European Journal of Operational Research

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1 European Journa of Operationa Research 198 (2009) Contents ists avaiabe at ScienceDirect European Journa of Operationa Research journa homepage: Production, Manufacturing and Logistics Muti-objective evacuation routing in transportation networks exander Stepanov *, James MacGregor Smith 1 Department of Mechanica and Industria Engineering, University of Massachusetts, mherst, M , United States artice info abstract rtice history: Received 17 December 2007 ccepted 29 ugust 2008 vaiabe onine 30 September 2008 Keywords: Evacuation Routing Queueing networks In this paper, the optima design and anaysis of evacuation routes in transportation networks is examined. n methodoogy for optima egress route assignment is suggested. n integer programming (IP) formuation for optima route assignment is presented, which utiizes M/G/c/c state dependent queueing modes to cope with congestion and time deays on road inks. M/G/c/c simuation software is used to evauate performance measures of the evacuation pan: cearance time, tota traveed distance and bocking probabiities. Extensive experimenta resuts are incuded. Pubished by Esevier B.V. 1. Introduction Natura and man-made emergency events, such as hurricanes, foods, widfires or chemica spis, can impose serious risk and threat on the we-being and safety of a popuation. In this case, massive evacuation or in site shetering (sheter-in-pace) has been used as means of protecting a popuation from potentia harm (Sorensen et a., 2004). o sheter-in-pace is a preferred option if there is a risk for a popuation to be exposed to some chemica agents or hazardous materias and if sheter provides adequate eve of protection. However, evacuation is a procedure most often used in case some infrastructure or community can be damaged by hurricanes, foods or widfires (Perry and Linde, 2003). Scientific or forma anaysis of the evacuation processes is very compex due to the arge number of evacuees and unforeseen oads on the transportation infrastructure. Such anaysis of emergency situations can require synthesis of knowedge from many different areas of expertise: socia and behavior science, transportation, heath care, teecommunication, security, poice, forestry, and so on Objectives of research he purpose of this paper is to design and to anayze egress route assignment agorithms for regiona evacuation in arge transportation networks. his probem is chaenging due to the fact that the speed of vehices decreases with an increase in the number of vehices on the road segments as we as of the potentia * Corresponding author. e.: E-mai addresses: astepano@ecs.umass.edu (. Stepanov), jmsmith@ecs.umass. edu (J.M. Smith). 1 e.: bocking on heavy utiized road segments. Stochastic arrivas of evacuees from affected areas and peak demands for transportation infrastructure add much compexity to the probem. n integer programming (IP) mode is suggested to compute an optima routing poicy for the popuation from the effected areas, where a set of feasibe and potentia egress routes is defined with the kth shortest path agorithm. n embedded M/G/c/c state dependent queuing mode is used to evauate the trave time aong the road inks. Finay, a state-of-the-art simuation mode MGCCSimu is utiized to evauate the resuting cearance time, trave distance, and eve of congestion for the designed evacuation pan. We iustrate the suggested approach with a case study and extensive experimenta resuts Outine of paper Section 2 provides a review of current research in the area of evacuation panning and modeing. Section 3 of the paper describes the overa probem. Section 4 outines the overa framework for the methodoogy and Section 5 discusses the soution methodoogy and agorithm. Section 6 describes the case study and the experimenta resuts and Section 7 concudes the paper. 2. Literature review In the iterature, reocation from areas at risk to areas of greater safety is referred to as an evacuation (Southworth, 1991; Zeinsky and Kosinski, 1991). Regarding its properties, evacuation may differ by scae, objects of reocation (peope vs. property), and by eve of contro by authorities. In addition, the evacuation can be mandatory, recommended, or vountary and shoud be conducted according to an evacuation pan. Evacuation is a compex process consisting of severa consecutive phases (Fig. 1). he first phase /$ - see front matter Pubished by Esevier B.V. doi: /j.ejor

2 436. Stepanov, J.M. Smith / European Journa of Operationa Research 198 (2009) Phase I Phase II Phase III Phase IV Phase V Phase VI Phase VII Incident detection Evacuation Order Issued Deiver order/ message to pubic via transmitters Preparation for evacuation Movement through Evacuation network rriva at safety zone Verification Phase Decision time Notification time Preparation time Response time (Cearing ime) Evacuation ime Fig. 1. Evacuation phases. is detection of an incident. In Phase II, decision makers have to evauate the risk and potentia threat for specific areas. n evacuation order shoud be issued for these areas if the risk is significant and there are no sheters to provide adequate in-pace protection (Linde, 1995). (hese areas constitute origins of evacuation or evacuation panning zones (EPZ)). Phase III, the aert, has to be communicated to the popuation. During the next phase, the popuation makes a decision to evacuate or not to evacuate depending on their perception of danger. his phase aso impies preparation for eaving. Phase V impies movement of popuation through a transportation network to designated safe areas (destinations) or centers of refuge. his step invoves cearing of occupants from affected areas. Finay, in Phase VI, evacuees arrive to areas outside of danger (destinations) and a verification that a evacuees have made it safey must be carried out, which is Phase VII. It is important to note that time intervas for Phases III VI represent average time for a groups of evacuees, as these steps may have different durations for each cass of evacuees. Depending on the scae of a emergency situation, neighborhoods, towns or regions may have to be evacuated. he evacuation time may range from hours to weeks or even months (Church and Sexton, 2002). Generay, capacity of transportation networks cannot satisfy the intense demand for transportation during the evacuation. Even for sma neighborhood-scae evacuations, transportation networks impede the fast cearing of the popuation from an effected area (Church and Sexton, 2002; Cova and Johnson, 2002). he magnitude of the probem is greater for regiona evacuations. o manage such emergencies effectivey, the decision makers may benefit from having in-pace evacuation pans (EP). he aim of an evacuation pan is to define optima evacuation poicies for the popuation from areas under risk and uncertainty. Decision makers design and update evacuation pans for scenarios which are most ikey to happen. his approach is known as prepanning, though rea-time design or re-evauation of evacuation pans are required in cases of infrastructure faiure, road bocks, and unexpected changes in direction and speed of emergency event propagation (exander, 2002) Emergency evacuation panning s we pointed out in the previous section, controed evacuation is governed by a specific evacuation pan, which defines the actions of the popuation and authorities. his action pan is a resut of a purposefu emergency evacuation panning (EEP) process. n emergency evacuation panning is considered as a five-step process (Southworth, 1991). If the evacuation panning process occurs before Phase I of the evacuation (Fig. 1), then it s known as prepanning. In this case, an action pan is designed prior to an emergency event for a finite set of potentia scenarios. If it occurs during Phases II V, then the evacuation panning process can be cassified as rea-time. We wi discuss these EEP steps in more detai in the foowing sections. he fact that many pans are designed prior to an evacuation and for a imited number of scenarios is defined by the compexity of the regiona evacuation process, imited resources and ack of we-defined protocos. In practice, assumptions made during such panning can be made invaid during an emergency due to changes in weather conditions or in an emergency event behavior, and destruction of transportation infrastructure. Rea-time emergency evacuation panning requires that an integrated methodoogy be avaiabe for decision makers. Such a methodoogy combines efficient anaytica methods for modeing emergency event behavior, optima routing assignment, and aows one to manage the evacuation process in rea-time Step 1: traffic generation Evacuation panning zones (EPZ) or areas, predicted to be affected by an emergency event, act as origins of evacuee traffic. he utimate purpose of this step is to define a number of vehices eaving the areas and oading an evacuation network. First, it is necessary to evauate the popuation size presented in a specific area at a particuar time. hen, this popuation size shoud be transated into the size of the vehice feet used for evacuation. Popuation distribution differs significanty between day and night time, as peope commute to work, schoos, shopping and service centers. Major data providers (the US Census program) have accurate information on night time popuation distribution, in other words this is information on where peope ive. n evauation of popuation size during daytime is a chaenging issue (Southworth, 1991). Daytime popuation comprises popuation at work, at schoo, at service centers (hospitas, medica care faciities, sport gyms) and popuation, which stay at home (retirees, sma chidren, etc.). In addition, there are specia groups of popuation with imited mobiity (hospitas, nursing homes, correctiona/penitary faciities). hese popuation groups are transit dependent. Finay, through traffic may generate additiona oads on the evacuation network. Linde and Prater (2007) provide detaied discussions of trip generation techniques based on empirica behaviora studies Step 2: evauation of traffic oading rate he purpose of this step is to evauate the time distribution of the evacuees departure process. Southworth (1991) outined four major approaches to define traffic oading curves, which represent the fraction of evacuated popuation at a specific time. he assumptions about evacuees behavior are based on empirica data and historica reviews, surveys of intentions and expert judgment, simuations of aarm message propagation and perception in the community (Southworth, 1991; Cohn et a., 2006; Stern and Sinuany-Stern, 1989). Linde and Prater (2007) provide a comprehensive overview of factors affecting departure time distribution, anaysis of distribution used in practice, as we as comparison with empirica data. Vaid assumptions on evacuees behavior for a particuar type of hazardous events is crucia for accurate evacuation modeing. Evacuees behavior is a compex phenomenon, for instance, many peope don t eave the region after being warned (Sorensen et a.,

3 . Stepanov, J.M. Smith / European Journa of Operationa Research 198 (2009) ). In some situations, on the contrary, a significant part of the popuation have decided or aready have eft prior to the moment when the evacuation was officiay decared (Linde et a., 2005). Cohn et a. (2006) provide some insight on popuation behavior using post surveys of three major widfire evacuations. Using empirica data, Linde et a. (2005) anayze househods response during hurricane Lii. he decision to eave depends on many factors: person s perception of danger, famiy and socia status, existence of famiy/friends at the destination points, or even type of emergency event (Stern and Sinuany-Stern, 1989). Peope tend to take foods and chemica spis more seriousy, than hurricanes and widfires. Some peope decided to stay to protect their property from fire or because of pets. In practice, the timing response is described with traffic oading S-shaped curves. For instance, Eq. (2), suggested by Radwan and modified by Southworth and Chin, expains the cumuative percentage of tota traffic P t oaded into evacuation network at time t (Radwan et a., 1985; Southworth and Chin, 1987; Southworth, 1991; Radwan, 2005) 1 P t ¼ 1 þ d expðatþ ; ð1þ d ¼ 1 P 0 P 0 ; ð2þ where a is a parameter to caibrate and d is ratio of vehices to be evacuated to vehices which are aready in the road network at time t = 0. here are reported appications of the Rayeigh distributions (weedie et a., 1986) and exponentia distributions (Hobeika and Kim, 1998) for estimating time evacuees departure time. In their mode, Cova and Johnson (2002) view evacuees departure as a Poisson process Step 3: definition of refuge centers t this step of the evacuation panning process, the destinations shoud be identified. Linde and Prater (2007) point out that for evacuation panning purposes both evacuees utimate and proximate destinations shoud be considered. Utimate destinations are pubic faciities and private estabishments where evacuees stay before returning back home. he majority of evacuees stay with reatives, friends or in hotes. In case of pubic refuge centers, they shoud provide enough space to accommodate evacuees and be ocated in areas with adequate ingress routes to deiver humanitarian aid, medica assistance, etc. From a practica point of view, sport arenas, pubic schoos or universities buidings may be used as they were designed to hod arge numbers of peope. In turn, proximate destinations are viewed as boundaries, after crossing, in which evacuees are safe. For more discussion on destination choice, pease refer to Linde and Prater (2007) Step 4: traffic route assignment Fig. 2. Evacuation network G e (N,) connects EPZs and destinations. t this step of the evacuation panning, we assume that EPZs, destinations and evacuation transportation networks have been defined. Demand and oad patterns of evacuee s have been evauated as we. hen, the purpose of this step is to determine optima routes and assign vehice feets to them (Fig. 2). his is a compex Emergency Event Propagation Mode Onine data/imagery Road/Street Network Sources/Destinations Popuation Distribution (1) (2) Visuaization of a proposed Evacuation pan EP(x) (9) Output of Routing Modue: Evacuation Pan EP(x) Performance measures: c, D and Pc (3) Geographic Information System (GIS) Modue (8) M/G/c/c Simuation Program (4) K-th shortest path agorithm (6) LESLM M/G/c/c naytica Program Input for Routing Modue: Evacuation Network G (5) E(N,) Egress Route Set x Occupants cass/size P (7) EEP IP mode Parameters of time deay functions Set of egress routes Evacuation Pan EP(x) GIS Modue Routing Modue Fig. 3. Soution framework.

4 438. Stepanov, J.M. Smith / European Journa of Operationa Research 198 (2009) Speed (mies/hr) Density (veh/mie/ane) Fig. 4. Speed-density curve: road ink performance decays with increase in traffic density (Source: (Smith, 2001)). network optimization modes to aocate fows optimay without exceeding evacuation network capacity. Methods, which appy user equiibrium (UE) techniques, minimize trave time for each vehice (gers et a., 1997; Sheffi et a., 1982; Hobeika and Kim, 1998). Evacuation modes, which represent traffic as fows, beong to macro-eve modes. ransportation engineering approaches often consider traffic at more detaied eves of an individua vehice or a group ( patoon ) of vehices. Such approaches can be cassified as micro and meso modes. he evacuation modes can be cassified aso regarding their route assignment procedure. Static assignment impies that vehices stay on a specific route during the evacuation, which can be a reaistic feature of mandatory evacuation with imposed route contros (roads cosure, poice reguating traffic). Other modes consider dynamic assignment, where a vehice chooses a future direction at every road crossing. Modeing techniques can be grouped by computationa techniques into anaytica and simuation techniques. abe 1 summarizes characteristics of severa evacuation modes. optimization probem, which has NP-hard compexity (Kerbache and Smith, 2000). One of the inherited properties of the probem is that the evacuation network performance depends on traffic oad. Veocity of vehices on particuar road inks decreases with increasing of number of vehices (Fig. 4). Over-utiization of a particuar road ink can bock traffic through the ink competey. Such bocking wi affect a upstream inks and can cause significant congestion through the evacuation network (Cruz and Smith, 2005; Cruz et a., 2005; aebi and Smith, 1985). he design of an optima evacuation poicy comprise the foowing three major sub-probems. First, it is necessary to define the functiona criteria of an evacuation pan and suggest quantitative measures of the evacuation. he second is to deveop routing poicies (as a set of egress routes). Finay, evauation of evacuation pan s effectiveness shoud be performed. In the practice of evacuation modeing, tota cearance time, tota traveed distance, queue size and bocking probabiity are the most common criteria which shoud be extremized Step 5: verification of an evacuation pan his step in the evacuation panning procedure concudes the design of an evacuation pan with its performance evauation. If the designed pan is prone to traffic congestion or fais to provide timey evacuation, some aternative scenarios or appication of traffic contro toos shoud be considered. Decision makers may consider reversing direction of traffic fow on some road inks and using contrafow anes (Woshon, 2001; Woshon and Lambert, 2006), disabing ramps and exists, changing routing at intersections to minimize traffic conficts (Cova and Johnson, 2003). hen Step 4 of the procedure shoud be repeated. his paper focuses mainy on Step 4 traffic route assignment and uses information from the previous steps as input for anaysis, as we assume that dynamics of emergency event propagation, origins, popuation sizes and departure characteristics, and transportation network were defined Evacuation modes review Historicay, evacuation modeing approaches were infuenced greaty by fieds of operations research (OR) and transportation engineering (E). OR approaches aim to minimize tota evacuation time for the whoe system or for individua user. System optimization methods consider traffic as non-interrupted fows, which satisfy demand existing in destination nodes, and appy different abe 1 Cassification of evacuation modes Mode ype Route assignment type Computationa techniques uthors (year) 1 Cear Micro Simpe Moeer et a. (1982) 2 DYNEV Meso Static Simuation with embedded KLD ssociates (1984) I-DYNEV UE agorithm 3 DYMOD Macro Dynamic Simuation with embedded optimization routine Southworth et a. (1992) 4 MSSVC4 Macro Dynamic Simuation with embedded UE and Dia s agorithm Hobeika and Kim (1998) 5 NEVC1 Macro Dynamic Sheffi et a. (1982) 6 RNSIMS Micro Dynamic Parae simuation Pickert and Nage (2001) 7 CEMPS Micro Dynamic C++ discrete event simuation couped with GIS Pidd et a. (1996) 8 M/G/c/c state dependent queueing mode Micro Static naytica technique 9 MGCCSimu Micro Dynamic Simuation with M/G/c/c statedependent queues. Support modeing of vehice and pedestrian traffic 10 OREMS (by ORNL) Micro & dynamic Static Jain and Smith (1997) Cruz et a. (2005) Rathi and Soanki (1993), Bhaduri et a. (2006), Franzese and Sorensen (2001) Micro 11 DYNSMR Micro Dynamic Mahmassani et a. (1995) 12 EMBLEM2 Micro Dynamic Empirica behaviora mode Linde (2008) 13 ane-based evacuation routing mode Macro Static Embedded OR procedure to contro intersections Cova and Johnson (2003)

5 . Stepanov, J.M. Smith / European Journa of Operationa Research 198 (2009) From a review of the evacuation modes (Southworth, 1991; gers et a., 1997; Smith, 2001; Santos and guirre, 2004), two approaches are evident. he first approach defines a set of optima routes and evauates performance measures simutaneousy. he second approach uses an anaytica optimization technique to offer a routing poicy, and then this poicy is evauated with a traffic simuation mode (aebi and Smith, 1985; Smith, 1991, 1994). ccording to mode reviews, the first approach is prevaent in practice. However, the second approach is overooked and shoud be examined in more detai. 3. Probem formuation his probem is situated within the domain of optima routing probems (ORP) (Kerbache and Smith, 2000; Smith, 2001). he cass of ORP considers the issues of routing of evacuees through an existing transportation network and addresses potentia congestion and deays which coud be created with the routing decision. his probem is known to be NP-Hard (Kerbache and Smith, 2000; Smith, 2001), due its muti-commodity nature, as evacuees can be differentiated by their origin destination pairs as we as by their cass or group. During an evacuation, we need to safey guide and reocate the occupants (popuation) P i from the affected areas (sources) S i to destinations D j through a transportation network G(N,) over time t. Each popuation cass/group P i shoud be assigned to an egress route (Fig. 2). he objective here is to find a set of egress routes to safey evacuate the occupant popuation P over G. Further we wi refer to a set of potentia routes as to an evacuation pan (EP), which we denote with the foowing notation: EP ¼fR S1 ; R S2 ;...; R Sn g, where R Si is an egress route from source S i. his probem is compex due to its stochastic, integer programming, and muti-criteria nature. s occupants attempt to use shortest routes some of the traffic wi over utiize various road segments, causing congestion and potentia bocking aong those routes. One of the ways to attain popuation safety during the evacuation process is minimization of tota cearance time C, tota traveed distance D and congestion on road inks (Smith, 2001; Kerbache and Smith, 1987, 1984). herefore, the method described in this artice aows one to generate an evacuation pan EP(x) (a vector of egress routes x for popuation P) that minimizes simutaneousy tota traveed distance (D ), bocking (P C ) and tota cearance time ( C ) aong the egress routes Notation In this section, we propose an integer set-packing formuation of the stochastic EEP probem. he notation used in the probem formuation is introduced beow: Indexes i an index for popuation sources (origins) S i ; j an index for destinations D j ; k an index for kth shortest paths; an index for a set of road inks ; Decision variabes x ijk 1 if popuation was assigned to kth shortest egress route connecting source i and destination j (otherwise x ijk =0) Parameters w a weight coefficient w 2 [0, 1] and shoud be set by a decision maker. D tota distance traveed by occupants cearance time for the system; C tota distance when a occupants are assigned to the 1st shortest routes min C the best cearance time of the system (in absence of congestion) and a occupants trave aong shortest egress routes with maxima veocity d ijk tota ength of route x ijk P i tota number of occupants from source i; k i an arriva input rate into the system from source i K a vector of arriva rates (k 1, k 2,...) k L a tota arriva rate into road ink k Lmax a maximum arriva rate into road ink such that P C ðk Lmax Þ 6, where is a threshod vaue of bocking probabiity t ð1þ minimum expected traverse time through a road ink t ðk L Þ expected traverse time through a road ink given an arriva rate k L V ðnþ a maximum veocity of occupants on road ink, where n occupants present on the road ink L distance (ength) of road ink a ikj 1 if egress route x ijk incudes road ink (0 otherwise) C j capacity of destination refuge center D j D min 3.2. Mathematica formuation he muti-objective mode to sove the EEP probem is suggested beow: min Z ¼ w 1 X X X P x D min i i j k X X X P i x ijk a ijk i j k x ijk d ijk þð1 wþ 1 min C X t ðk L Þ subject to: X X x ijk ¼ 1 for 8i ðsourcesþ; ð4þ j k X X P i x ijk 6 C j for 8j ðshetersþ; ð5þ i k X X X k i a ijk x ijk 6 k Lmax for 8 ðroadinksþ; ð6þ i j k x ijk ¼ 0 or 1 for 8i and 8k ðroutesþ: ð7þ he objective function Z is a inear combination of excess tota trave distance D and excess cearance time C. he excess trave distance evauates deviation of tota traveed distance D (which is the performance measure of a routing assignment x) from the best or minima trave distance D min D min ¼ Xn i¼1 P i X m j¼1 x ij1 d ij1 : he occupant popuation wi trave D min mies if and ony if the evacuees are assigned the 1st shortest egress routes, therefore the is the ower bound for any evacuation pan EP(x) (Eq. (8)). Eq. (9) represents the minimum cearance time min C, which vaue of D min defines the ower bound on cearance time C for any egress routing x and corresponds to the best case scenario, when popuation is evacuated aong 1st shortest egress routes with maximum veocity V ð1þ min C at any road ink ¼ X X t ð1þ n X m P i x ij1 a ij1 : i¼1 j¼1 his above probem is difficut to sove since an optima soution is a trade-off among three specified criteria: congestion, cearance time and tota traveed distance; ð3þ ð8þ ð9þ

6 440. Stepanov, J.M. Smith / European Journa of Operationa Research 198 (2009) the parameters of the mode t and P C depend upon the stochastic mode which evauates a given set of routes. In the above mode, Eq. (6) defines a constraint, which captures the traffic congestion minimization. he probabiity of bocking P C on a road segment a depends on evacuees arriva intensity k L.We discuss an approach to compute an upper bound on input arrivas k Lmax which wi assure the bocking probabiity wi not exceed the threshod vaue, therefore P C ðk Lmax Þ 6. By setting a threshod vaue on the maximum bocking probabiity in the evacuation network G e (N,), we assure that evacuees fows on egress routes are non-interrupted. Constraint set, defined by Eq. (4), ensures that a popuation from a source node i is evacuated aong one egress route. his formuation aows one to mode mutipe cass evacuation with introduction of dummy source nodes with specific popuation size and arriva rates. Eq. (5) represents the constraint set which captures aggregate capacity restrictions for destination D j. We assume that destination node D j is an utimate destination for some part of evacuees and a proximate destination for the rest of peope. herefore, C j is an upper imit on tota number vehices which wi arrive at D j and stay there at sheters, hotes or at reatives, and vehices, which wi go through D j to other areas. he suggested mode formuation has two unique features, which require thorough expanation. he first feature ensures the non-interrupted movement of evacuees via the evacuation network and absence of bocking on any road segment. he second feature copes with time deay modeing aong egress routes given that vehices veocity is decreasing function of number of vehices at the road segment and that arrivas as we as number of vehices on the road segment is random vaue. Both features are based on a M/G/c/c state-dependent decay service queueing mode to mode traffic congestion on a road ink, which is a core of the proposed IP formuation. he next sections wi describe treatment of time deay functions t (k) and congestion eve in detai. 4. Soution framework he exporation of vehice routing probems in transportation networks is performed with the foowing framework, see Fig. 3. his system is based on modues deveoped in the Dynamic Faciities Layout and Simuation Modeing Laboratory, UMass. he system incudes three major components: (i) Geographic Information System (GIS) component (ii) Routing (evacuation) panning component and (iii) a set of utiities, which combine two components together. he GIS component is based on rcview GIS 9.2Ó. he GIS modue (bock 3 in Fig. 3) is responsibe for entering, storing, querying and representing spatia information. he foowing information is used for evacuation panning: popuation ocation and characteristics, origin destination data, road network topoogy, road inks characteristics, etc. he kth shortest path agorithm (Jimnez and Marza, 2003; Eppstein, 1998) is utiized to compute an egress route set for each origin destination pair. herefore the road inks, which comprise egress routes, define an evacuation network G E (N,). his information, together with occupants data, is used as input data for the routing modue. he Routing (traffic assignment) modue incudes three major sub-modues. he first one is a LESLM modue, which uses road ink data to define time deay functions t(k) for each road segment. he second, EEP IP mode, soves the route assignment probem and finds a Pareto optima evacuation pan (set of evacuation routes) for specified scenario. he third modue is a state-of-the-art vehicuar evacuation simuation mode (Cruz et a., 2005), which is used to evauate evacuation pan s performance. he foowing measures of effectiveness are computed: estimated cearance time C, and tota traveed distance D, as we as detai statistics for each road ink, such as: congestion eve (probabiity of bocking P C ), throughput h, average number of vehices on the road segment E(Q) and average traverse time S. fter that the resuting soution is visuaized with the GIS modue and an evacuation panner (a decision maker) can obtain an evacuation pan map aong with evacuation pan performance characteristics. GIS has been an important too for emergency management and panning (de Siva and Egese, 2000; Church and Sexton, 2002; Pa et a., 2003; Zerger and Smith, 2003), which provides adequate and effective ways to represent and manage transportation network modes, as we as to integrate such modes with required socio-demographic data for the modeed area. he road networks of different scae and detai eve (from major highways to streets) and demographic data can be obtained from the Department of ransportation and from State cartographic agencies as we as from private data providers. In addition, an abiity of GIS to incude highresoution imagery and onine sateite data, aids a decision maker in preparing a reaistic network mode, as some specifics of road configurations, topoogy, avaiabe ramps or exists can be verified. 5. Soution methodoogy We foow an approach suggested by Smith et a. (Smith, 1991; Smith, 1994; Karbowicz and Smith, 1984; Cruz et a., 2005; aebi and Smith, 1985) to mode traffic and congestion on a singe road ink and Markovian arrivas into the road ink. his M/G/c/c queueing mode aows one to capture traffic congestion and its stochastic nature on a road ink with specific physica characteristics as ength L, number of anes W and maximum density K max (Fig. 5) and tota capacity of c = K max LW. he queueing mode updates service rate with changes in number of vehices in the system (Fig. 4) and aows one to evauate system in term of expected service time, expected number of vehices in the system, probabiity of n-vehices being in a road ink, etc. herefore, a road segment can be modeed as a queueing service faciity with the foowing physica characteristics: Road segment rriva rate λ Bocked vehices Pc Width W (# anes) Length L (mies) Capacity C = Kmax*L*W Effective rate λ* = (1-Pc) λ Fig. 5. Representation of a road ink as M/G/c/c queueing system.

7 . Stepanov, J.M. Smith / European Journa of Operationa Research 198 (2009) (i) (ii) (iii) (iv) (v) (vi) queue has a finite size since a road segment has finite ength and width(capacity), or number of anes and can accommodate a imited number of vehices. Capacity c of a road segment can be computed as c = K max LW, where K max, L, W are jam density, ength and number of anes, respectivey. Vehices arrive according to Markovian Process. rave time aong the road segment is considered as a service time in the queueing faciity. he queue can be cassified as state-dependent because the service rate is a function of number of vehices aong the road segment (Fig. 4). Number of servers is imited with capacity c of the road segment. he queue faciity cannot serve more than c vehices. In other words there is no buffer or waiting space, therefore as soon as a vehice enters the road ink, it starts obtaining the service. Service rate distribution is genera (G) due to service rate decay. abe 2 Congestion modes Linear congestion mode Exponentia congestion mode h c i V ðnþ ¼ Vð1Þ c ðc þ 1 nþ V ðnþ ¼ V ð1þ exp n 1 b whereh i. c ¼ n nðva=vð1þ Þ n a 1 nðv b=v ð1þ Þ b 1 a 1 b 1 b ¼ ½nðV ð1þ =VaÞŠ 1=c ¼ ½nðV ð1þ =V bþš 1=c h c i f ðnþ ¼ VðnÞ ¼ cþ1 n V ð1þ c f ðnþ ¼ V ðnþ ¼ exp n 1 V ð1þ b h ½kL=V P n ¼ ð1þ Š Q n n. Qn h kl j 1 c ii n P 0 P n ¼ j½ðcþ1 jþ=cš V ð1þ j¼1j exp b P 0 j¼1 where P 1 0 ¼ 1 þ P c i¼1 " # ½kL=V ð1þ Š i Q i j¼1 j½ðcþ1 jþ=cš where P 1 0 ¼ 1 þ P c i¼1 kl V ð1þ i Qi h j 1 c i j¼1j exp b Where c is the capacity of a road ink ; n is the number of vehices with a road ink n 2 (0,...,c); a is the density of vehices K dens = a; b is the density of vehices K dens = b; V a is the average trave speed at vehice density K dens = a; V b is the average trave speed at vehice density K dens = b; V (n) is the average trave speed for n vehices within a road ink; V (1) is the maximum trave speed in free fow conditions (speed imit); b, c are the scae and shape parameters for exponentia congestion mode. he assumptions above aow one to anayze road segments with an M/G/c/c Erang oss queueing mode. he equations summarized in abe 2 are a forma description of the physics of a singe road segment, modeed as an M/G/c/c queue. he equations for V (n) describe the veocity of n vehices moving aong a road segment. Service rate of a road segment can be evauated with expression for f(n). Equations for P 0 and P n can be used to compute probabiity distribution functions of service rates for specific road segments naytica and simuation mode he M/G/c/c queueing modes were impemented as two appications: LESLM and MGCCSimu. n appication LESLM was deveoped in DyMS ab by Smith and Jane, this appication numericay soves equations (abe 2) for M/G/c/c state dependent queue (Jain and Smith, 1997). LESLM aows one to evauate performance parameters of a road ink with specific input parameters and arriva rate of vehices. he second appication the simuation mode which is utiized to evauate a route assignment poicy, was deveoped by Cruz et a. (2005) and impemented as C++ program MGCC- Simu. he simuation mode considers a road segments a as M/G/c/c state dependent queue, with Poisson arriva rate k, genera service rate G and imited capacity c. ransportation network with detaied ink characteristics, an origin destination matrix, assigned egress routes and arriva rates for each source are used as an input for this mode. he mode updates service rates on the road segments dynamicay during simuation, appying inear or exponentia congestion mode for singe road segments (abe 2). he M/G/c/c simuation mode evauates tota cearance time C and tota traveed time D for specific evacuation pan. In addition, the mode provides detai information for every road ink a of the evacuation network G e (N,), such as bocking probabiity P C, expected trave time to traverse a road ink E( S ), expected number of evacuees on a road ink E(Q) and throughput h Modeing time deay on road inks Modeing time deays aong egress routes is a chaenging probem due to the fact that vehices veocity depends on number of evacuees in the system, which is a random vaue by definition of arriva process. o get some insight into reation between trave time, properties of a road ink and properties of arriva process (arriva rate k) we performed series of simuations. Using the MGCCSimu Fig. 6. Expected trave time E[ S ] and probabiity of bocking P C as a function of an input rate k (L = 5.0 mies, W = 1 ane, K max = 200).

8 442. Stepanov, J.M. Smith / European Journa of Operationa Research 198 (2009) program, we evauated expected time E[ S ] to traverse a road ink (L = 5 mies, W = 1 ane and K max = 200 vehice/mie/ane) and probabiity of bocking for a given arriva rate k2[0,..., 10,000]. Fig. 6 presents resuts of simuations. naysis and comparison of trave time and bocking probabiity curves (Fig. 6) ead to severa concusions and observations. Expected time t to traverse a road ink a of ength L and capacity C L is bound by ower ðt b Þ and upper ðt ub Þ vaues (Eqs. (10) and (11)) t b t ub ¼ t ð1þ ¼ L V ð1þ ¼ t ðcl Þ ¼ L V ðcl Þ ; ð10þ ; ð11þ where V ðcl Þ is veocity of vehices on fuy utiized road and can be computed with inear or exponentia congestion modes (abe 2). Bocking before service (P C P ) occurs when an input arriva rate vaue exceeds some critica vaue k Lmax. We need to note that a steep increase in trave time occurs in the vicinity of k Lmax as we. he vaue of k Lmax for a road ink a and specific bocking probabiity threshod ðp c ðc L ; klmax Þ 6 Þ can be obtained with the M/G/c/c state dependent queueing mode and LESLM appication. herefore, we propose (i) to set an upper imit on arriva rate of incoming evacuees to any road ink a and (ii) to approximate trave time on a road ink a which is a part of egress routes with a inear function (Fig. 7) t ðk L Þ¼tð1Þ L þ tðc Þ t ð1þ k Lmax k L : ð12þ he aggregate arriva rate into a ink a (which composed by arrivas of evacuees traveing different egress routes which share the same road inks) can be evauated as k L ¼ X X X k i a ijk x ijk : ð13þ i j k he proposed formuation of trave time deays for stochastic EEP assures no bocking during evacuation and prevents over utiization of shortest routes. he trave time function can be approximated with a inear piece-wise approximation as we. We use a inear function to approximate t ðk L Þ to not overy compicate the objective function Z (Eq. (3)) Description of the agorithm Fig. 8. regiona transportation network: popuation centers and road infrastructure. Step 2.0 Step 2.1 Step 2.2 Step 3.0 Step 3.1 Step 3.2 egress routes for each source i (Eppstein, 1998; Jimnez and Marza, 2003). Using kth shortest path routes, define subsets of road inks and nodes N which comprise evacuation network G e (N,), where 2 and N 2 N. naysis step: Compute upper bound on arriva rates k Lmax for each road ink a 2 to ensure absence of bocking within the evacuation network G e (N,). Compute maximum trave time t ðcþ a 2 for a arcs. he vaue of t ðcþ for a singe road ink corresponds to a case and t ðcþ when a road ink fuy utiized. Given k Lmax, t ð1þ define inear function t ðk L Þ for a road arcs a. Synthesis step Formuate and sove the EEP IP mode according to Eqs. (3) and (4). he soution of the EEP IP mode wi generate a route assignment vector x. Evauate the route assignment pan for the evacuation network G e (N,) with the MGCCSimu program to obtain tota cearance time C, tota traveed time D and performance measures for road inks. he proposed approach to sove a stochastic EEP can be summarized with the agorithm beow. Step 1.0 Representation step: Given a regiona transportation network G(N, ), where N is a finite set of nodes, is a finite set of edges. Reconfigure the network into a directiona one if necessary. Use the k-shortest path agorithm to determine 1st, 2nd, 3rd,..., kth shortest 6. Experimenta resuts In this section, a sma case study is utiized to demonstrate the proposed methodoogy. he approach was tested on the foowing sampe evacuation probem, where the number of evacuees, arriva rates and road network characteristics were chosen to be Fig. 7. Simpe mode of veocity and time as function of arriva rate.

9 . Stepanov, J.M. Smith / European Journa of Operationa Research 198 (2009) adequate to ensure congestion during peak demand and to not obscure the exampe, though the suggested mode do not have an upper imit on number of evacuees or network size. herefore, it is necessary to evacuate 4500 househods from three neighborhoods S 1, S 2 and S 3 to refuge center D 1 (Fig. 9). We assume that each househod wi use one vehice and that evacuees depart from EPZs and arrive into evacuation network according to Poisson Process with arriva rates k 1 = k 2 = k 3 = 1200 vehice/h. he task is to design an optima evacuation pan (EP) and evauate the pan s measures of effectiveness, such as cearance time C, tota traveed distance D and probabiity of bocking P C. road network, which topoogy was prompted by the arger network shown in Fig. 8, is presented on Fig. 9. Physica characteristics of the sampe network, such as ength L, number of anes W, speed imit V ð1þ and maximum density K max for each road ink a, are presented in abe 7. he kth shortest paths (k = 15) from each source to refuge center were computed for the sampe network with Eppstein s agorithm (Eppstein, 1998; Jimnez and Marza, 2003). hese potentia egress routes for each source are presented in abe Benchmark evacuation poicy o estabish a benchmark for an optima routing poicy, we tested the evacuation pan when a evacuees assigned to the 1st shortest egress routes (Fig. 10). his poicy (which we wi denote SP poicy) is an exampe of common sense soution, when evacuees choose the shortest (and as perceived the fastest) routes to eave the affected areas. We denote it with the foowing notation: EP 1 ¼fR S1 ;1; R S2 ;1; R S3 ;1g; where egress routes are R S1 ;1 ¼fa 1 a 5 a 12 a 19 a 22 a 26 g; R S1 ;1 ¼fa 2 a 5 a 12 a 19 a 22 a 26 g; R S3 ;1 ¼fa 3 a 5 a 12 a 19 a 22 a 26 g: he evacuation pan was evauated with the M/G/c/c simuation software, and according to the simuation resuts the popuation was evacuated in 2.95 h and traveed 162,000 mies ð SP C ¼ 2:95 h; D SP ¼ 162; 000 miesþ. abe 3 summarizes road ink performance measures for the SP poicy (Probabiity of bocking P C, abe 3 Evauation of SP route assignment poicy with M/G/c/c simuation Road ink P c H E(q) E( S ) E( 1 ) a (vehice/h) (vehice) (h) (h) a a a a a a a a throughput h, average number of vehices on the road segment E(Q), average traverse time S ). here is significant bocking at entering road inks a 5 and a 19. For exampe, 38% of evacuees arriving from road segments a 1, a 2 and a 3 were not abe to proceed to road segment a 5 (due to its saturation) and have to stop and wait. Bockage occurs at a 19 because of its sma capacity. In particuar we need to compare the parameters of the ink eading to a 19 and a 19 itsef (abes 4 and 7). Link a 19 has capacity of 600 vehices and critica arriva rate 2720 vehice/h. Link a12 which eads to a 19 has a capacity of 1600 vehices and a critica arriva rate of 2870 vehice/h. his particuar configuration of these two inks and difference in their properties may cause 10% bocking at a 19. his poicy iustrates that routing soutions beong to the NI-set, where shortest routes does not ensure the safest evacuation EEP IP poicy he second poicy (IP) was designed with the proposed EEP IP mode. For the given network G e (N,), an upper bound on arriva rates k Lmax were computed for a road inks a with a program LESLM. LESLM is a program which aows one to numericay compute performance parameters for a singe road ink a using M/G/c/c state dependent queueing modes (abe 2). fter that parameters t ð1þ, tðk Lmax Þ, and t ðcþ were derived (abe 4) for each type of road inks. hese parameters are required to define a inear approximation of time deay cost functions t ðk L Þ for each road ink in accordance with Eq. S 1 S D S Fig. 9. Sampe evacuation network G e (N,). Fig. 10. First shortest path route assignment poicy.

10 444. Stepanov, J.M. Smith / European Journa of Operationa Research 198 (2009) abe 4 Road ink characteristics Length L Capacity C k Lmax t ð1þ tðk Lmax Þ t ðcþ P max C V 1 V ðk Lmax Þ V ðcþ (mies) (vehice/mie) (vehice/h) (h) (h) (h) (mie/h) (mie/h) (mie/h) W = 1 and K max = 200 for a road inks. abe 6 Performance measures for the routing poicies First SP poicy EEP IP mode Reative change Evacuation time C (h) % ota distance D (mies) 162, , % Number of bocked inks 2 0 Max. bocking probabiity 38% 0 100% R S1 ;1 ¼ a 1 a 5 a 12 a 19 a 22 a 26 ; R S2 ;10 ¼ a 2 a 5 a 11 a 14 a 17 a 24 a 26 ; R S3 ;11 ¼ a 6 a 9 a 12 a 19 a 22 a 26 : Fig. 11. Famiy of time deay functions for road inks with different ength L. (12). Fig. 11 presents origina noninear time deay functions. he EEP IP mode was soved with SS9.1.3 optimization sover. he optima decision vector x for the probem is x ={x 1,1,1 =1, x 2,1,10 =1, x 3,1,11 =1} he EEP IP routing poicy is to evacuate popuation from the first source node aong the 1st shortest route, the popuation from source node S 2 shoud be assigned to the 10th shortest route and 11th shortest route shoud be used to evacuate popuation from source node S 3 (Fig. 12). his poicy was evauated with the M/G/c/c simuation program as we. abe 5 summarizes performance measures for the road inks: such as probabiity of bocking P C, throughput h, and average trave abe 5 Evauation of EEP IP route assignment with M/G/c/c simuation Road ink P C H E(Q) E( S ) E( 1 ) a (vehice/h) (vehice) (h) (h) a a a a a a a a a a a a a a Percent of evacuated vehices "Shortest path" Poicy EEP IP Poicy Expected cearance time C Fig. 13. Cearance time curves for SP and EEP IP poicies. Fig. 12. IP evacuation poicy.

11 . Stepanov, J.M. Smith / European Journa of Operationa Research 198 (2009) abe 7 Sampe road network characteristics Road ink Length Road ink Length Road ink Length a (mies) a (mies) a (mies) a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a W =1,V 1 ¼ 55 mies=h and K max = 200 for a inks abe 8 First fifteen shortest paths from a origins to destination D 1 S i k Egress route R Si;1;k Length D i1k (mies) S 1 1 a 1 a 5 a 12 a 19 a 22 a a 1 a 5 a 12 a 16 a 18 a 22 a a 1 a 5 a 12 a 16 a 17 a 24 a a 1 a 5 a 12 a 19 a 22 a 25 a a 1 a 5 a 12 a 19 a 22 a 27 a a 1 a 5 a 12 a 16 a 18 a 22 a 25 a a 1 a 5 a 12 a 19 a 22 a 23 a 28 a a 1 a 5 a 12 a 16 a 18 a 22 a 27 a a 1 a 5 a 12 a 16 a 18 a 22 a 23 a 28 a a 4 a 8 a 12 a 19 a 22 a a 4 a 8 a 12 a 16 a 18 a 22 a a 1 a 5 a 11 a 14 a 18 a 22 a a 4 a 8 a 12 a 16 a 17 a 24 a a 1 a 5 a 13 a 15 a 20 a 22 a a 1 a 5 a 11 a 14 a 17 a 24 a S 2 1 a 2 a 5 a 12 a 19 a 22 a a 2 a 5 a 12 a 16 a 18 a 22 a a 2 a 5 a 12 a 16 a 17 a 24 a a 2 a 5 a 12 a 19 a 22 a 25 a a 2 a 5 a 12 a 19 a 22 a 27 a a 2 a 5 a 12 a 16 a 18 a 22 a 25 a a 2 a 5 a 12 a 19 a 22 a 23 a 28 a a 2 a 5 a 12 a 16 a 18 a 22 a 27 a a 2 a 5 a 12 a 16 a 18 a 22 a 23 a 28 a a 2 a 5 a 11 a 14 a 18 a 22 a a 2 a 5 a 13 a 15 a 20 a 22 a a 2 a 5 a 11 a 14 a 17 a 24 a a 2 a 5 a 13 a 15 a 20 a 22 a 25 a a 2 a 5 a 11 a 14 a 18 a 22 a 25 a a 2 a 5 a 13 a 15 a 20 a 22 a 27 a S 3 1 a 3 a 5 a 12 a 19 a 22 a a 3 a 5 a 12 a 16 a 18 a 22 a a 3 a 5 a 12 a 16 a 17 a 24 a a 3 a 5 a 12 a 19 a 22 a 25 a a 6 a 9 a 12 a 19 a 22 a a 3 a 5 a 12 a 19 a 22 a 27 a a 6 a 9 a 12 a 16 a 18 a 22 a a 3 a 5 a 12 a 19 a 22 a 23 a 28 a a 3 a 5 a 12 a 16 a 18 a 22 a 25 a a 3 a 5 a 12 a 16 a 18 a 22 a 27 a a 6 a 9 a 12 a 16 a 17 a 24 a a 3 a 5 a 12 a 16 a 18 a 22 a 23 a 28 a a 6 a 9 a 12 a 19 a 22 a 25 a a 6 a 9 a 12 a 19 a 22 a 27 a a 6 a 9 a 12 a 16 a 18 a 22 a 25 a time E( S ). For purposes of comparison, abe 5 contains E( 1 ), which is traversa time aong a ink for an idea case if a vehices move up to speed imit. hough expected trave time E( S ) aong road inks a 5, a 12, a 19 and a 22 is amost twice greater than E( 1 ), this routing poicy copes with the congestion eve (P C = 0 for a road inks a ). Moreover, this IP routing soution provides evacuation time IP C ¼ 2:30 h and tota traveed distance DIP ¼ 183; 260 and no bocking at any road ink due to more baanced road utiization by evacuees. his EEP IP evacuation pan can be considered as an optima one, as it achieved no bocking and minimum cearance time, therefore it wi ensure safe evacuation process naysis of experiment abe 6 compares two approaches. Both anayzed evacuation poicies are Pareto optima. hough the poicy derived with EEP IP mode eads to greater tota traveed distance, it simutaneousy minimizes bocking and congestion eve, as we as cearance time. In comparison with the origina SP poicy, EEP IP poicy resuts in 22% decrease in cearance time. he cearance time curves for both poicies presented in Fig. 13. hese curves were obtained during evauation of anayzed poicies with simuation. he suggested EEP IP mode provides anaytica soution fast and requires ony one evauation with simuation program. 7. Summary and concusions In this paper, we have presented a methodoogy for designing optima routing poicies for emergency evacuation panning (EEP). n integer program (IP) mode formuation was presented to generate an optima route assignment for a stochastic EEP. his IP mode utiizes state-dependent decaying service rate M/G/c/c queueing modes to capture time deays functions on road inks. he performance measures for the generated evacuation poicy, such as cearance time, tota traveed distance and congestion eve, are evauated with the MGCCSimu simuation software. 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