 Black Saturday bushfires in Australia highlight the importance of both strategic and tactical planning to enhance operational efficiency of transit-based, short-notice emergency response. Given the uncertainty of bushfire propagation and the resulted network and supply disruptions, the transfer of people from bushfire-prone areas to safe shelters within a short-time window has remained a key challenge for emergency service agencies. Simultaneous decisions are to be made in bushfire situations to assign and allocate evacuees to safe shelters, selecting the right evacuation vehicles and choosing the optimal route. These decisions are to be optimised considering risk and cost minimisation, efficient resource utilisation, while at the same time maximising the safety of evacuees. The development of a decision support system capable of simultaneously considering multiple objectives and constraints under various bushfire scenarios is urgently needed to effectively respond to the perennial threat of bushfires in Australia. Hence, this study aims to develop a multi-objective integer-programming model to tackle the key operational challenges associated with timely evacuation, shelter assignment and routing in a short-notice evacuation during bushfire events. The model’s parameters are based on real observations from the Black Saturday bushfire events to mimic the potential impact of observed risk of network disruptions and the available clearance time on resource allocation. The developed model is an integer program with two objectives – maximising number of evacuated people using the safest routes and optimally determining the number of shelters and the required number of vehicles. The model therefore considers capacitated multi-location, multi-routing, and multi-vehicle types (high and medium capacity vehicles). A solution approach based on e-constraint method is implemented in the model. Section 2 of this paper presents the literature review. This is followed by a brief description of the short-notice evacuation problem in Section 3. Section 4 introduces the mathematical model formulation. Section 5 offers the solution approach, along with a discussion of the computational results and practical implications of the model. Finally, Section 6 presents the key findings and conclusions.

2. Understanding the late evacuation procedures An emergency evacuation system is a set of procedures and processes that guide the systematic evacuation of evacuees from disaster-affected areas to safe shelters. It is a complex, multi-component system, which requires multi-agency response to an emergency. Fig. 1 illustrates the interrelated and interlocking sequences and procedures involved in a typical emergency response. The system contains four major stages: (I) disaster (bushfire) impact evaluation (II) evacuation estimation (III) evacuation plan generation and (IV) evacuation plan implementation. In stage (I), the scale, intensity and magnitude of a bushfire are evaluated. This step requires the initial inputs (e.g. affected areas, bushfire direction, and transportation network data) to evaluate the potential threat of a bushfire for the next steps. Stage (II) estimates the operational inputs, which include the number of late evacuees, the optimum location of candidate shelters (based on capacity, accessibility and other risk factors), accessibility of routes and most importantly the time window. Stage (III) generates evacuation plans and actions, including help assign rescue vehicles, allocation of shelters, and identification of the safest and shortest routes. Generated plans are re-assessed and moderated by decision-makers in terms of feasibility and soundness in responding to real world situations. Finally, in stage (IV), the generated evacuation plans are implemented by emergency service agencies including broadcasting evacuation-warning messages, transferring evacuees to shelters, assignment of vehicles, and delineation of routes. In evacuation planning, the decision of individuals/households plays a critical role. The ability to anticipate and estimate the number of people who potentially decide to leave early or stay in the area to protect property has a strong bearing on evacuation planning and resource allocation (Lindell et al., 2011). There are a wide range of behavioural factors that influence the evacuation decisions, including the age; physical capacity, mobility and health; responsibility of children, pets and livestock; and perception of risk and the degree of preparedness. Murray-Tuite and Wolshon (2013) developed one of the most comprehensive evacuation models in which a broad range of behavioural parameters were incorporated. Their research identified key factors that influence evacuation decision-making. However, these factors were considered in modelling only in specific types of disaster such as hurricanes (Hsu and Peeta, 2013; Pel et al., 2012). Only a few studies have considered incorporating these factors in routing or resource allocation modelling.

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Fig. 1. Interdependent and interlocking components of the short-notice emergency evacuation process.

Late evacuation is a bushfire is life-threatening (Handmer and Tibbits, 2005; Krusel and Petris, 1992; Whittaker et al., 2013). Late self-evacuation using a personal vehicle in particular escalates the risk. Unknown road conditions and hazards such as flames, amber, smoke, strong winds, fallen trees, and road blockade significantly increases the fire risk (Tibbits and Whittaker, 2007; Victorian Bushfires Royal Commission Report, 2009). Another concern with late evacuation decision is a growing evidence showing that many homeowners do not intend to evacuate for bushfire (Cohn et al., 2006; Paveglio et al., 2010; Stidham et al., 2011). The tendency for people to wait until a fire arrives before deciding whether to stay and defend their properties remains as a perennial challenge for fire agencies (Rhodes, 2005; Teague et al., 2009; Whittaker et al., 2009). Behavioural factors such as the stress of being away from house, uncertainty on when they could return home, and lack of information on status of homes can lead to residents stay and defend their properties (Cohn et al., 2006; Stidham et al., 2011). Women and families with young children are more likely to evacuate early while men are more likely to stay (Eriksen et al., 2010). Proudley (2008) found that a greater proportion

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of women (54%) left their homes and properties before or during the fires compared to 35% in men. Women are thus more likely evacuate when confronted by bushfire (Eriksen et al., 2010). Evacuation difficulty of people with limited mobility or special needs is another reason which makes people opt to take shelter in their places till last minutes (Cohn et al., 2006; Paveglio et al., 2010). Perkins et al. (2001) discussed the use of buses to evacuate people (elderly and disabled) under a no-notice scenario. A significant proportion of evacuees reported that there were members of households who needed looking after during bushfire, including infant or children (23.7%), elderly personns (4.1%), disabled personns (2.4%), ill personns (1.9%), distressed personns (5.1%). Emergency evacuation modelling has attracted considerable attention in recent years (e.g. see (Southworth, 1991; Murray-Tuite and Wolshon, 2013; Caunhye et al., 2012). While the evacuation process has been widely examined from operational and behavioural perspectives, the integration of underlying operational complexities associated with short-notice emergency evacuation has yet to be thoroughly modelled. While it is obvious that pick-up locations or assembly points could significantly affect the responsiveness of transit-based evacuation services, this issue has not attracted much attention until recent years. In addition, most studies have focused on minimising the total evacuation time, without considering other objectives such as resource utilisation or area coverage. There has been no comprehensive attempt to model short-notice evacuation in the policy context that addresses key operational challenges and combines the problems of timely evacuation, shelter assignment and routing. This research attempts to bridge gaps by developing a reliable optimisation multi-objective mathematical model in order to generate possible solutions to enhance emergency evacuation response in bushfire planning. An optimisation problem minimises or maximises a certain objective function, subject to supply, demand and time constraint. Cohon (2004) identified a number of objectives, such as clearance time, number of evacuees, and allocation of finite resources to be critical in emergency evacuation decision-making. Time is considered as the most common objective function, which was represented through parameters such as minimum evacuation travel time (Liu et al., 2007; Tuydes and Ziliaskopoulos, 2004), network clearance time (Sattayhatewa and Ran, 2000), or mobilisation time (Sbayti and Mahmassani, 2006). Multiple objective functions have been formulated with the aim of improving the efficiency of the evacuation process, by minimising travel or clearance time (Abdelgawad and Abdulhai, 2010; Sayyady, 2007). However, real-time location–allocation and routing of late evacuees to shelters are subject to a range of other factors which should also be simultaneously optimised while operating within a range of stringent constraints (Li et al., 2012; Negreiros and Palhano, 2006). These factors include real or perceived risks, capacity constraints, travel time and network distances, and susceptibility or vulnerability to disaster. Ultimately, the most challenging objective for the emergency is to safely transfer late evacuees to shelters within the shortest period of time, with limited resources. From an operation perspective, short-notice emergency evacuation therefore necessitates quick response to multiple operational requirements. These are grouped in two key operational areas which are discussed as following. 2.1. Shelter location–allocation problem Shelter allocation, in terms of optimum number, location, availability and capacity, is the key component of short-notice evacuation (Lim et al., 2012) and emergency management (Alexander, 2000). The allocation of a facility as a safe shelter in a bushfire however is a complex, risky and time-sensitive decision Owen and Daskin (1998). Some studies (Chan, 2010; He et al., 2009; Mastrogiannidou et al., 2009) have applied location–allocation modelling to solve emergency evacuation location–allocation problems. Concerns being raised regarding the timely availability of both rescue vehicles and safe shelters. Uncoordinated pre-emergency planning and the computational hurdles posed by the complex facility allocation formulation problem and the potential disruption uncertainties have often resulted less efficient response to life threatening situations. Despite the significant contribution of previous studies in transit-based evacuation, due to the computational hurdle posed by the complex facility allocation formulation problem and the potential disruption uncertainties, there are limited attempts to integrate transit-based systems with emergency evacuation location–allocation planning. Coppola (2006) acknowledged that there is a gradual shifting of paradigm towards risk reduction-based disaster management. Chan (2010) considered utilising uncertainties in formulating solutions to both prior to, and in the aftermath of, a disaster outbreak. However, the uncertainty of the situated context of a bushfire in terms of its propagation rate and direction, as well as the potential for network disruptions and system failure on an in situ response to emergency evacuation has been largely discounted in most optimisation models. Furthermore, most optimisation models assume that emergency service facilities (e.g. shelters, hospitals) continue to provide services in an emergency situation without any disruptions (An et al., 2013). Furthermore, it is often assumed that majority of public evacuation shelters provide enough space to host the evacuees and are located in areas where there is adequate ingress routes to deliver humanitarian, medial and other aids (Campos et al., 2012). Majority of studies consider shelters as destinations choice in modelling (Bish and Sherali, 2013; Campos et al., 2012; Lim et al., 2012; Sayyady and Eksioglu, 2010) while others do not even take into account the shelter in the mathematical formulations (Huibregtse et al., 2010; Pel et al., 2010). Nonetheless, there are reliable location models, which have addressed the location–allocation problem of emergency facilities in disaster situations (Shen et al., 2011; Chen et al., 2011; Berman et al., 2007). However, only a few models have explicitly considered maximising the transferability of late evacuees whilst simultaneously minimising resource utilisation (Shahparvari et al., 2015a,c,b).

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2.2. Transit mode of an emergency evacuation Short-notice evacuation is often operationalised using a transit-based system. A number of studies have developed mathematical formulations to examine the transit-based evacuation routine for both unpredictable (e.g. earthquake) Sayyady (2007) or more predictable disasters (e.g. cyclone) (Chan, 2010; Margulis et al., 2006). For example, Margulis et al. (2006) developed a binary integer-programming model to determine the assignment of buses to pick up points and to shelters during an evacuation. The objective of their model was to maximise the number of evacuee throughput in a given time period. However, their model would assume that buses are at the pickup points at the beginning of the evacuation, and regulate each bus to return to the same evacuation site. Sayyady (2007) formulated the car-less evacuation problem with a minimum cost flow model under additional side constraints. His model assumes that bus stops are pickup locations and the car-less evacuation is guided to the stop that is closest to their current location. A Tabu search is used to identify evacuation routes for buses. In that study, buses only carry out one single trip and do not return to pick up the car-less after leaving the affected area. Lim et al. (2012) considered a short-notice regional hurricane evacuation maximising the number of evacuees reaching safety weighted by the severity of the threat. They developed an evacuation-scheduling algorithm to expedite the solution process. Recent studies (Tunc et al., 2011; Sayyady and Eksioglu, 2010) have also developed mixed-integer linear programming models to identify optimal evacuation routes for transit. For example, Abdelgawad and Abdulhai (2011) developed a mass transit system that can be harnessed to evacuate transit-dependent travellers in no-notice evacuation events. Cavusoglu et al. (2012) addressed the importance of transportation needs of transit-dependent and car-less populations and suggested that well-coordinated utilisation of transit assets would lead to safer evacuation. This study also suggest that the behavioural characteristics of evacuees should be considered along with other determining factors to make transit evacuation an integral part of evacuation management plans. In the case of a bushfire short-notice evacuation the use of a personal vehicle is often discouraged due to highly hazardous environment such as risk of disorientation, road blocks, and threat from radiant heat (Teague et al., 2009). In addition, the tracking of personal vehicles is also difficult, which could exacerbate traffic congestion, resulting further hindrance in the evacuation process. The utilisation of larger commercial vehicles is recommended for a mass evacuation due to high capacity. A bus can carry up to six times as many passengers as a passenger car (Litman, 2006). However, the higher carrying capacity could also mean higher cumulated risk of exposing to a threat. Another important consideration is the choice of pick-up locations, which can significantly impact on the effectiveness of transit-based evacuation response. One way to solve this problem is to maximise the number of evacuated people while simultaneously minimising the allocation of resources Shahparvari et al. (2015c). Their model is applicable in generating emergency evacuation plans within short time windows. They have applied their model in group bushfire spread scenarios. However, their model has not considered factors such as multiple road connections, inflation time due to traffic, and the risk of road and segment disruptions. Consideration of these factors in modelling would enhance the effectiveness of emergency response and better utilisation of finite resources.

3. Problem formulation 3.1. Multi-objective integer programming approaches and methods The emergency evacuation problem formulated in this analysis deals with a range of objective functions, uncertainties and stringent constraints. An appropriate formulation of such a problem should eliminate symmetries and infeasible edges of the solution area as a pre-processing step. To achieve this, a few symmetries are eliminated by not labelling individuals’ vehicles. Elimination of these symmetries reduces the number of solutions that need to be considered when searching the solution space. Hence, the problem is formulated as a multi-objective integer linear-programming (MOILP) optimisation model. A number of solution approaches have been developed in operational research for solving multi-objective integer problems (MOIPs). These include goal programming (Wilson and Macleod, 1993), multi- and bi-level programming (Hansen et al., 1992), weighted um (Scalarising) (Coello Coello and Lechuga, 2002) and Pareto front optimisation approaches (Zitzler and Thiele, 1998) such as the lexicographic minimax method (Bazaraa and Goode, 1982) and the epsilon constraint method (Chankong, 1983). In goal programming, the goal is typically referred to as a planned objective. The goal programming method measures the deviation of objectives (optimality) from the planned objectives. The objective functions however are not simultaneously optimised (Abbass and Sarker, 2002). Multi-level programming is another MOIPs approach that hierarchically orders objectives and successively optimises objective functions. Multi-level programming problems are complex to solve (Hansen et al., 1992) as the last set of objective functions at the bottom of the hierarchy is often constrained, resulting in infeasible conditions. This could mean that the low-level ranked objective functions may have negligible impact on the derivation of final optimal solution (Vicente et al., 1996). The weighted sum (Scalarising) optimisation method is based on converting multiple objectives to a linear single normalised weighted scalar function, which is often solved by conventional methods. Determining the appropriate weight in this method however is challenging. The optimum solution is based on normalisation and the assigned weights.

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Overall, despite the broad range of multi-objective optimisation solution methods, multi-objective integer problem solutions are not usually straightforward. Typically, no feasible absolute optimal solution exists to optimise all objective functions concurrently (Ehrgott, 2006). To address this problem, Pareto-optimal solutions have attracted much more attention, and are often referred to as a ‘‘posteriori’’ or ‘‘non-dominated solution generation’’. Pareto-optimal solutions MOIPs that attempt to optimise the main objective by degrading at least one of the other objectives. Out of these, the lexicographic minimax method is one of the most extensively used Pareto optimal solutions to order the weighted average aggregation of objectives. The application of this method is suitable where all of the objective functions are equally important for decision-makers. The main disadvantage of this method is that some of results may not be Pareto-optimal because the solution is not typically unique (Kostreva et al., 2004). The epsilon constraint method (e-constraint) is another common general-purpose Pareto front solution technique, which has been widely utilised to solve multi-objective integer problems (Chankong, 1983). This method provides extensive flexibility for decision-makers by varying the lower or upper bounds ei to achieve Pareto optima (Zitzler and Thiele, 1998). 3.2. Assumptions The modelling is based on the following key assumptions: – – – – – –

The shelters are pre-designated by the fire agency – Country Fire Authority (CFA). Number and capacity of shelters and rescue vehicles is finite. The late evacuee population in each assembly point is known. Access to some routes and shelters is restricted by bushfire propagation. The availability of assembly points and routes is subject to rigid time windows. There is no background traffic outside the affected region.

3.3. Notations Sets I set of evacuation points (origins) J set of candidate shelters (destinations) V set of vehicle types K set of routes across evacuation point i and shelters j L set of route sections Indices i index of evacuation points ði 2 IÞ j index of candidate shelter ðj 2 JÞ v index of vehicle types ðv 2 VÞ l index of road sections ðl  LÞ k index of route k among evacuation points and shelters ðk 2 KÞ Parameters uv route’s occupied capacity by one unit of vehicle type v ðv 2 VÞ hv usage cost of vehicle type v ðv 2 VÞ kvijk capacity of route k along evacuation point i to shelter j for Vehicle v ði 2 I; j 2 J; k 2 KÞ Cj capacity of assigned shelter j ðj 2 JÞ Pi population of late evacuees at assembly point of township i ði 2 IÞ sijk traversal time of route k between assembly point i to shelter j ði 2 I; j 2 J; k 2 KÞ T time impedance parameter due to road congestions DT dwell time (vehicle preparation, boarding/alighting of evacuees) l ijk disruption risk of route k between node i and j ði 2 I; j 2 J; k 2 KÞ X number of candidate shelters w weighted sum coefficient TV v number of available vehicles type v ðv 2 VÞ VC v vehicle capacity of type v ðv 2 VÞ TW i availability time window of evacuation point i ði 2 IÞ TWRijk availability time window of k route ði 2 I; j 2 J; k 2 KÞ aj facilities functioning vector; 1, if shelter j is available; 0, otherwise ðj 2 JÞ bijk link/route disruption matrix; 1 if route k is accessible between point i and shelter j; 0, otherwise ði 2 I; j 2 J; k 2 KÞ

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3.4. Decision variables Two sets of decision variables are used to represent the maximal short-notice evacuation coverage problem and the equivalent minimal resource location–allocation during an emergency situation. These include: – X ijk Number of transferred late evacuees from assembly point i to functioning shelter j throughout k route. – NV vijk Quantity of vehicle type v in need to be routed to transfer evacuees via route k from the assembly point i to functioning shelter j. – cj If evacuees are transferred to the dedicated shelter j; 0, otherwise. – rj If a shelter is assigned at candidate place j; 0, otherwise. Variable cj ensures that late evacuees from any assembly point can be transported to the designated shelters until they are operational. Accordingly, the decision variable rj , measures if candidate place j is chosen as a safe shelter to transfer evacuees or not. Moreover, two integer auxiliary decision variables are also applied to improve the results: Mijk Number of times that vehicle v travels between assembly point i to candidate shelter j via route k. X 0ijk Unoccupied seats in the last trip of assigned vehicle. 3.5. Objective functions of problem The following model objectives are set out: – To maximise the number of evacuated people using the most reliable routes (minimum cumulative disruption risk) to the nearest functioning shelters within the rigid clearance times. – To minimise the allocation of functioning resources (the number of functioning shelters and rescue vehicles). 3.6. Problem formulation The proposed multi-objective model comprises two objective functions and a range of constraints as follows: 3.6.1. Objective functions Maximise ff 1 ; f 2 g where

f1 ¼

XXX  ijk Þ X ijk ð1  l i

j

f 2 ¼ w1

ð1Þ

k

XXXX v v X h NV ijk þ w2 rj i

j

v

k

ð2Þ

j

3.6.2. Constraints The objective functions are subject to the following constraints:

X

rj 6 X

ð3Þ

j

cj 6 aj rj 8j 2 J

ð4Þ

XX X ijk 6 C j cj i

8j 2 J

ð5Þ

k

X ijk 6 bijk Pi

8i 2 I;

j 2 J;

k 2 K;

v2V

ð6Þ

Mijk  ðð2  sijk ð1 þ TÞÞ þ DTÞ 6 MinimumðTW i ; TWRijk Þ X ijk þ X 0ijk X 0ijk <

X ¼ Mijk VC v NV vijk v

X VC v NV vijk v

8i 2 I;

8i 2 I; j 2 J;

j 2 J; k2K

k2K

8i 2 I;

j 2 J;

k2K

ð7Þ ð8Þ ð9Þ

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XX X ijk 6 Pi i

8i 2 I

ð10Þ

k

XXX v NV ijk 6 TV v i

j

8v 2 V

ð11Þ

k

X v

155

uv NV vijk 6 kvijk

8i 2 I;

X ijk ; X 0ijk ; Mijk ; NV vijk 2 Z þ

rj ; cj 2 f0; 1g

j 2 J;

Integer

8j 2 J

k2K

8i 2 I;

ð12Þ j 2 J;

k 2 K;

v 2V

ð13Þ ð14Þ

The first objective function (Eq. (1)) maximises the number of evacuees at assembly point i who must then be transferred to shelter j in the minimum time possible via the safest route. The second objective function (Eq. (2)) minimises the total number of designated shelters and rescue vehicles. The goal is to decrease the cost of allocating new facilities and utilising the minimum number of shelters and rescue vehicles needed for the evacuees. w1 and w2 are utilised as auxiliary coefficients in the weighted sum method, and are selected in the range of [0–1], summing to one to represent the weight of impression for each parameter. In Eqs. (3) and (4) are embedded in the model as the maximum number of shelters covering location–allocation constraints. Hence, constraint (3) restricts the number of allocated shelters to the maximum number available. Constraint (4) guarantees that the candidate shelter must first be both functioning and available in order to be allocated. Constraint (5) is a shelter capacity expansion constraint, and ensures that people at affected point i will only be evacuated and transferred to shelter j if shelter j is accessible and available. In addition, since each capacitated shelter is expected to serve the maximum number of late evacuees, it is assumed that each rescue vehicle will always travel directly between its assembly point and the designated shelter, rather than travelling to additional places for further evacuee boarding (An et al., 2013). This constraint therefore ensures that the number of evacuees transferred to the designated shelter j will not exceed the shelter’s capacity. Constraint (6) ensures that evacuees at assembly point i will only evacuate and transfer to shelter j if there is an accessible and available road connection between the origin and destination. Eq. (7) links transferred evacuees and assigned vehicles. This constraint mandates that the maximum evacuation time for a round trip of each vehicle from assembly point i (2  sijk ) must not exceed the available clearance time. The evacuation time for each trip has two components. The clearance time will be chosen based on minimum values of these two critical components – the route and the township availability. This constraint also ensures that each vehicle cannot carry more late evacuees than its seating capacity. In Eq. (8), X 0ijk is a dummy variable defined to compute the number of unoccupied seats in the last trip of assigned vehiclens. Eq. (8) therefore mandates that number of required trips to transfer the evacuees (Mijk ) is integer. Eq. (9) ensures that unoccupied seats in the last trip are less than the total capacity of assigned vehiclens. Upon realisation of infrastructure disruptions, the population of the area in each of the evacuee assembly points is determined. The quantity of late evacuees are then allocated to the designated shelters j, denoted as cj . Constraint (10) therefore ensures that the number of late evacuees to be transported from an assembly point i to the shelter j does not exceed the population of the area that each assembly point represents. Constraint (11) mandates that the total number of assigned rescue vehicles is TV v (the total number of available vehicles for the entire evacuation process). Constraint (12) is the route passage capability constraint, and limits the maximum number of assigned vehicles to each route based on the capacity of that route. Constraints (13) and (14) are non-negativity and integerality constraints. Constraint (13) expresses that negative numbers could not feasibly be considered as variables. Constraint (14) restricts the assignment of shelters and transferring issues to binary values, as rj ; cj are either allocated or not. 3.7. Route reliability module As noted earlier, there might be k different routes between assembly point i and the selected shelter j. Furthermore, as depicted in Fig. 2, each route may contain several segments that can be denoted as:

K 1;5;1 ¼ fl24  l25 g .. . K 1;5;2 ¼ fl20  l17  l16  l14 g .. . K i;j;k ¼ fli      lo      lk g

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Fig. 2. An overlay of the transportation network with the disruption risk map for the bushfire-affected Murrindindi Shire area on Black Saturday. Each road between the townships contains several segments that are indicated by dashed lines and numbered with an index of ‘‘l”. The arterial roads are numbered in the same manner and are highlighted as bold black lines. The disruption risks of the routes are marked in three colours. Red indicates a high-disruption risk, while orange and green respectively denote medium- and low-disruption risks. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Each evacuation plan (EP) may include a set of optimal progress routes and can be denoted as EP i ¼ fK i;1;1 ; Ri;1;2 ; . . . ; Ri;j;k g (Table A1, Appendix). In this research, we have adapted the VicRoads1 bushfire risk assessments as the source for determining the disruption risk for each route (Table A2, Appendix). Each segment in each route has a specific risk of disruption as measured by VicRoads. The output of the risk assessment is illustrated by the application of a standard deviation approach to the classification of risk levels. Fig. 2 classifies the disruption risk into three groups based on the level of risk: low-risk roads (marked in green), moderate-risk roads (marked in orange), and high-risk roads (marked in red). Based on the length of each road segment, the weighted average sum method is applied to calculate the cumulative risk of routes as follows:

l ijk ¼

PL

l

l¼1 ð l  PL l¼1 Dl

Dl Þ

8i 2 I;

j 2 J;

k2K

ð15Þ

l ijk denotes the average capacity of route k for vehicle type v, ll denotes the risk rating assigned to section l in route k, and Dl denotes the geometric distance of section l in route k (Table A3, Appendix). 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 focused registration and licensing services.

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3.8. Route passage capability The road passage capability plays a key role in the appropriate distribution of vehicles to prevent traffic congestions and reduce in vehicle’s speed. Evacuation plans must consider transportation network capacity restrictions and should account for evacuating passenger vehicles and transit buses if they are involved in evacuation (Cox, 2006). In this research, the weighted sum method is used to precisely evaluate passage capacity of the transportation routes as follows:

kv ¼ ijk

PL

v

l¼1 ðkl  PL l¼1 Dl

Dl Þ

8v 2 V;

k2K

ð16Þ

where  kvijk represents the average capacity of route k for vehicle type v, kvl is the previously predicted capacity of section l of

route k for vehicle v, and Dl denotes the geometric distance of section l for route k. It should be mentioned that in calculation of overlapping routes capacity, the minimum capacity of the route with overlapping segments is considered for all the common routes (Table A3, Appendix). 3.9. Route resistance function During emergency evacuation, timing is cited as the most critical issue. When traffic increases, the travel time increases accordingly. This in turn may slow down the evacuation process (Zhao et al., 2010). There is a wide range of road congestion modules which can be utilised (for example see (Sisiopiku et al., 2004; Han and Yuan, 2005) to tackle different levels of road traffic. An appropriate route resistance function, for instance, should reflect the resistance to traffic flow across various transit components that may direct or indirectly impact on the efficiency of bushfire evacuation. The following formulation is therefore applied to calculate the inflation of time factor:

sijk  ð1 þ TÞ þ DT

ð17Þ

where sijk denotes travel time of route k between assembly point i and shelter j, T defines percent time inflation of trips due to road congestions (time impedance factor) and DT represents the dwell time. DT consists of the time lost prior to opening and after closing the transit vehicle doors, and the time required for boarding/alighting of passengers at most heavily used doors. Factors affect the calculation of dwell times include: vehicle floor height and platform height, number of boarding/ alighting channels (doors), and fare type and fare collection. Vuchic (2005) developed a formulation to calculate the dwell time as below: 0

DT ¼ t o þ b xb þ a0 xa

ð18Þ

0

b ¼ ðbi =n0 Þ  Xb 0

where t o denotes the time lost before/after doors are opened. b denotes the number of boarding riders through the most heavily used boarding door. a0 is the number of alighting riders through the most heavily used alighting door. xb defines boarding times per person. xa denotes alighting times per person. n0 represents the number of doors per vehicle and Xb is a coefficient of distribution among doors. It should be noted that in emergency evacuation cases, fares are not collected; thus, the time spent for boarding and alighting per person should be lower than the standard values. However, due to the chaos coupled with emergency evacuation xa and xb are assumed to be, at least, equal to the typical values. It is worth noting that in case of emergency evacuation, passengers would be either boarding at evacuees’ pickup points or alighting at safe destinations. On average, the dwell time (vehicle preparation boarding/alighting of evacuees) takes no longer than 12.5 s/ person in the emergency evacuation (Kittelson et al., 2003). In this research, it has therefore assumed that DT is known and pre-defined. 3.10. Availability of routes (time windows) module The impact of bushfires varies differently on various segments of a transport route depending upon the intensity, direction and vulnerability. Therefore, each segment of a route has different time window, within which evacuation has to be carried out. Hence, once bushfire front hits a segment, all the routes containing the impacted segments will be no longer accessible (e.g. in the following network, by blockage of l3 , all the egress routes from assembly point i2 to shelters are influenced and disrupted (Fig. 3). The behaviour of bushfire propagation across a geographic area is highly unpredictable. Generally, Bushfire Spread Rate (BSR) is influenced by three fundamental factors as below:

BSR ¼ fðfuel; topography; weatherÞ More importantly, fire outbreak changes in each element (e.g., wind speed or direction vegetation density, slope) can extremely deteriorate the consequences. Most developed modules also have adapted in a way to be applicable in their respective case study region (due to strong topographical and GIS related settings (see (Papadopoulos and Pavlidou, 2011) for a comprehensive review on wildfire simulators). In this study, we simulate the propagation of the 2009 Black Saturday bushfires

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Fig. 3. Definition of timely impact of bushfire spread on the route segments.

using the historical data provided by the Victorian Bushfire Royal Commission (VBRC). Therefore, the minimum time window for each segment on a route is considered as the total time window of the route. This is indicated in Eq. (19) as follows:

TWRijk ¼ MinfTWRl g

8i 2 I;

j 2 J;

k 2 K;

l2L

ð19Þ

where TWRijk denotes the time window (availability time) of route k between assembly point i and the designated functioning shelter j. TWRl denotes time window of segment l for the same route (Table A3, Appendix).

4. Solution approach In this study, the e-constraint approach has been chosen as an efficient method to solve the research problem. Using the econstraint method, the main objective of the problem will be optimised while the remaining objectives are converted into inequality constraints, by assigning allowable levels of epsilon as upper or lower bound target values. All other objective functions will be minimised while constrained to the first objective function value. By varying the constraint bounds, a rich representation of the efficient elements of the Pareto front can be obtained. This iteratively solves single-objective versions of the multi-objective problem, with additional e-constraint, in order to enumerate all Pareto-optimal solutions. In mathematical terms, when the decision-maker lets f n ðxÞ be the objective function selected from among N objective functions to be optimised, the multi-objective problem is transformed as follows:

Maxx2u s:t:

f n ðxÞ f p ðxÞ P ep

8p 2 f1; 2; . . . ; Ng n fng

where u defines the feasible solution space of the problem.

ð20Þ

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4.1. Solution framework Algorithm 1 illustrates a general solution framework for the bi-objective optimisation problem. All Pareto-optimal solutions to a bi-objective minimisation problem are calculated and stored into set Z. The algorithm frequently runs in order to solving single-objective forms of the multi-objective problem computing all Pareto-optimal solutions. In each run, the mixed-integer multi objective problem M.I.M.O.P is solved a first for the first objective function ðf 1 Þ, using an additional constraint bounding the second objective ðf 2 Þ, so-called as e-constraint. The value of e is also considered small enough in comparison to the differences between the values of objective functions f 1 and f 2 along the Pareto front. Afterward, using the new found optimal value for f 1 , f 2 is optimised. The generated Pareto-optimal solution is thus computed, and added to the solution set Z. The e-constraint then is adjusted so that the next point to be compiled is optimal than all previous results regarding the second objective (f 2 ). Algorithm 1. Bi-objective

e -constraint solution approach

1: Set 2: Z = ø f2 6 1 3: e-constraint 4: Insert e-constraint to M.I.M.O.P 5: While there is no feasible solution for M.I.M.O.P, do: 6: x Maxðf 1 Þ 7: Objective.Bound f 1 ¼ f 1 ðxÞ 8: Insert Objective.Bound to M.I.M.O.P 9: x Minðf 2 Þ 10: Z Z [ fxg 11: Clear Objective.Bound from M.I.M.O.P 12: Amend e-constraint f 2 6 f 2 ðxÞ  e 13: End while Output: Set of Pareto-optimal solution Z

As noted above, the main objective of the model is to evacuate the maximum number of people within the short time windows. Therefore, the first objective ðf 1 Þ is selected as the main objective to be optimised. While the second objective (minimising the resources), ðf 2 Þ, is converted into a hard constraint by applying the mentioned e-constraint algorithm. 5. Case study – Murrindindi Mill fire Black Saturday The Shire of Murrindindi is located approximately 100 km north-east of Melbourne in Victoria, Australia. It covers an area of 3889 km2. At the 2011 Census, the shire had a population of 41,860 with a population density of 3.5 people/km2. It includes the towns of Alexandra, Buxton, Eildon, Flowerdale, Kinglake, Marysville, Molesworth, Strath Creek, Taggerty, Yarck and Yea. 46% of the total land area of the Municipality is forest (1788 km2), which includes State Forest, Parks and Reserves and other public land. A large proportion of this land is mountainous and densely forested (Fig. 4). The shire experienced a series of bushfires on 7 February, 2009, the Black Saturday, with the first fire starting in the north of a saw mill in Wilhelmina Falls Road in Murrindindi at approximately 2.55 pm. The bushfire traversed rapidly and by 4:30 pm it reached Narbethong, a distance of more than 50 km. Following a wind change that occurred at approximately 6:15 pm, the fire swept through the communities of Marysville, Buxton and Taggerty. The bushfires burnt 168,542 ha of land (40% of the Shire) and, disrupted Melbourne’s water reservoirs, among many other things. The Murrindindi bushfire resulted in 40 deaths, 71 casualties, and the evacuation of more than 500 people, mainly in the areas of Marysville, Narbethong and Buxton (Fig. 5). 5.1. Network data The study area compromises six main townships where were under fire risk including Narbethong ði1 Þ, Marysville ði2 Þ, Taggerty ði3 Þ, Buxton ði4 Þ, Cambarville ði5 Þ, and Rubicon ði6 Þ). Approximately 2160 people (Table 1) were to be evacuated from major townships. Approximately one-half (51%) of residents affected by the Black Saturday bushfires were classified as late or very late evacuees (Whittaker et al., 2009). In this study, we assumed that the number of late evacuees is half of the population of the affected townships, which is about 1100 (Table 2). Five shelters in adjacent townships (destinations) are nominated by Country Fire Authority (CFA). They are Goughs Bay Fire Station in Alexandra ðj1 Þ, a recreation reserve oval in Thornton ðj2 Þ, a basketball court in Eildon ðj3 Þ, a skate park in Yea ðj4 Þ, and a racecourse track in Yarra Glen ðj5 Þ. The shelters are identified on a wide range of criteria, including capacity, accessibility, vulnerability, and availability (CFA, 2014). Each shelter has specific capacity and associated constraints to safely shelter the evacuees (Table 2).

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Fig. 4. Case study region – Murrindindi Shire, Victoria, Australia.

Two types of capacitated rescue vehicles (buses and vans) are considered for late evacuation. Buses are capable of carrying up to 45 people, with vans having a maximum capacity of 15 passengers and being an alternative for smaller-sized evacuation demands. Route travel time and rescue vehicle passenger capacity are critical factors in emergency evacuations, having a drastic impact on the performance of time–space networks. Travel time and rescue vehicle passenger capacity between any two nodes on the proposed network are derived from actual geographical data and travel speed zones maps (VicRoads, 2014), input into the route reliability and passage capability modules. The VicRoads speed zones map (VicRoads, 2014) indicates that the travel speed for the areas in the entire case study is 100 km/h. The time windows, bushfire spread directions and disruptions data have all been derived from the recorded figures for the 2009 Black Saturday Murrindindi bushfire (Victorian Bushfires Royal Commission Report, 2009). The time impedance parameter is set as 10% for each traversed trip in order to allow for the impact of traffic congestion (e.g. early evacuees, rush hour, and collisions). We parameterised the proposed model based on this case study and implemented it using CPLEX solver 12.6 on a PC with a CPU capacity of 3.40 GHz and 8 GB RAM. 6. Display of results 6.1. Sensitivity analysis of optimal evacuation plans by number of functioning shelters There is no report that indicates an inadequate number of vehicles being available in the region on Black Saturday in 2009. We therefore assume the maximum number of available evacuation (rescue) vehicles, TV V , is TV Bus ¼ 20 and

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Fig. 5. Murrindindi 2009 Black Saturday bushfire propagation map. The fire ignited on the north-western side of the area adjacent to the Murrindindi Sawmill at approximately 2:55 pm. The grey and yellow arrows show the wind direction on Black Saturday. The white and yellow arrows respectively show bushfire direction before and after the wind direction change that occurred at approximately 6:30 pm. After around 9 h, the fire reached nearby Rubicon. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 1 Demographical information of the case study area (ABS, 2011). Affected townships

Narbethong Marysville Taggerty Buxton Cambarville Rubicon Total

Population

Families

Private dwelling

Vehicle per private dwelling

Age

Carless

One

Two

Three+

Average

Median

Children (0–14)

Old (65+)

474 518 330 257 220 361

130 143 91 64 50 99

288 192 226 161 141 225

5.8% 3.5% 7.2% 6.5% 4.5% 8.2%

24.6% 34.3% 29.3% 28.3% 39.3% 36.7%

38.0% 43.3% 41.4% 48.4% 36.0% 38.8%

31.6% 18.9% 22.1% 16.8% 20.2% 16.3%

2.2 1.8 2.1 1.9 1.8 1.9

47 51 53 45 40 51

15.6% 15.9% 15.7% 13.6% 18.0% 17.4%

17.8% 27.7% 19.3% 23.7% 8.6% 26.8%

2160

577

1233

5.95%

32.08%

40.98%

20.98%

1.95

48

16.03%

20.65%

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Table 2 The case study population and capacity data.

i1 i2 i3 i4 i5 i6

Assembly points [i]

Population [Di]

Time windows [TWi]

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]

Capacity ½Capj 

Alexandra Thornton Eildon Yea Yarra Glen

1500 500 500 1000 1000

TV Van ¼ 15. Furthermore, based on Black Saturday historical data, in this case study the total number of candidate shelters is five (X ¼ 5). However as mentioned earlier, determining the optimal allocation of reliable shelters significantly influences the evacuation routing arrangement pattern. Hence, a sensitivity analysis of the optimal evacuation plans (e.g. optimal transportation, routing and trips) in relation to the maximum number of available shelters X is provided in Table A4 (Appendix). As shown in Table A4 in Appendix, the optimal number of shelters is four (X ¼ 4), where five buses and twelve vans are used. Fig. 6 illustrates the optimal deployments and the evacuee-to-shelter allocation results, with functioning shelters indicated by green triangles on the map. Each assigned vehicle follows its route, and the number of required trips is listed above each vehicle. Second routes are highlighted via dashed lines. The results show that the high capacitated shelters have been used most frequently to centralise the routing plan distribution. In addition, we observe that heavily-populated areas tend to be evacuated to denser shelters (for example, Marysville ði2 Þ is mostly evacuated to Alexandra ðj1 Þ and Thornton ðj2 Þ). Accordingly, the dispersion of vehicle trip patterns are more centralised (routing via single route alternatives), as the number of shelters decreases. Contrary to our insights, it is interesting to note that increasing the number of shelters does not drastically increase vehicle assignment. It indicates that for as long as possible, the model resists the spread disruption of evacuees by increasing the number of shelters (which, of course, increases the expected number of required travels) (Fig. 7). We note that for as long as possible, the model tends to decrease the number of assigned vehicles according to the sufficiency of time to travel (which, of course decreases the objective function). For example, in the case that X ¼ 4, Rubicon late evacuees are planned to be moved by only one van travelling seven times between Rubicon and Eildon, due to sufficient time being available. The results also show that as the number of available shelters ðXÞ increases, the number of rescue vehicles tends to increase slightly (i.e. when X ¼ 2, 3, 4, and 5, the respective number of assigned vehicles are 6 buses–13 vans, 7 buses– 11 vans, 5 buses–12 vans, and 6 buses–12 vans). However with X ¼ 1, more available vehicles (10 buses–11 vans) are used to reduce the risk of disruptions for longer trips from more distant townships. Interestingly, in this case, in order to avoid

Fig. 6. Optimal routing evacuation plan via the assignment of four functioning shelters, five buses and twelve vans. In the figure, the evacuation routes for each township are depicted by different colours. A solid line indicates that ‘‘route 1” is chosen for the transportation of evacuees, while the dashed line is used as a symbol of ‘‘route 2”. In the left figure, the numbers above each assigned vehicle (a bus or van) shows the number of trips required to evacuate the entire assigned evacuee population to each of the shelters shown in front of the vehicles. The right figure illustrates the evacuation plan in its real context.

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Fig. 7. Optimal routing pattern by different number of functioning shelters. Different line colours are utilised for paths to show the evacuation paths of each township. In each scenario, the evacuation process is limited to the number of available functioning shelters as indicated in the pattern above each.

traffic congestion and reduce the rescue fleet traversal speed, vehicles are distributed across most alternative routes in the network. Using this approach, all the evacuees could be transported before the bushfire disruptions. In addition, the shelters further away are backed up by those closer, to be utilised only in the case of contingent functioning failures of closer shelters. (e.g., the Yea shelter is not allocated in cases where X 6 4). In reality, a fewer number of shelters is more manageable. However, in the event of a severe bushfire, contingent outbreak failures and disruptions in resources and infrastructures may exacerbate the evacuation mission. Therefore, such a strategy may not be appropriate. This finding highlights the need to consider backup shelters in emergency evacuation planning. This finding also highlights the need for a rigorous system-level approach to transit-based evacuation system planning, rather than relying on intuition. Furthermore, to investigate the impact of vehicles on the optimal number of required shelters, another sensitivity analysis has been conducted. For simplicity and illustration, it is assumed that only buses are used. The results shows that the optimal number of required vehicles initially increased in different scenarios (changing the number of X), then subsequently became independent. While via utilising fourteen buses the optimal number of required shelters remained as four, however the objective function increased to 73.9. In addition, as illustrated in Fig. 8, any further increase in the number of buses has not affected the evacuation plan of the entire impacted network. Sensitivity analysis with respect to the time impedance has also carried out on the optimal evacuation plan to investigate the effect of traffic on evacuation plan. Interestingly results show that minimum number of required vehicles to evacuate all the evacuees increases by time impedance factor. In case that there is no traffic in the network (T = 0), minimum 13 buses can evacuate all the evacuees while by T = 80% at least 17 buses are required to cover all evacuees (Fig. 9)

Sensitivity analysis of number of shelters with T.V 1100

Number of transported people

1000 900 800 700

Ω=1

600

Ω=2

500

Ω=3

400

Ω=4

300

Ω=5

200 100 0

0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20

Number of required buses Fig. 8. Sensitivity analysis of the optimal number of functioning shelters with the total number of one vehicle type ðTV bus Þ. Different colours are used to depict the related graphs based on the number of available shelters ðXÞ. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Sensitivity analysis of time Impediance factor 1100

Number of transported people

1000 900 800 700

T= 0%

600

T= 20%

500

T= 40%

400

T= 60%

300

T= 80%

200 100 0

0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20

Number of required buses Fig. 9. Sensitivity analysis of the time impedance factor with the total number of one vehicle type ðTV bus Þ. Different colours are used to depict the related graphs based time impedance factors ðTÞ. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Sensitivity analysis of Dwell Time with T.V 1100

Number of transported people

1000 900 800 700

DT = 7.5"

600

DT = 10"

500

DT = 12.5"

400

DT = 20"

300

DT = 30"

200 100 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20

Number of required buses Fig. 10. Sensitivity analysis of the dwell time factor with the total number if required buses. Different colours are used to show related dwell times. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

To show the effect of dwelling time on the overall out of model, a sensitivity analysis is carried out based on the selection of various dwelling times. As shown in Fig. 10, in this case with the dwell time values around the considered average dwell time (12.5 s/person) there is no significant difference on the overall output. However, in some cases higher dwell time values may lead in significant delay in the evacuation process. For instance, in the worst case by considering 30 s/person as dwell time, there is a significant increase in number of required buses to cover all the evacuation demands. Furthermore, in order to analyse the effect of different evacuation demands on the output plan, a sensitivity analysis is conducted with lower and higher population (Fig. 11). As it is shown, increase in population has a direct relation to the number of required buses to evacuate all the population within the predefined time windows. 6.2. Disruption analysis In a bushfire emergency evacuation, there are key parameters, such as wind direction and speed, the accessibility and availability of resources, and evacuee behavioural factors, that all may heavily impact the evacuation process. Among these parameters, abrupt variation in the availability and accessibility of resources is known to be the most significant issue that

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Sensitivity analysis of Lower population with T.V 2100

Number of transported people

1950 1800 1650 1500 1350 1200 1050 900

T.V

750 600 450 300 150 0

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

Number of required buses Fig. 11. Sensitivity analysis of the dwell time factor with the total number if required buses.

can be mitigated by applying an appropriate decision tool. Hence, the two following scenarios are designed to evaluate the reliability of the model in generating an optimal evacuation plan in the case of unforeseen road disruptions and shelter availability. 6.3. Disruption in the central shelter availability In our case study, most late evacuees had been evacuated to Alexandra. However, due to bushfire spread direction and acceleration, emergency services were uncertain of the availability/accessibility of the Alexandra township. Given that shelter in the Alexandra township plays such a key role in serving evacuees in our model, we are interested to test the model for the eventuality that Alexandra is not available. Fig. 12 depicts the percentage of usage capacity of the designated shelters for each such case. In this case, the other four shelters are assumed to remain available. The optimal plan to save all the evacuees within the rigid time windows is to route late evacuees to the three shelters in Thornton ðj2 Þ, Eildon ðj3 Þ and Yarra ðj4 Þ, as prescribed in Table 3. As Table 3 demonstrates, due to the unavailability of Alexandra ðj1 Þ, the objective function is increased by 7%. This result highlights that the assignment of fewer shelters does not inevitably decrease the total cost function, although it may slightly

Shelter capacity usage

Total population coverage

100%

10%

8% 18%

45%

75%

8% 10% 10%

44%

Yea 50%

38% 100%

37%

Yarra Eildon

55%

25%

0%

Ω=1

Ω=2

Thornton 46%

Ω=3

36%

35%

Ω=4

Ω=5

Alexandra

Shelter deployment Fig. 12. The result of shelter capacity usage dispersion. The y-axis denotes the total evacuee population covered by the utilisation of different numbers of available shelters in different scenarios (x-axis). The usage capacity of each shelter under each scenario is indicated on the related bar graph and plotted in a different colour. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

166

Number of available shelters

X=4

Optimal assignment of the resources

Assigned shelters –Thornton –Eildon –Yarra Number of vehicles –Bus –Van Objective function value 56.85

From

6 12

To

Evacuees routing plan

Number of required vehicles

Route 1

Route 2

Route 3

Vehicle

Route 1

Route 2

Route 3

Route 1

Route 2

Route 3

Bus Bus Van Van Bus Van Van Van Van Van Bus Van Van

1 1 2 2 1 – 1 1 1 – 1 1 –

1 – – – – 2 – – – 1 1 – 1

– – – – – – – – – – – – –

2 1 3 5 2 – 6 3 7 – 2 7 –

2 – – – – 7 – – – 5 1 – 6

– – – – – – – – – – – – –

Narbethong

Thornton Yarra

80 100

60 –

– –

Marysville

Thornton Eildon

75 85

– 100

– –

Buxton

Thornton Eildon Thornton Eildon Eildon Thornton Eildon

85 45 100 – 65 100 –

– – – 70 45 – 90

– – – – – – –

Taggerty Cambarville Rubicon

Number of trips

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Table 3 Optimal evacuation plan when Alexandra (j1 ) is unavailable.

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Fig. 13. Optimal evacuation plan under a central shelter (Alexandra ðj1 Þ) disruption. In the figures, the evacuation routes for each township are depicted by different colours. A solid line indicates that ‘‘route 1” is chosen for the transportation of evacuees, while the dashed line is used as symbol of the ‘‘route 2”. In the left figure, the numbers above each assigned vehicle (bus or van) shows the number of required trips to evacuate the entire assigned evacuee population to each path shown in front of each vehicle. The right figure illustrates the evacuation plan in its real context.

increase the number of used vehicles. The illustration of the evacuation disruption in Fig. 13 demonstrates that the unavailability of Alexandra has not significantly impacted the optimal routing pattern. 6.4. Disruptions in both major road and shelter It is vital to evaluate how our model reacts against different levels of road accessibility. The results show that more than half of the evacuees (59 per cent) could be evacuated to the designated shelters in northern parts of the region via Maroondah Highway (B360 that is coded as link l17 in Fig. 2). Maroondah Highway and its related road links thus play a critical role in the evacuation. Let us assume that Maroondah Highway is no longer accessible. We set X ¼ 4, and infinite vehicles in order to consider the extreme case. In contrast with what we expected, the result shows that the optimal result is to use three out of four functional shelters, as shown in Table 4. In addition, the objective function increases due to an increase in the number of allocated vehicles (14 buses-10 vans). This may be partially due to the longer routes. Therefore, this indicates that neglecting the vulnerability of contingent risks (e.g. a collision, or network disruption), which are prevalent in most of the past incidents, may provide sub-optimal allocation solutions. Fig. 14 shows how the evacuee dispersion and routing change with varying degree of disruptions. Fig. 14, moreover, shows the results of the scenario where the main road is disrupted by bushfire. It demonstrates that the denser and higher capacity shelters (in this case Alexandra and Eildon) can accommodate more evacuees, while demanding more emergency vehicles. The modelling results indicate the importance of real-time information about the shelters and route disruptions, which has the potential to influence the optimised solution. A number of available shelters should be allocated adjacent to the central bushfire-prone shelters, while the more bushfire-prone shelters can be allocated closer to the impacted region’s boundaries. The model tends to allocate shelters in areas of high demand. In addition, the allocated shelters are likely to support each other when unforeseen failures occur in the coordination of emergency evacuation. 7. Conclusion In this research, an optimisation evacuation decision support tool was developed for short-notice emergency evacuation problems. Key assumptions and constraints were derived using a real case study of the 2009 Black Saturday bushfires in Victoria. A bi-objective integer-programming problem was formulated which simultaneously maximised the number of evacuees and minimised resource utilisation. The model was solved using the e-constraint method. Sensitivity analysis was carried out to test different bushfire scenarios, using a combination of shelter accessibility and road disruptions. The results indicated that the short-notice evacuation of late evacuees in the case of the Black Saturday bushfires could be operationalised using existing resources. Sensitivity analysis demonstrated the possibility for timely evacuation of all late evacuees from the region, even under stringent constraints of shelter and road disruption. In all scenarios, the town of

168

Number of available shelters

X=4

Optimal assignment of the resources

Assigned shelters –Alexandra –Eildon –Yarra Number of vehicles –Bus –Van Objective function value 92.85

From

To

Evacuees routing plan

Number of required vehicles

Route 1

Route 2

Route 3

Vehicle

Route 1

Route 2

Route 3

Route 1

Route 2

Route 3

Bus Bus Van Bus Van Bus Bus Van Bus Bus Van Van Bus Van Van Bus

– 1 1 – – – 2 1 – 2 – – – – 1 1

– 1 – – – 2 – – – – 1 – 2 1 1 –

1 – – 1 2 – – – 1 – 1 1 – – – –

– 2 1 – – – 1 1 – 1 – – – 1 3 2

– 2 – – – 1 – – – – 7 – 1 – 4 –

1 – – 1 1 – – – 1 – 5 1 – – – –

Narbethong

Eildon Yarra

– 105

– 90

45 –

Marysville

Alexandra

–

–

70

Eildon Yarra

– 100

90 –

– –

Alexandra Yarra Alexandra Alexandra Eildon

– 90 – – –

– – 100 – 100

40 – 70 10 –

Alexandra Eildon

40 90

60 –

– –

14 10 Buxton Taggerty Cambarville

Rubicon

Number of trips

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Table 4 Optimal routing plan under intense disruption.

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Fig. 14. Optimal evacuation plan under both road and shelter disruptions. It is assumed that the main road segment is disrupted and no longer accessible as well as the Thornton township (j2 ). In the figures, the evacuation routes for each township are depicted by different colours. A solid line indicates that ‘‘route 1” is chosen for the transportation of evacuees, while the dashed line is used as symbol of the alternative ‘‘route 2”. In the left figure, the numbers above each assigned vehicle (bus or van) show the number of required trips to evacuate the entire assigned evacuee population to each path shown in front of each vehicle. The right figure illustrates the evacuation plan in its real context. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Alexandra plays a key strategic role in providing refuge for evacuees, because of its geographic centrality in the region. In addition, Maroondah Highway is identified as the widely used arterial link, capable of connecting different parts of the region. The results show that more than 59 per cent of short-notice Black Saturday evacuees could have being effectively transferred to key shelters via Maroondah Highway, due to its high capacity, high-speed road connectivity. Further, the results imply that the selection of shelter be based on lower risk road networks, even if there are other shelters which require shorter travel time. The model has been proven reliable in generating optimal routings and evacuation solutions for problems involving complex evacuation scenarios. In each scenario, the model identified the shortest and safest routes to transfer late evacuees in bushfire- affected areas within the set time frame, taking into account road accessibility and the available resources. The model has optimally assigned the available shelters to absorb all 1100 late evacuees from the affected areas. The findings demonstrate that short-notice evacuation is manageable via advanced planning and the efficient allocation of limited resources. The model is capable of defining the optimal allocation of resources to evacuees in any emergency evacuation planning objectives. An important consideration is to ascertain the minimum number of safe shelters in adjacent areas to the affected townships. In addition, the optimal number of rescue vehicles required for each affected area can be assigned by the model. The model then assigns the safest available routes to reach the allocated shelters, taking into consideration the pre-defined bushfire spread rate. Ranking the usage of the assigned routes and resources also provides insights on developing risk mitigation plans for each road link in the network, as a part of the emergency evacuation plan. However, similar to other research, it should be noted that there are a number of key limitations. Complexity in simulating people’s behaviour and bushfire propagation are the key limitations that may influence an entire evacuation process. The evacuees’ decision behaviour and actions can significantly impact the loads and the allocation process. Despite these limitations, we argue that the generated model provides a decision support tool for bushfire agencies to help analyse the modelling outcomes under various possible scenarios. To strengthen the proposed model future research will consider stochasticity in parameters, which could offer richer and deeper insights into evacuation behaviour. Relaxation of other restrained assumptions, such as rescue flow allocation and direction, may also be appealing and lead to the development of a much more versatile decision tool. Considering shadow evacuation and background traffic as a key caveat to the emergency evacuation modelling could be an interesting future study as well. With appropriate model calibration and adjustment, this modelling approach could also be applied to other disasters such as flooding and cyclones, which are also widely prevalent in Australia and other countries. Appendix See Tables A1–A4.

170

Table A1 Network information. From

To

Route 1 Road segments (links)

Narbethong Yea

74

85

0.54

30

34

47

0.55

240

30

38

0.55

240

Eildon

Taggerty

112

63

0.5

30

48

52

0.52

195

40

45

0.52

195

41

52

0.56

240

51

58

0.54

195

36

35

0.48

270

61

62

0.53

90

L19-L18-L17L16-L14-L39L40-L42 Alexandra L19-L18-L17L16-L15 Thornton L19-L18-L17L12-L6 Eildon L19-L18L17L12-L6-L2 Yarra L21-L24-L25

71

71

0.5

270

L19-L18-L17L16-L15-L41L42 L19-L18-L17L12-L6-L13 L19-L11-L8L7-L6 L10-L1

57

73

0.5

240

L19-L18-L32L33-L37

34

42

0.5

270

41

45

0.5

240

30

33

0.49

270

124

53

0.49

240

41

46

0.52

270

19

17

0.41

280

46

47

0.51

130

L21-L24-L26L28-L27

71

74

0.54

90

Yea

L17-L16-L14L39-L40-L42 Alexandra L17-L16-L15

59

59

0.5

300

37

49

0.51

30

0.51

500

29

33

18

21

0.49

400

33

L17-L12-L6-L2

29

34

0.53

345

L16-L15-L41L42 L17-L12-L6L13 L17-L16-L15L13 L17-L12-L6-L2

22

Thornton L17-L12-L6 Eildon Yarra

L20-L24-L25

48

53

0.53

240

37

49

0.51

500

14 10 21 56

18 9.4 21 64

0.52 0.5 0.62 0.52

500 400 500 130

76

89

0.49

2;40

53

58

0.48

240

L20-L17-L12-L6L2 L24-L25

Yea

Yea

L16-L15-L41L42 Alexandra L16-L15 Thornton L12-L6 Eildon L12-L6-L3 Yarra L17-L20-L24L25

Cambarville Yea

L10-L19-L18L17-L16-L15L41-L42 Alexandra L10-L19-L18L17-L16-L15

L22-L30-L35L36 L21-L19-L18L17-L16-L14 L21-L19-L18L17-L12-L6 L21-L19-L18L17-L12-L6-L2 L24-L26-L28L27

Road segments Distance Disruption Time (km) risk windows (links) (min)

Travel time (min)

80

0.53

130

50.7

0.55

130

65.1

0.51

195

90.5

0.41

195

99

42.9

0.55

90

140

69.2

0.45

50

L19-L11-L8-L7135 L6-L13 L19-L18-L1749 L16-L14-L13 L19-L11-L8-L7135 L6-L2 L21-L23-L28-L27 106

65.5

0.5

240

52.8

0.5

240

66.5

0.51

240

58.8

0.58

90

500

L32-L33-L37

75

34.4

0.5

50

0.5

320

L32-L38-L39

112

52.3

0.48

270

42

0.51

30

62.3

0.47

270

29

34

0.53

320

54.4

0.53

320

L19-L21-L24l25

67

66

0.7

195

L18-L11-L8-L10- 127 L6 L17-L16-L1448 L13-L2 L20-L23-L28-L27 108

64.1

0.6

90

L16-L14-L39L40-L42 L16-L14 L16-L15-L13 L12-L6-L2 L17-L20-L24L27-L28-L27

51

48

0.5

500

L17-L32-L33-L37 136

69

0.45

50

18 25 21 78

17 31 23 92

0.51 0.52 0.56 0.55

500 500 500 90

L12-L6-L13 L16-L14-L13 L16-L15-L13-L2 L16-L15-L42L41-L36-L29-L27

21 29 36 78

21.8 29.4 43.9 107

0.52 0.52 0.55 0.53

500 500 500 90

101

0.48

300

L10-L19-L18L23-L33-L37

179

81.6

0.46

50

64

0.46

300

L1-L2-L13

154

87.5

0.43

345

L9-L8-L7-L12- 163 L16-L15-L41L42 L9-L8-L7-L6- 145 L13

L20-L17-L1763 L14-L41-L42 L20-L17-L12-L641 L13 L21-L19-L11-L8- 134 L7-L6 L21-L10-L1 161

Distance Disruption Time (km) risk windows (min)

L23-L7-L27

S. Shahparvari et al. / Transportation Research Part E 93 (2016) 148–176

Buxton

Route 3

Road segments Travel Distance Disruption Time time (km) risk windows (links) (min) (min)

L24-L26-L28L29-L36 Alexandra L20-L17-L16L15 Thornton L20-L17-L12-L6

Yarra Marysville

Route 2 Travel time (min)

Table A1 (continued) From

To

Route 1 Road segments (links)

Rubicon

Route 2 Travel time (min)

Route 3

Road segments Travel Distance Disruption Time time (km) risk windows (links) (min) (min)

Road segments Distance Disruption Time (km) risk windows (links) (min)

Travel time (min)

Distance Disruption Time (km) risk windows (min)

49

50

0.46

240

L9-L8-L7-L6

134

52

0.45

300

L1-L2

143

75.1

0.41

345

60

63

0.49

240

L1

132

62

0.37

345

L9-L8-L3-L6-L3

146

71.6

0.53

300

65

64

0.48

195

L10-L21-L26L28-L27

67

70

0.52

90

L10-L21-L23L28-L27

125

75.6

0.55

90

Yea

47

53

0.55

460

L43-L4-L13L39-L40-L42

59

58

0.54

460

73

0.53

460

26

27

0.58

460

39

42

0.55

460

–

–

–

15 147 81

15 76 88

0.63 0.42 0.54

460 345 130

L43-L4-L6L12-L16-L15 – L43-L4-L2 L43-L5-L8L11-L19-L2124-L25

– 26 176

– 28 107

– 0.61 0.62

– 460 195

– – 116

– – 0.55

– –

L43-L4-L13-L13L42

Alexandra L43-L4-L13 Thornton L43-L4 Eildon L43-L4 Yarra L43-L4-L6-L12L17-L20-L24L25

L43-L4-L6-L12L16-L15-L39L40-L42 –

72

–

– – – – L43-L4-L13-L4190 L42-L36-L29-L27

90

S. Shahparvari et al. / Transportation Research Part E 93 (2016) 148–176

Thornton L10-L19-L18L17-L12-L6 Eildon L10-L19-L18L17-L12-L6-L2 Yarra L10-L21-L24L25

171

172

S. Shahparvari et al. / Transportation Research Part E 93 (2016) 148–176

Table A2 Bushfire routes risk prioritisation module. Risk 100% ðIP l  PP l Þ  ðRRC l  LC l Þ

ðIP l  PPl Þ Probability

50%

ðRRC l  LC l Þ Consequence

50%

IP l (Ignition Probability)

45%

Ignition potential Fuel on road reserve Ignition history

50% 20% 30%

PP l (Propagation Probability)

55%

Fire Behaviour Propagation ability

50% 50%

RRC l (Road Reserve Consequence)

10%

Economic assets Cultural assets Environmental assets CFA precincts

40% 5% 5% 5%

LC l (Landscape Consequence)

90%

Economic assets Cultural assets Environmental assets Human assets

40% 10% 10% 50%

Table A3 Route segments data. Code Road name

Time window (min)

Travel time (min)

Distance (km)

Disruption risk (%)

Road capacity for bus

Road capacity for van

L1

345

132

61.7

0.37

2

3

600 600 460 400 500 420 340 300 280 600 500 600 600 600 500 320 270 240 130 195 100 300 270 270 190 90 220 240 60 90 270 135 30 30 600 50 300 500 500 600 600 600

11 11 12 16 4 10 52 68 19 49 6 11 12 8 6 8 3 9 12 10 25 75 23 13 17 14 7 13 66 70 55 20 15 7 14 53 45 12 11 13 10 3

13.4 11.7 10.7 6.4 3 3.6 19.9 25.3 16.8 18.7 6.4 12.4 8.5 9.6 8.5 11.6 3.9 7.9 17.3 12 11.1 22 21 14.2 16.1 17.3 7.5 18.8 24.8 26.3 26.4 8 7.3 5.5 21.3 23 17 8.9 8.1 17.2 13.8 4

0.60 0.71 0.64 0.50 0.50 0.50 0.50 0.39 0.41 0.47 0.50 0.53 0.51 0.54 0.50 0.48 0.50 0.50 0.63 0.58 0.50 0.57 0.49 0.47 0.48 0.54 0.73 0.63 0.51 0.51 0.50 0.50 0.50 0.50 0.50 0.36 0.44 0.50 0.50 0.51 0.50 0.59

4 2 3 0 3 1 1 0 3 0 3 4 2 4 4 4 3 3 4 3 0 0 4 3 3 4 3 4 2 1 1 1 2 2 4 2 1 2 2 4 4 3

5 3 4 1 3.6 2 1 1 4 1 3.6 5 3 5 5 5 4 4 5 4 2 2 5 4 4 5 4 5 3 2 2 2 3 3 5 2 2 3 3 5 5 4

L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 L13 L14 L15 L16 L17 L18 L19 L20 L21 L22 L23 L24 L25 L26 L27 L28 L29 L30 L31 L32 L33 L34 L35 L36 L37 L38 L39 L40 L41 L42 L43

Eildon-Warburton, Eildon Jamieson Rd Goulburn Valley Hwy(B340) Back Eildon Rd Rubicon Rd Blue Range Rd Taggerty-Thornton Rd (C515) Bulls Ln Blue Range Rd Lake Mountain-Royston River Rd Marysville-Woods Point Rd(C512) Mount Margaret Rd Taggerty-Thornton Rd(C515) Goulburn Valley Hwy(B340) Breakaway Rd-Hoban Rd Maroondah Hwy (B360) Maroondah Hwy (B360) Maroondah Hwy (B360) Buxton-Marysville Rd (C508) Buxton-Marysville Rd (C508) Maroondah Hwy (B360) Marysville Rd (C512) Plantation Rd Stony Creek Maroondah Hwy (B360) Healesville-Yarra Glen Rd (C726) Healesville-Kinglake Rd (C724) Melba Hwy(B300) Healesville-Kinglake Rd (C724) Melba Hwy(B300) Murrindindi Rd Black Range Rd Cameron Rd Simmonds Track Myles Rd Murrindindi Rd Melba Hwy(B300) Limestone-Ginter Rd Scrubby-Black Range Rd Whanregarwen Rd Whanregarwen Rd Goulburn Valley Hwy(B340) Goulburn Valley Hwy(B340) Rubicon Rd

Table A4 The objective function and decision variables results are presented in the left column and the optimal routing transportation pattern results in the right columns. Number of available shelters

Optimal assignment of the resources

From

To

X¼1

Shelters –Alexandra Vehicles –Bus –Van Objective function value

Narbethong

Alexandra

Marysville 10 11 72.95

Shelters –Alexandra –Thornton Vehicles –Bus –Van Objective function value

Number of required vehicles

Route 1

Route 2

Route 3

Vehicle type

Route 1

Route 2

Route 3

Route 1

Route 2

Route 3

83

98

59

Alexandra

100

100

60

Buxton Taggerty Cambarville Rubicon Narbethong

Alexandra Alexandra Alexandra Alexandra Alexandra Thornton

67 – 84 90 64 26

63 70 – 100 50 100

– 100 26 – – –

Marysville

Alexandra Thornton Alexandra Thornton Alexandra Thornton Alexandra Thornton Alexandra Alexandra Thornton Alexandra Thornton Eildon Alexandra Thornton Alexandra Thornton Thornton Eildon Alexandra Alexandra Thornton Yarra

96 78 – 82 – 100 19 91 90 71 69 60 100 – 48 82 – 81 68 – 90 69 81 90

86 – 48 – 70 – – – 100 – 100 – – 100 – – 89 – – 42 100 – – –

– – – – – – – – – – – – – – – – – – – – – – – –

Alexandra Thornton Eildon Alexandra Thornton Alexandra Thornton Alexandra Thornton Alexandra Eildon

85 75 – – 75 70 100 20 90 95 –

– – 100 55 – – – – – – 95

– – – – – – – – – – –

Bus Van Bus Van Van Van Bus Van Bus Bus Van Bus Van Van Van Van Van Van Bus Van Bus Bus Van Bus Van Van Van Van Van Bus Bus Van Bus Bus Bus Van Bus Van Van Van Van Van Van Van Bus Van Van

1 1 1 – 1 – 1 1 1 – 1 1 2 – 1 – 1 1 1 1 1 1 2 1 – 1 1 – 1 1 – 1 1 1 1 1 1 2 – – 1 1 1 1 1 1 –

2 – 1 1 2 1 – 2 1 1 2 1 – 1 – 1 – – – 2 – 2 – – 2 – – 1 – – 1 2 – – – – – – 2 1 – – – – – – 1

1 – 2 1 – 1 1 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –

2 6 2 – 4 – 2 6 1 – 2 2 3 – 5 – 7 1 2 6 2 2 2 2 – 3 5 – 5 2 – 6 2 2 1 3 2 5 – – 5 5 7 2 2 7 –

1 – 2 7 2 5 – 3 1 2 3 2 – 3 – 5 – – – 3 – 1 – – 3 – – 6 – – 1 3 – – – – – – 7 4 – – – – – – 7

1 – 1 4 – 7 1 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –

6 13

Buxton

59.95

Taggerty Cambarville

X¼3

Shelters –Alexandra –Thornton –Eildon Vehicles –Bus –Van Objective function value

Rubicon Narbethong Marysville

7 11

Buxton Taggerty

57.85 Cambarville

X¼4

Shelters –Alexandra –Thornton –Eildon –Yarra Vehicles –Bus –Van Objective function value

Rubicon Narbethong

Marysville 5 12

Buxton

52.8

Taggerty Cambarville Rubicon

Number of trips

173

(continued on next page)

S. Shahparvari et al. / Transportation Research Part E 93 (2016) 148–176

X¼2

Evacuees routing plan

174

Table A4 (continued) Optimal assignment of the resources

From

To

X¼5

Shelters –Alexandra –Thornton –Eildon –Yarra –Yea Vehicles –Bus –Van Objective function value

Narbethong

Alexandra Thornton Yarra

Marysville

Alexandra Thornton Eildon Alexandra Thornton Alexandra Thornton Alexandra Thornton Eildon Yea Alexandra

6 12

Buxton Taggerty

58.75 Cambarville

Rubicon

Evacuees routing plan

Number of required vehicles

Number of trips

Route 1

Route 2

Route 3

Vehicle type

Route 1

Route 2

Route 3

Route 1

Route 2

Route 3

71 79 90

– – –

– – –

98 62 – 48 82 – 100 9 87 – – –

– – 100 – – 70 – – – 14 – 85

– – – – – – – – – – 100 –

Bus Bus Bus Van Bus Van Van Van Van Van Van Van Bus Van Bus Van

1 1 1 1 1 2 – 1 1 – 1 1 1 – – 1

– – – – – – 2 – – 1 – – – 1 – –

– – – – – – – – – – – – – – 1 –

2 2 2 6 2 2 – 3 5 – 7 1 2 – – 6

– – – – – – 3 – – 5 – – – 1 – –

– – – – – – – – – – – – – – 2 –

S. Shahparvari et al. / Transportation Research Part E 93 (2016) 148–176

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