Mathematical Modelling of Evacuation Problems: A State of Art

H. W. Hamacher, S. A. Tjandra

Mathematical Modelling

of Evacuation Problems:

A State of Art

© Fraunhofer-Institut für Techno- und

Wirtschaftsmathematik ITWM 2001

ISSN 1434-9973

Bericht 24 (2001)

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Prof. Dr. Dieter Prätzel-Wolters

Institutsleiter

EVACUATION PROBLEMS :

A STATE OF THE ART

Horst W.Hamacher

1;2

and Stevanus A. Tjandra

1

1

FraunhoferInstituteTechno-undWirtschaftsmathematik(ITWM),

D-67663Kaiserslautern,Germany.

2

Fachbereich Mathematik,UniversitatKaiserslautern,Germany.

Abstract

Thispaperdetailsmodelsandalgorithmswhichcanbeappliedto

evacu-ationproblems.Whileitconcentratesonbuildingevacuationmanyofthe

resultsareapplicablealsotoregionalevacuation. Allmodelsconsiderthe

timeasmainparameter,wherethetraveltimebetweencomponentsofthe

buildingispartoftheinputandtheoverallevacuationtimeistheoutput.

Thepaperdistinguishesbetweenmacroscopicandmicroscopicevacuation

modelsboth of which are able to capture the evacuees' movementover

time.

Macroscopicmodelsaremainlyusedtoproducegoodlowerboundsfor

theevacuationtimeand donot considerany individualbehaviorduring

the emergency situation. These bounds can be used to analyze

exist-ingbuildingsorhelpinthedesignphaseofplanning abuilding.

Macro-scopicapproacheswhicharebasedondynamicnetwork owmodels

(min-imum cost dynamic ow, maximum dynamic ow, universal maximum

ow,quickestpathandquickest ow)are described. A specialfeatureof

thepresentedapproachisthefact, that traveltimesof evacuees arenot

restrictedto beconstant, but may be density dependent. Using

multi-criteriaoptimization priority regions and blockage dueto re or smoke

maybeconsidered. Itisshownhowthemodellingcanbedoneusingtime

parametereitherasdiscreteorcontinuousparameter.

Microscopicmodelsareabletomodeltheindividualevacuee's

charac-teristicsand theinteractionamongevacuees whichin uencetheir

move-ment. Duetothecorrespondinghugeamountofdataoneusessimulation

approaches. Some probabilistic laws for individual evacuee's movement

are presented. Moreover ideas to model the evacuee's movement using

cellularautomata(CA)andresultingsoftwarearepresented.

Inthispaperwewillfocussonmacroscopicmodelsandonlysummarize

someoftheresultsofthemicroscopicapproach. Whilemostoftheresults

areapplicabletogeneralevacuationsituations,weconcentrateonbuilding

Evacuation asoneaspectof emergencyprocessescanbesimplydened as the

removal of residents from a given area that has been considered as a danger

zoneto safetyasquicklyas possibleandwith utmostreliability. Twodierent

evacuationscenarioscanbeconsidered.

(i) Precautionary

Inthistypeofevacuationtheestimationoftheevacuationtimecompared

tothehazardpropagationtimeandtheestimationoftheriskcanbedone

apriori. Hence,time and potentialrisks arethekeycomponentsof this

typeofevacuation.

(ii) Life-savingoperations

ThistypeofevacuationoccurswheninsuÆcientwarninghaspreventedthe

organizerfromconductingapre-emergencyevacuationplanning.Hereitis

morelikelythatproblemsastherescueofinjuredevacueesinandaround

thedamagedarea,routeclearance,etc. haveto bedealtwith.

Evacuationproblems mayarisein dierenttypesofsystems,suchas

build-ings,citiesorregions,ortransportationcarriers(e.g. train,shipandairplane).

The system structures ( e.g. population and the behavior of people at risk,

hazard propagation speedand charateristics) essentiallyin uence the optimal

planningin the correspondingsystem. In general,the followingis alist of

in-formationswhich wouldbeidealtohavein evacuationplanning(andneedsto

beestimatedsomehow,ifnotavailable):

 Typeofsystemdenedbylayout/geographicinformationandfamiliarity,

forexample: oÆcebuilding,shoppingmaland airport

 Behaviourestimationoftheoccupantsunderpanicsituation.

 Occupantsdistribution(includesage,genderanddisability).

 Source and location of hazard, hazard propagation speed/charateristics

andfactorsaecting thehazardpropagation.

 Safedestinations(refugeplaces).

 Availabilityofemergencyservicefacilitiesandpersonel.

These informations areused to dene the dynamic severity matrix [r

it

] where

vectorr

it

statestheestimatedseverityorhazardlevelofareaiattimet. The

number of components of vector r

it

is determined bythe numberof

informa-tionswhichareconsidered. Equation(1)showsanexampleofdynamicseverity

matrix. [r it ]= 0 B B B B B B B B @ . . . ::: r it = 0 B B @

relevelofareaiat timet

temperaturelevelofareaiat timet

smokedensitylevelofareai attimet

toxicitylevelofareaiattimet

1 C C A ::: . . . 1 C C C C C C C C A (1)

it

toascalarr

it

. Ifhisthenumberofvectorcomponentsandw

l

istheweightof

componentl,then r

it

canbescalarizedasfollows.

r it = h X l=1 w l r itl

This matrix will be used to determine the evacuation priorities that will be

discussedin moredetailin Section8.

The evacuation time, that is the time needed to complete an evacuation

process, basicallyconsistsofthreemaintimecomponents([30], [47])namely:

 Thetimeevacueesneedto recognizeadangeroussituation. Thistime is

in uencedmainlybythereliabilityofthealarmsystemandthefamiliarity

ofevacueeswithemergencysignals.

 The time evacuees need to decide which course of action to take. This

timeis in uencedbythe experienceof evacueesin facingtheemergency

situation.Thiscan,forinstance,begeneratedthroughemergencypractice

andtraining.

Behaviouralandorganizationalfactorsarethemaincontributorstotheduration

ofthesetimes. Thesubsequentdecisionsaremadeduringthemovementtothe

safety, especially whenevacueesencounter a"hazard"route, i.e. onewhich is

aectedbyre,smoke,etc.

 Thetimeevacueesneedtomovetowardsthesafetyarea,whichisknownas

egress time. Thelatterisin uencedbytheavailabilityofemergencyexit

signs,wellplannedevacuationprocedures,constructionalfactors(eective

width of walkway, slope of stairs), and human behaviour during panic

situations.

Since the behaviouraland organizationalfactorsare the main contributorsto

the rst two time components, it is hard to predictanalytically the duration

of those time components. Therefore, most evacuation modelsemphasize the

calculation of egress time and treat the result asthe lowerbound of the real

evacuationtime. This willalsobetheapproachin thispaper.

Thenextsection describesthe ideaofmacro- andmicroscopic models and

give a list of classes of existing models which can be used in the evacuation

problems. Informations on various specic mathematicalmodels are given in

Section3-6,likedynamic ows(Section 3),maximumdynamic ows(Section

4), universal maximum ows(Section 5), quickest path and ows(Section 6).

InSection 7weshowthe interrelationbetweenthese models byestablishinga

tripleoptimizationresult. Section 8dealswithmultiplecriteria. Thefactthat

the travel time may be dependend on the evacuationdensity is considered in

Section9. Astrongermodel,butsoundsmorediÆculttosolveisthecontinuous

time dynamic network ow model which is the subject of Section 10. Before

concluding the paper we discuss microscopic models, in particular simulation

Approacheswhichare usedto model evacuationproblems maycome from

dif-ferentproblem elds, such as,network owproblems, traÆcassignment

prob-lems, and simulation. Table2listsclassesof models which canbeused in the

evacuationproblemmodelingalthough someofthemmayhavebeenoriginally

developedfordierentpurposes. Ingeneral, thereare twoapproachesusedto

model evacuation problems which emphasize on the estimation of the egress

time,namelymacro- andmicroscopic models .

Macroscopicmodelsaremainlybasedonoptimizationapproachesanddonot

consider individualdierencesanddecisionsforselectingegressroutes,i.e.

oc-cupantsaretreatedasahomogeneousgroupwhereonlycommoncharacteristics

are taken into account. Since the timeis a decisive parameterin the

evacua-tionprocess,mostmacroscopicapproachesarebasedondynamic network ow

models (see e.g. [14], [18], [19], [41], [22], [12], [43], [50]). The common idea

of these models isto representa building andthe attributes of the building's

components in a static network G. The modeling of evacuation over time is

then donein adynamic network G

T

which is thetime expandedversionof G

andthenetwork owscorrespondtoevacuationprocesses(see,forinstance,[1]

foranoverviewonnetwork owtheory).

ThestaticnetworkGisusedtomodelsupplyanddemandpoints,androutes

which areused to transfersuppliesto demands. These routesmayhavesome

intermediate transshipmentpoints. Inthestaticnetwork owmodels, supply,

demand andtransshipmentpointsaremodeledbynodeswhileroutesare

mod-eledbypathsofthegraph. Apathofthegraphiscomposedbynodesandarcs,

whereanarcconnectstwoadjacentnodes. Theinterrelationbetweennodesand

arcs can, for instance, be described bythe node-arc incidence matrix. In the

representationofabuildingusingastaticnetwork,nodesmayrepresentrooms,

lobbies orintersection points,while arcs canbe usedto modelcorridors,

hall-ways,stairwaysoraconnectionbetweentwointersectionnodes. Somelocations

in the building that house a signicant number of evacuees are considered to

be source nodes in the network. The supply of a source node is given by an

estimate of the number of evacuees in the location that the node represents.

Thebuildingexitsorsafetylocationsthatmightbeconsideredasthenal

des-tinationofevacuees'movement,areconsideredassinknodes. Intheevacuation

problem we have only one sink node by connecting all the exit nodes to one

articialnodeandassignthetotalnumberofevacueesasthedemand valueof

this node. Hence, evacuation problemscanbemodeledas multi-source/single

sinknetwork owproblem. Eachnodehasacapacitywhichistheupperbound

of the number of evacuees simultaneously allowed to stay in the node. This

nodecapacitycan bedetermined,forinstance, by

nodecapacity :=minf

oorspacearea

minimumrequiredareaperperson

;

maximumallowableweight

averageweightperperson

g

Arcshaveotherattributes,suchas owcapacityandtraveltime. Thearc ow

capacity istheupperbound ofthe numberof evacuees perunit time thatcan

traversethe arc. The traveltime is the time needed to travel from one node

process, theconnection betweentwopositions mayonlybetemporary dueto,

for instance, blocking by re or smoke. In this case, the arc that represents

theconnectionmustalsobetemporary,i.e. thearccapacitycanbesettozero

after sometimes. These timeconstraintscannotbeproperlymodeled bythe

staticnetwork owmodelsbut thedynamicones. Moreover,wecanformulate

thedynamicnetwork owproblemintwowaysdependingonwhetherweusea

discreteorcontinuousrepresentationoftime.

Intheareaofdynamicnetwork owproblems,someoftheexistingmodels

assumeconstantattributes, e.gconstanttraveltimefrom onenodeto another

and constantarc owcapacity. Theconstanttraveltimemightbedetermined

according to some predetermined queuing levels such that the model can be

solvedeÆcientlybutstillabletogivequiterealisticresults. Inthispaperitwill

beshown that themodelsmay be extended,thus providing a better estimate

ofthenalevacuationtime.

Microscopicmodels,in which theindividualevacuees'movementis

empha-sized, are based on simulation. These models consider individual parameters

(e.g. walking speed, reaction time, physical ability) and interaction of each

evacuee with other evacueesduring the movement. In recentyears there is a

growinginteresttousecellularautomataasthebaseofmicroscopicsimulation

in theeldof pedestriansand traÆcmovement(seeforexample[8], [42],[51])

whichhavecloseinterrelationswithevacuationproblems.

ModelClass EvacuationModel References

StaticNetwork Shortestpath [22], [68]

Minimumcostnetwork ow [68]

Quickestpath [15], [16],[38],[61]

Discrete Time

Dy-namicnetwork

Minimumturnstilecost [14], [18],[41],[50]

QuickestFlow [12], [23]

Universallymaximum ow [34], [48],[67]

Minimumweightpath(multi

ob-jectives)

[43]

Lexicographicallyminimalcost [33]

Flowdependentexitcapacity [18], [19]

Continuous Time

Dynamic Network

Constant capacity and travel

time

[24]

Timedependentcapacity

(maxi-mal ow)

[2],[55]

Universally maximum ow with

zerotraveltime

[25], [52]

TraÆcassignment Transportationnetwork [62], [69]

Density dependent travel time

(singleobjective)

[13], [17], [35], [37], [39],

[60]

Simulation Probabilisticmodels [21], [46],[47]

A discretetime dynamic network owproblemis adiscretetime expansionof

astaticnetwork owproblem. Inthiscasewedistribute the owoveraset of

predeterminedtimeperiods t=1;2;:::;T.

Denition3.1 Let G=(N;A) be adirectednetwork with N the set of nodes

and A the setof arcs(the staticnetwork). On each arc (i;j)2A travel times

 ij

aregiven which areassumedtobeconstant. The timeexpansionof Gover

a timehorizon T denes the dynamic networkG

T =(N T ;A T ) associated with Gwhere N T :=fi(t) ji2N ; t=0;1;:::;Tg andA T

consists ofthe setof movementarcs A

M A M :=f(i(t);j(t 0 ) j(i;j)2A ; t 0 =t+ ij T ; t=0;1;:::;Tg

andthe setof holdover arcsA

H A H :=f(i(t);i(t+1)) ji2N ; t=0;1;:::;T 1g i.e. A T :=A M [A H

Figure2showsaT-timeexpansionofthestaticnetworkofFigure1,withT =4.

The time period t is dependent on the basic unit  in which travel times

aremeasured. Thus, ifwechoose5secondsasthelengthofthebasicunit (i.e.

=5),thenspecifyingthreetimeperiods(i.e. t=3)fortraversinganarcmeans

weneedfteensecondsto doso. Thenumberoftime periodsT isobtainedby

dividingtheevacuationplanninghorizonofinterestbythelengthofthebasic

unit. The smaller themoreacurately themodel representstheactual ow's

evolution. Choosing  too small, however, will result in undesirable size of

the network and may have fractional arc capacities which make the problem

diÆculttosolve. Hence,thechoiceofisacompromisebetweenmodelrealism

andmodelcomplexity.

Sincethedynamicnetworkhas(T+1)copiesofeachsourcenodeandeach

sink node, thedynamicnetwork willhavemultiple sourcesand multiple sinks.

Therefore in order to handle many sources and sinks, oneintroduces a super

source s and a super sink d to create asingle source/single sink network (see

Figures2and3). Inevacuationproblems,thesupersinkcanbeinterpretedasa

commonsafetyarea. Howthesupersourceisconnectedtothesourceisactually

problem-dependent. Inthenetworkclearingproblem(clearingthenetworkfrom

initial occupancies), the supersource is connected onlyto thetime zero copy

ofthesourcenodes(seeFigure 2). Inthiscase,wemayhaveholdoverarcsfor

sourcenodes. Arcs fromthesupersourcehavezerotraveltimeand capacities

are equal to initial occupancies. In the maximumdynamic ow problem (see

Section4), thesupersourceisconnectedtoalltime-copiesofthesourcenodes.

Inthis case,wedonothaveholdoverarcs forsourcenodeswhich donothave

predecessors(e.g. node1inFigure1)asshowninFigure3. Arcsfromthesuper

sourcetoothernodeshavezerotraveltimeandinnitecapacities. Ontheother

zerotraveltimeandinnite owcapacities.

By constructingthe dynamic network as dened above, dynamic network

owproblemscanalwaysbesolvedasstatic owproblemsintheexpanded

net-work.Also,itmaybenotedthattheequivalentstaticproblemdoesnotrequire

keepingarccapacitiesandtraveltimesxedovertime,asassumedinDenition

3.1. ButtheseassumptionsareessentialforbuildingeÆcientalgorithmstosolve

the problem. The upper bound for the number of nodes and arcs in discrete

timedynamic networkisdescribedbyProposition3.1.

Proposition3.1 If n:=j N j and m:=j Aj then n(T+1) and (n+m)T+

m P

(i;j)2A 

ij

are the upper bound for the number of nodes andarcs in G

T

withoutconsidering super sourceandsuper sink, respectively.

Sincewedonotuseanyarcinthepathfromthesupersourcetoanysinknode

attimegreaterthanT,wecanreducethesizeofthetime-expandednetworkby

eliminatinginessentialarcsincludingthecorrespondingnodes(seeFigure3).

1

2

3

4

{3,5}

{3,20}

_{0,

_∞

_}

{4,8}

(1,2)

(1,2)

(1,2)

(1,2)

(2,3)

{initial contents, node capacity}

(travel time, arc capacity)

1 := room 1

2 := room 2

3 := lobby

4 := safety exit

Figure1: StaticNetworkGofaSimpleBuildingLayout.

Inthedynamicnetwork owmodels,wedenotebyx

ij

(t)the ow(e.g. the

number of evacuees moving at time t) that leave node i at time t and reach

nodejattimet+

ij

. Flowsfromnodeiattimettothesamenodewithtravel

time

ii

=1representthenumberofevacueeswhoprefertostayinthebuilding

componentrepresentedbynodeiattimetforatleastoneunittime. This ow

isdenoted byy i (t+1), i.e. y i (t+1):=x i(t);i(t+1)

Thecapacityofmovementarcs(i(t);j(t+

ij ))2A M isdenotedbyb ij (t)where

weassumewithoutlossofgeneralitythat

b ij (t):=minfb ij (t 0 ):t 0 =t;t+1;:::;t+ ij g

Thecapacityofaholdoverarc(i(t);i(t+1))2A

H

is determinedbythe node

capacity a

i

(t), and represents how many evacuees can stay in the node at a

given time. With (X;Y) as the general objective and with q

i

as the initial

10

11

12

13

14

20

21

22

23

24

30

31

32

33

34

40

41

42

43

44

time

0

1

2

3

4

s

Super source

{10}

[0,3]

[0,4]

[0,3]

[0,5]

[0,5]

[0,5]

[0,5]

[0,8]

[0,8]

[0,8]

[0,8]

[0,20]

[0,20]

[0,20]

[0,20]

[cost, arc capacity]

[0,2]

[0,2]

[0,2]

[0,2]

[0,2]

[0,2]

[0,2]

[0,2]

[0,2]

[0,2]

[0,2]

[0,2]

[0,2]

[0,2]

[0,2]

[0,2]

[0,4]

_[0,4]

[0,4]

{supply > 0 or demand < 0}

{0} = transshipment node

{0}

{0}

{0}

{0}

{0}

{0}

{0}

{0}

{0}

{0}

{0}

{0}

{0}

{0}

{0}

{0}

{0}

{0}

{0}

d

SuperSink

[1,

∞

]

{-10}

{0}

[2,

∞

]

[3,

∞

]

[4,

∞

]

Figure 2: Dynamic Network G

T

of the Static Network G of Figure 1, with

T =4.

modelforevacuationprocessescanbeformulatedasfollows.

min/max T X t=0 (X;Y) (2) y i (t+1) y i (t) = X k 2pred(i) x k i (t  k i ) X j2succ(i) x ij (t) ; t=0;:::;T; 8i2N; (3) y i (0) = q i ;8i2N; (4) 0y i (t)  a i (t);t=1;:::;T 1;8i2N; (5) 0x ij (t)  b ij (t);t=0;:::;T  ij ;8(i;j)2A (6) where

pred(i):=fjj (j;i)2Ag ; succ(i):=fjj (i;j)2Ag

arethenodeswhich arepredecessorsandsuccessorsofnodei,respectively.

Inorder to measure the time whenevacuees reach theirnal destinations,

so-calledturnstilecost([14], [33])isdenedoneach arcasfollows.

Denition3.2 If D isthe set of sink nodes of the static network G andd is

the supersink node ofthe associateddynamic network G

T

,the(turnstile) cost

of any arc (i(t);j(t

0

=t+

ij

))2A

T

isdeneddierent from 0if and only if

i2D andj(t

0

)=d. In thiscasec(i(t);d)=t.

LetusdenoteS N astheset ofsourcenodesofthestaticnetworkG. Using

10

11

12

20

21

22

30

31

32

33

41

42

43

44

time

0

1

2

3

4

s

Super source

[0,3]

[0,4]

[0,3]

[0,8]

[0,8]

[0,20]

[0,20]

[0,20]

[cost, arc capacity]

[0,2]

[0,2]

[0,2]

[0,2]

[0,2]

[0,2]

[0,2]

[0,2]

[0,2]

[0,2]

[0,2]

[0,2]

[0,4]

_[0,4]

[0,4]

d

SuperSink

{-10}

[0,3]

[0,3]

[0,4]

[0,4]

[0,3]

[0,3]

[0,3]

[1,

∞

]

[2,

∞

]

[3,

∞

]

[4,

∞

]

Figure 3: Dynamic Network G

T

of the Static Network G of Figure 1, with

T =4,withoutInitial Contents,andbyDeletingInessentialArcs.

averageevacuationtimerequiredbyanevacueetoleavethenetwork,i.e.

(X;Y):= P T t=0 P i2D tx id (t) P i2S q i

Sincethedenominatorisconstantanddependsonlyonthe owvariables,one

just needstodenetheobjectivefunction as

(X;Y):=(X)= T X t=0 X i2D tx id (t)

The initial occupancies are modeled by using ow from the super-source s

to each source node. Under assumptions of constant capacity (i.e., b

ij (t) = b ij ;8(i;j)2Aanda i (t)=a i

;8i2N;8t)andconstanttraveltime,the

min T X t=0 X i2D tx id (t) (7) x si (0) = q i ;8i2S; (8) T X t=0 X i2D x id (t) = X j2S q j ; (9) y i (t+1) y i (t) = X k 2pred(i) x k i (t  k i ) X j2succ(i) x ij (t) ; t=0;:::;T; 8i2N (10) y i (0) = 0;8i2N; (11) y i (t) = 0;8i2D;t=0;:::;T (12) 0y i (t)  a i ;t=1;:::;T;i2N D (13) 0x ij (t)  b ij ;t=0;:::;T  ij ;8(ij)2A (14)

We can treatthe time-expandednetwork asdened in theDenition 3.1 asa

staticnetworkandthenapplyanyminimumcoststaticnetwork owalgorithm

(see,e.g.,[1])to obtainthesolution.

Basedonthisminimumcostdynamicnetworkoptimization,KiskoandF

ran-cis [41] developedEVACNET+, anevacuationsoftware which can beused to

determinetheegresstimeandpossiblebottlenecklocations. Theminimumcost

dynamic network optimization problem is solved as a staticnetwork by using

theNETFLOcode[40].

In public buildings where the number of evacuees is diÆcult to estimate,

one can model the evacuation problem asa maximum dynamic network ow

problem. Inthenextsectionwewillgiveageneraldescriptionofthemaximum

dynamicnetwork owproblemsand thealgorithmstosolvethem.

4 Maximum Dynamic Flows Problem

Giventhetime horizonT,maximumdynamic owproblems(MDF) maximize

the dynamic ows reaching the sink. These problems can be used to model

evacuation processeswhich have noreliable information about thenumber of

evacuees. As already mentionedin Section 3, thesuper sourcenode in G

T is

connectedtoeverytime-copyofeverysourcenodeandthereisnoholdoverarc

forsourcenodesandsinknodes. Arcsfromsuper-sourcehavezerotraveltime

andinnitecapacities.

TheobjectivefunctionofMDFisdened asfollows.

(X):= t=T X t=0 X i2D x id (t)

max t=T X t=0 X i2D x id (t) (15) y i (t+1) y i (t) = X k 2pred(i) x k i (t  k i ) X j2succ(i) x ij (t) ; t=0;:::;T;8i2N (16) y i (0) = 0; 8i2N; (17) y i (t) = 0; 8i2S[D;t=1;:::;T (18) 0y i (t)  a i ; t=1;:::;T;8i2N S[D; (19) 0x ij (t)  b ij ; t=0;:::;T  ij ;8(i;j)2A (20)

ThesolutionofMDFdenedbyEq. (15)-(20)canbeobtainedbyrepeating

the feasible ows along somechains (see Denition 4.1) of the static network

from thesource to the sink. The owson these static chains are repeated in

the dynamic network for everytime period within the time horizon T. This

approachiscalledtemporally repeated ow technique(seeDenition 4.2).

Denition4.1 (Chain, chain ow and chain decomposition)

 A chain is a sequence of nodes P = fi

1 ;i 2 ;:::;i k g; k  2, such that (i j ;i j+1 ) 2 A and i j 6= i j 0 whenj 6= j 0 ; for j;j 0 = 1;:::;k 1 , i.e. a

chainhas norepeatednodes.

 Achain ow =<jP j;P >isastatic owofvalue jP jalongthe chain

P.  Let =fP 1 ;P 2 ;:::;P l

gbeaset ofchain owsand letjP

i

jbethe chain

ow along path P

i

. is a chain decomposition of the static ow f if

P i=l i=1 jP i j=f.

Itiswell-knownthat anynetwork owcanbedecomposedintochain ows

(plus possiblysome owsoncycles).

Denition4.2 (Temporallyrepeated ows) Let =<jPj;P >beachain

ow. The temporally repeated ow

T

is adynamic ow obtainedby repeating

(T+1 (P))timesthechain ow ,i.e.,bysendingjP junitsof ow every

timeperiodfromtimezerototimeT+1 (P)along thesamepath(static) P.

Thenexttheorem showsthat themaximumdynamic owproblem canbe

solved as a minimum cost ow problem (MCFP) in the static network. The

readermayreferbooksonnetwork owtheoryformoredetailsonMCFP (see

forinstance[1]).

Proposition4.1 ([26]) Finding a maximum dynamic ow is equivalent to

solving a MCFP. In particular, the temporally repeated ow obtainedfromthe

maximal number of persons, which can be evacuated within T time periods

from a given building, we only have to solve amin cost ow problem in the

small,staticnetworkG.

Algorithm4.1

step1 Apply a minimum cost ow algorithm to the original static network G.

Letx



bean optimal solution.

step2 Decomposex



intok chain owson P

1 ;P 2 ;:::;P l suchthat x  = i=l X i=1 jP i j

step3 Repeateachchain owP

i

from time0till timeT (P

i ).

Example4.1

Figure4showsastaticnetworkofasimplebuildinglayoutwithtraveltimeand

capacity parametersattached on everyarc. Node1and 6are sourceand sink

nodes,respectively. Itisdesiredtocalculatethemaximumnumberofevacuees

who can reach the safety during T = 7 time units. The optimal solution of

MCFP isobtainedasshownintable4.1 (onlythepositive ows).

1

2

4

5

3

6

(3,6)

(1,5)

(0,5)

(1,1)

(0,5) (1,5)

(3,6)

(1,1)

Figure4: StaticNetworkGforExample4.1.

Table4.1: OptimalMaximumFlowsforStaticNetworkinExample4.1.

Arc (1,2) (1,3) (2,4) (2,6) (3,6) (4,3)

Flow(f) 6 1 5 1 6 5

Step2ofthealgorithm givesthefollowingchain ows:

 P 1 =(1;2;6);(P 1 )=4;jP 1 j= 1; P 1

must be repeated four times for

t=0;1;2;3.  P 2 =(1;2;4;3;6);(P 2 )=7;jP 2 j=5;P 2

mustberepeatedonlyonceat

timet=0.  P 3 =(1;3;6);(P 3 )=4;jP 3 j= 1; P 3

must be repeated four times for

13unitof owinwhich2unitsarriveatthesinkattimet=4;5;6and7units

at time t = 7. It means that oneneeds at least 7 time units to evacuate 13

evacueeswhoareatthelocationrepresentedbynode1inthebeginningofthe

evacuationprocess.

time

0 1 2 3 4 5 6 7

1

node

2

3

4

5

6

1 1 1

_{1 1 1}

₆

₁

Figure5: AmaximumDynamicFlowforExample 4.1.

The solutionof themaximum dynamic ow in this exampledoes notuseany

holdoverarcs, i.e. x

ii

(t)=y

i

(t+1) =0;8i2N;8t=0;:::;T 1. Infact, it

canbeshown,thatthemaximumdynamic owpoblemwithconstantcapacities

andtraveltimesneverrequireshold-overatanynodes[34]. Therefore,variables

y i

(t) canbeeliminatedfrom theproblem formulation.

5 Universal Maximum Flow Problem

The(discrete)Universalmaximum owproblem(UMF)wasintroducedbyGale

[28] asavariantofthemaximumdynamic owproblem. UMFis theproblem

of ndingmaximumdynamic owsreachingthesink ateverytime period t=

1;:::;T. Hence,theoptimalsolutionof UMFisthesolutionof themaximum

dynamic owproblem,notonlyfortheallotedtimehorizonT,butalsoforany

smaller time horizons. Such a ow is also known in the literature asearliest

arrival dynamic ow(seeHoppeandTardos[34]). Itcanbeformulatedas

max t=T 0 X t=0 X i2D x id (t) ;8T 0 =1;:::;T (21)

(2-ndrow). Time 0 1 2 3 4 5 6 7 Arc (1,2) 6 1 1 1 6 1 1 (1,3) 1 1 1 1 1 1 1 1 1 (2,4) 5 5 (2,6) 1 1 1 1 1 1 1 1 1 (3,6) 1 1 1 6 1 1 6 (3,5) 1 1 (4,3) 5 5 (5,2) 1 1 arrivalat6 2 2 2 7 1 1 2 2 7

The relevance of UMF for the evacuation problem is obvious. In every time

periodthemaximalnumberofevacueesisbroughttosafety,suchthatan

evac-uationmodeledbyuniversalmaximum owsisaverysafeone.

Bydenition,everyuniversal maximum owis amaximumdynamic ow,

but the reverseis nottrueas is illustratedbythe ow distribution in Table2

using thedataofExample 4.1. InTable2therstline ofeacharc owshows

the optimal solution of maximum dynamic ow problem and the second line

showstheoneofUMF.

Thefollowingalgorithmndsauniversalmaximum owinthecaseofstatic

networkswithsinglesourceand singlesink.

Algorithm5.1 ([36])

step0: Identify K, the value of the maximum dynamic ow with respect to the

timehorizon T,andsetK asthetotal capacity ofarcsfromsupersource

stotime-copiesof sourcenode,i.e.

K= T  X t=0 b si(t)

with i(t) the t-th time-copy of source node i of the static network and 

theshortestpossibletimetoreachthe sinknodefromthesourcenode. Set

T

the capacity of arcs (i(t

0

);d);i2D;t 0

>t to zero (i.e., close temporarily

thosearcs) usingthe Ford-Fulkerson labeling algorithm([26]).

step2: Ift=T stop. Otherwise,increasetby1, open arc(i(t);d)andgo tostep

1.

The validity of the algorithm has been proved in Minieka [48]. Note that

this algorithm aswell asalternativealgorithms proposed byMinieka [48] and

Wilkinson[67]arenon-polynomial(bothauthorsuseshortestaugmentingpath

algorithm). Moreover,theyallworkwiththe timeexpandednetwork,the size

ofwhichisdependendonthetimehorizonT.

HoppeandTardos[34]proposedapolynomialapproximationalgorithmfor

the discrete UMF by introducing ageneralization of the chain decomposition

introduced in the previous section, the non-standard chain decomposition. It

is achain decomposition of thestatic ow which canuse arcs in theopposite

direction of the static ow, i.e. it is allowedto send ows in the direction of

arc (j;i) where (i;j) 2 A and (j;i) 62 A. In this case the travel time along

(j;i)isthe negativevalueoftheoneon(i;j)2A. Byusingthenon-standard

chain decomposition,one can again usethe ideaof temporally repeated ows

to produce UMF.Figure 6showsdynamic owsinduced bythe nonstandard

chain decomposition. Consider the static network of Example 4.1. Let =

time

0 1 2 3 4 5 6 7

1

node

2

3

4

5

6

1

1

1 1

1 1

6

1

Figure 6: UniversalMaximal FlowInducedbyNon-standardChain

Decompo-sition fP 1 ;P 2 ;P 3

gbetheset ofchains,with

P 1 =f1;3;5;2;6g;P 2 =f1;2;5;3;6g;P 3 =f1;2;4;3;6g

isanon-standardchaindecompositionsinceP

2

2 usesarcs(5;2)and(3;5)

the opposite direction of the static ow. In order to keep the feasibility due

tocapacityconstraints,theremustbeanotherchain owP



thatalsousesarc

(i;j) but sends ow in the opposite direction of P to cancel chain owP on

(i;j). Sincethe traveltime of the opposite arcis nonpositive, ifchain owP

arrives at node i at time t, then the chain ow P



must arrive at i at time

t 

t. Similarly,ifchain owP stops usingarc(i;j)attime t,thenP

 must

continuesending owfromjuntiltimet



t. InFigure6,chain owP

2 starts

usingarc (3;5)in theoppositedirection (5;3)at timet=3andstops usingit

at timet=4. On theother hand,chain owP

1

startsusingarc (3;5)at time

t=1andstopsusing itattimet=6.

Byusing thisnon-standardchaindecompositionand applyingthecapacity

scaling shortestaugmentingpathalgorithm, HoppeandTardos[34]developed

therstpolynomialapproximationalgorithm,withtimecomplexityO(

m 

(m+

nlogn)logU), whereU isthemaximumarccapacity. It isproved tobewithin

(1+)ofoptimality.

6 Quickest Path and Quickest Flow

The quickest path problem as introduced by Chen and Chin [15] is another

variantoftheshortestpathproblem. Theobjectiveisto sendapredetermined

numberof units from their initial position (i.e. thesource node) to the

desti-nation(i.e. the sinknode) asquickly aspossiblealong asinglepath. Butthe

notionof"shortest"dependsnotonlyonthetraveltimebutalsoonthenumber

of unitsthat haveto bedeliveredalongthepath. Flowsaresentcontinuously

overthe time. The quickest path problem is relevantto a special evacuation

problemwhere evacueesmayuseonlyasinglepathortunnelfrom theirinitial

position,thatwillnotbeinterferedbyevacueesfrom otherplaces. Anexample

of this problemis the evacuationof spectatorsfrom asportsstadium. In this

case, the quickest path model is applied independently to each network that

modelsthe evacuationof each standin thestadium. Another problemknown

asquickest ow[12]issimilartothequickestpathproblem. Here,itisallowed

to send owsalong multiple paths. Thelatterproblem is knownasthe

mini-mum timenetwork clearing problemwith its obvious relevance forevacuation

problems. Inthissectionwerstdiscussthequickestpathandthenthequickest

owproblem.

6.1 Quickest Path Problems

Denition6.1 Given apath P :=(i

1 ;i

2 ;:::;i

k

)inthe staticnetwork G.

 Thecapacityof pathP,b(P)isdenedas

b(P)= min 1ik 1 b(i i ;i i+1 )

 Thelengthof path P (in timeunit) (P)isdenedas

(P):= i=k 1 X i=1 (i i ;i i+1 )

1 k

T(;P)=(P)+



b(P)

Dierentfromtheclassicalshortestpathproblemtheconcatenationproperty

(i.e. thepropertythateverysubpathofashortestpathisalsoashortestpath)

is no longer true for the quickest path problem, asis shown by the following

example.

s

c

d

b

a

(5,10)

(5,5)

(6,20)

_(6,20)

(2,4)

(travel time, flow rate capacity)

Figure 7: ExampleforQuickestPathProblem.

Example6.1 Refering to the example of Figure 7, we want to send out 20

evacuees from source node s to sink node c. Then thequickest path is P

1 =

(s;b;c) with egresstime T(20;P

1

) = 12+

20 20

=13. Suppose instead of c we

take dasthe nal destination, thenthe quickest path is P

2 =(s;a;c;d) with T(20;P 2 )=17. ButP 3 = (s;a;c) P 2

is notthe quickest path from s to c,

violatingtheconcatenationproperty. Weseealsothat P

1

isnotthe(classical)

shortestpath(withrespecttotraveltime)fromstoc. Thenumberofevacuees

reaching the destination d overtime t is shown as function I

d (t) in Figure 8 with I d (t)=  4(t 12) ;t12 0 ,otherwise

12

17

20

t

)

(t

I

d

Figure8: ThenumberofEvacueesReachingtheDestinationd.

The following two theorems show the interrelation between quickest and

shortestpath in Gwhichturn outto beusefulin developinganalgorithm for

solvingthequickestpathproblem. Forthispurposewedeneforgivennetwork

Gsending unitsof ow then

 P isashortests-dpathin G(b(P)).

 Anysubpath ofP itself mustbe ashortestpath inG(b(P)).

Theorem6.2 (Rosen [61]) Let r be the number of distinct capacity values

andletP j beashortests dpath inG(b j );j=1;:::;r. If (P l )+  b(P l ) = min 1jr f(P j )+  b(P j ) g; then P l

isthe quickests dpathin Gsending unit of ows.

Basedonthesetwotheorems,Rosen,etal. [61]developedasimplealgorithmas

follows. Intheinitialstepwecomputeforeachj=1;:::;rashortests dpath

in G(b

j

) and then apply Theorem 6.2. Using Fredman and Tarjan [27], each

computationofP

j

requiresO(m+nlogn)timesuchthattheoveralcomplexity

ofthealgorithmisO(rm+rnlog n).

This result can be extended by considering the more realistic assumption

thatthetraveltimeisdensitydependent(i.e. owdependent). Hereweassume

that thetraveltimeis astepfunction ofthe ow,which isnondecreasing,and

constantineachunit of ow. Letk

ij

bethenumberofdistincttraveltimesof

arc (i;j)and k



:=max

(i;j)2A fk

ij

g. Thetraveltime of arc(i;j)then canbe

dened asfollows.  ij (x ij )= 8 > > < > > :  1 ij ;0x ij b 1 ij  2 ij ;0x ij b 2 ij ::: ;:::  k ij ij ;0x ij b k ij ij

For each arc (i;j) we create k

ij

articial nodes denoted ij

1 ;:::;ij

k ij

. Then,

weconnectnodeito nodeij

l

andnode ij

l

tonodej foreach l=1;:::;k

ij as

shownin Figure9. Thecapacityof arc(i;ij

l ) isb

l ij

with traveltime 

l ij

. The

capacityandtraveltimeforarc(ij

l

;j)are1and0respectively. Themodied

network will havemaximum(n+mk

 ) nodes, 2mk  arcsand mk  numberof

dierent capacities. The quickest path then can be obtainedby applying the

previousalgorithmtothemodiednetwork.

Proposition6.1 ([64]) Let P is the quickest path with arc (i;j)2 P. Then

arc (i;j)willuse the closestcapacity totheb(P), i.e. if b

l 1 ij <b(P)b l ij then (i;ij r

)2P for r=land(i;ij

r

)62P for r6=l.

6.2 Quickest Flow Problems

Unlike the quickest path problem, the quickest ow problem (QFP) relaxes

the limitation of a singlepath to multiple paths. QFPis a dynamic network

owproblem with singlesource and single sink that clear the network in the

minimum possible time ([12], [23]). The objective function of QFP can be

formulatedasminimizingthetimehorizonT =:T(v)wherev isthenumberof

i

j

i

j

ij

k

ij

1

ij

2

ij

( )

1

1

,

ij

ij

b

λ

( )

2

2

,

ij

ij

b

λ

(

ij

k

ij

)

ij

k

ij

,

b

λ

( )

0

,

∞

( )

0

,

∞

( )

0

,

∞

Figure9: Network ModicationforDensityDependentTravelTime.

intheminimumturnstilecostdynamicnetwork owmodeldiscussedinSection

3. The following theorem states properties of the value v(T) of a maximum

dynamic owovertime period T, that canbe used to derive analgorithm to

solvethequickest owproblem.

Theorem6.3 ([12])

 Let T

0

be the length of the shortestpath from the sourceto the sink with

respecttothe traveltime. v(T)isamonotoneincreasingfunction andfor

T T

0

itincreasesstrictly.

 4(T):=v(T) v(T 1) isfor T >0monotonouslyincreasing, i.e.

4(T+1)4(T);8T >0

 4(T)attainsitsvalueonlyfromthesetf0;1;:::;jx

max

jg,wherejx

max j

isthe valueofmaximum static ow inthe network G.

The interrelations betweenmaximum dynamic ow and quickest ow are

de-scribedbythefollowinglemma.

Lemma6.1 ([12])

 T(v) = minfT j v(T)  vg where v(T) is a maximum ow over time

periodT.

 Letxbeamaximumdynamic owofvaluevinthetimeinterval[0;T];T 

0. Ifv(T 1)<vthenxisaquickest owofvaluev,andfortheminimum

egresstime T(v), wegetT(v)=T.

 v(T(v) 1)<v

The solution is obtained by applying an iterative process with two main

Burkard, et. al. [12] developed several polynomial and strongly polynomial

algorithms forthequickest owproblem usingacontinuousversionof v(T)in

caseofsinglesourceandsinglesink.

7 Tripple Optimization Result

UsingtheturnstilecostdenitionintroducedinSection3,veryinteresting

prop-ertiescanbederivedinterrelatingtheevacuationtimeandthemaximumnumber

ofevacueesthatcanbesentouttosafetyineverytimeperiod. Thisinterrelation

isdescribedinthefollowingtheoremknownastrippleoptimizationtheorem. It

says that the ow pattern which minimizes the average evacuation time (see

Section 3) also maximizes thenumber ofevacuees reaching thesafety at each

time period (see Section 5) and vice versa. Moreover,the solutionof the

sec-ondproblemalsominimizesthetotaltimeneededtoevacuateallevacuees(see

Section 6.2)but theconverseingeneralisnottrue.

Theorem7.1 ([36]) Let F

t

be the ow vector of arcs connectedto the super

sink at time t and let c

t

is the associated weight (or cost) vector where the

weight c isincreasing overthe timet. Consider three dierent problems under

the assumption thatthere existsafeasible ow of K units,i.e. the value ofthe

maximumdynamic ow withinatimeT isnotlessthanK.

(a) Universalmaximum ow problem:

max T 0 X t=0 F t ;8T 0 T

(b) Minimumweightedsum ow problem :

T X t=0 c t F t

(c) Quickest owproblem of initial occupanciesK :

minfT jF T 0 =0;8T 0 >Tg

Then the solution for eitherproblem (a)or (b)isalsothe solution ofthe other

two problems, i.e.

(a),(b))(c)

As a consequenceit suÆces to solve either of the twoproblems using the

turnstilecostapproachortheUMFapproachtoobtainabestevacuationplan

accordingtothethree goalslistedatthebeginningofthissection!

8 Multiple Objectives

Usingthevaluescontainedinthedynamicseveritymatrix(seeSection 1),one

level. Obviously,evacueesshouldnotmovefromlowerpriorityregionstoregions

withhigherpriority. Inthemodelingonecanenforcethisbycharginghighcosts

to correspondingarcs. With this prioritysystem, theevacuationprocessmay

haveseveralobjectivesthat mustbesatisedsimultaneously. Forexample,let

P 1

;:::;P k

be a partition of the underlying system into k dierent parts such

that the evacuation of P

1

has the highest, the evacuation of P

2

has the next

highest,andnally,theevacuationofP

k

hasthelowestpriority. Theobjective

oftheevacuationprocessisto minimizetheevacuationtimesuchthat ([33])

(1) TheevacuationtimeofP

1

isassmallaspossible.

(2) Amongallplans optimizing(1), theevacuationtimeof P

2 isassmallas possible. . . .

(k) Amongallplansoptimizing(1)until(k 1),theevacuationtimeofP

k is

assmallaspossible.

Hamacher and Tufekci ([33]) consider this multiple priority level problem as

lexicographicalminimumcostdynamic owproblem.

Thelexicographicalorderingisdened asfollows.

Denition8.1 Supposewehavetwovectorswith k-components, candc.

c= 0 B @ c 1 . . . c k 1 C A , c= 0 B @ c 1 . . . c k 1 C A

Vector c islexicographically smaller than cif and only if c

i

isstrictly smaller

thanc

i

for the rstcomponenti,wherec

i

andc

i

aredierent.

Givenaset K ofvectorswithk-componentsits lexicographicminimumis

de-notedaslexmin K.

Inthemultipleobjectiveevacuationproblem,wereplacetherealvaluedcost

c ij

(t)ofthesingleobjectiveproblembythevector

c ij (t)= 0 B @ c 1ij (t) . . . c k ij (t) 1 C A

for each arc (i;j) 2 A and for t = 0;:::;T. The cost value for each priority

levell=1;:::;kcanbedenedby

c lij (t)= 8 < : t 0 =t+ ij ; ifi(t)2P l andj(t 0 )2P l 0 ; l 0 >l M ; ifi(t)2P l andj(t 0 )2P l 0 ; l 0 <l 0 ; otherwise.

Theobjectivefunctioncanbedenedas

T X

k

oflexicographically shortestaugmentingpath . Denition8.2 Let P =fs=i 0 ;i 1 ;:::;i k

=dg beapath from super source s

tosuper sinkdinthe time-expandednetworkG

T andlet P + =fe2P je=(i j ;i j+1 ); j=0;:::;k 1g P =fe2P je=(i j+1 ;i j ); j=0;:::;k 1g If x isa ow inG T dene  + (P)= minfb e x e je2P + g  (P)= min fx e je2P g (P)= minf + (P); (P)g and c(P)=c(P + ) c(P )= X e2P + c(e) X e2P c(e)

P iscalled alexicographically (lex) shortestaugmentingpathif (P)>0and

c(P)=lex minfc(P 0 )j(P 0 )>0; 8pathP 0 in G T g

Inthe denition,thetime-expandednetwork G

T

isconsidered asastatic

net-work. The proposed algorithm ([33]) to solve the lexicographically minimum

cost owproblem needsthe notionoflexextreme ow asexplainedin

Deni-tion8.3.

Denition8.3 A owxisalexextreme owinG

T

ifxisafeasible ow with

value v andifthe cost

c(x)= X e2AT x e c(e)

isminimal amongall owswith the same ow valuev.

The algorithm starts with nding a lexicographically shortest augmenting

path andthen augmentsthecurrent owalong thispath. Theorem8.1 shows

that theupdated owaftertheaugmentationisagainalexextreme ow.

Theorem8.1 Letx be alexextreme ow with ow value v andletP bealex

shortestaugmentingpath. Then the new ow x

0 denedby x 0 e = 8 < : x e +(P) ; if e2P + x e (P) ; if e2P x e ; if e62P

isalexextreme ow withrespect to owvalue v+(P).

The search for lex shortest augmenting paths is repeated until no more

aug-mentingpathis available. Ifthis situation isreached,thecurrent owisalex

mayhaveseveralattributes which aretimedependent. Travelcost,travel

dis-tanceandriskareexamplesofattributeswhichmaybeattachedtoeacharc. We

maywanttoconsider,forinstance,theeectofre,smokeortoxicalchemicals

on theavailabilityof arcsovertime. If thedevelopmentis such thatevacuees

can not walkas fast as usual or even may not be able to pass this arc then

thecostchangeovertimein allorsomecomponentsofthecost vector. If, for

instance, the arc (i;j) becomes impassable at time t

0

, then the travel cost of

goingfromnodei tonodej canbemodiedtobe

c ij (t)=  c ij (t) ; ift<t 0 M ; iftt 0

withM alargepositivenumber.

Let fF

i

g be the set of non-dominated evacuation routes from node i to

safety d, and let c

ij

be the vector of attributes associated with the arc from

nodeitonodej. Denesalsovminasthevectorminimizationtondthe

non-dominated evacuationroutes. The dynamic programmingformulation to nd

thenon-dominatedevacuationroutesgivenby([43],[44]).

fF i g= vmin j6=i;(i;j)2A ffF j g+c ij g ;8i2N fdg fF d g=f0g

Here,0isthenullvector. KostrevaandWiecek ([43])proposed backwardand

forwarddynamic programmingalgorithms to solve this problem. They allow

discontinuityinthecostfunctions,anasumptionwhichiscertainlyrelevantfor

evacuationplanning,asseenabove.

9 Dynamic Network with Density Dependent

Travel Time

So far, we have mainly considered travel times which are either constant or

dependentontime. Inreality,theyarehoweverdependentonthedensityofthe

ow. Duringanykindofmovement,andthisisinparticulartrueforevacuation

movement, thespeed (i.e. traveltime)will growwithhigher density, until we

encounterslowdownandqueuingphenomenaatcertaindegreesofdensitiy. In

this section we consider thereforetravel times

e (t)=g e (x e (t)) where g is an

appropriatelychosenfunctiondependingbothontimeandthe owatthistime.

Thismodelismorerealistic,but alsomorediÆculttohandlefroma

math-ematical point of view. If we consider, for instance, the ow augmentation

process, the amountof time aspecic arc in asource-sink path is available is

depending on the amountof ow sent through this arc. Moreover, since the

densitydependenttraveltimeisingeneralnonlinear,thedynamicconservation

owcontraints(seeconstraint(3))arealsononlinear,whichmakestheproblem

muchmorediÆcultto solve.

Several references on the dynamic network ow problem with density

de-pendenttraveltimescan befound in theeld of traÆc assignment(see Table

problem, system optimum and user optimum. The former objective tries to

minimizethetotaltraveltime. Itconsidersthewillingnesofthecommunityto

share thelateness. The latter tries to optimize everyindividual's travel time

whereitisassumedthateachindividualbehavesegotistical,Ifweassumesingle

destinationandallowexogenous owsonlyatthebeginning(i.e. attimezero),

the dynamic traÆc assignment problem becomes an evacuation problem with

knowninitialoccupancydistribution.

IntraÆcassignment,thenonlinearityofthetraveltimefunctionishandled

bythefollowingapproaches.

 Use linearapproximationof the travel time by introducing 0-1 decision

variablesforselectingtheappropriatetraveltime(Kaufmannetal. [39]).

 Usepiecewiselinearapproximation(CareyandSubrahmanian[13]).

 Applyaniterativeprocesswhereineachiterationthetraveltimeisxed

temporarilyaccordingto thecurrent ow(see[37],[60]).

TherstapproachusesthedecisionvariableÆ

ts ij

withvalueequalto1ifthe

ow entering arc (i;j) at time t needs arc travel time s and zero otherwise.

Thefree- owtraveltimegivesthelowerboundtotheparameterswhereasthe

upperboundisgivenbythevalue(T t). The owvariablex

ts ij

representsthe

owsenterarc(i;j)attimetandexitsattimet+s. The owvalueisbounded

bythe arccapacitycorrespondsto theselected traveltime determinedby Æ

ts ij .

Theproblemisthusformulatedasamixed integerprogrammingproblem.

The second approach uses a piecewise linear approximation of the travel

times. A timeexpandedarc isobtainedbyjoining node iat timet with node

j at time (t+k) with k = 0;1;:::;K integer breakpoints of the travel time

function. Thearc owmustliebetweenatmosttwoneighbouringbreakpoints,

i.e. thearc owis represented asaconvexcombination of two ow valuesat

twoneigbouring breakpoints. Using this approach, theproblem is formulated

asalinearprogrammingproblem.

Inthe last approach, the valueof the travel time is approximatedusing a

2-leveliterativeprocess. Intherstlevel,thetraveltimeistemporarilyxedin

eachiteration. Inthe secondlevel,anonlinearoptimization problemis solved

iterativelyasa convex programmingproblem. This approach is equivalentto

theproblemofndingthexed-pointoftwointerdependentalgorithmicmaps.

One algorithmic map is foradjusting thetraveltime and the other oneis for

ndingtheoptimal owunderatemporarilyxedtraveltime.

An even morerealistic modelling can be achievedby considering a

depen-denceof thetraveltime notonlyon theexisting ow, but on three ow

com-ponentsoneacharc(i;j)2Aat timet,namely: incoming owu

e

(t), existing

owsx

e

(t) andoutgoing owv

e

(t). Hence,thetraveltimeofanyarceat any

timet isgivenby :AT !R  e (t)= g e (u e (t);x e (t);v e (t));

wheregisassumedtobeanondecreasingfunction,whichisconvex,continuous

situation under congestion. Instead of the node-arc ow formulation used so

far,theirapproachusesanarc-path ow formulationintheconstraints. Letus

deneO(j)asthesetofarcsleavingnodej andI(j)asthesetofarcsentering

nodej. Theobjectiveofthemodelistominimizetheaverageevacuationtime,

i.e. weminimize (U):= T X t=0 X e2I(d) (t+ e (t))u e (t)

Constraintsinthemodelinclude owconservation, owpropagation,

nonnega-tivityandboundaryconstraintsasfollows.

 LetA(p)bethesetofallfeasiblepathsfromsourcestosinkd. Ifepisan

arceonpathP 2A(p), thentheinterrelation amongincoming, existing

andoutgoing owsofarcealongpathpis

x ep (t+1)=u ep (t)+x ep (t) v ep (t);8ep; 8t (22)

 Evacueeswhoare in thesource s at thebeginning candirectly moveto

arcsleavingsorwaitinthenodeiftheythinkthosearcsaretoocrowded.

Assumingasingle source s and asinglesink d, thetotal movementout

ofthe the source must be equalto theinitial contentsq. Therefore the

supply-demandconstraintscanbeformulatedas

T X t=0 X e2O(s) u e (t) T X t=0 X e2I(s) v e (t)=q (23) T X t=0 X e2I(d) v e (t) T X t=0 X e2O(d) u e (t)=q (24)

 Flowsconservationconstraints:

X e2O(i) u e (t) X e2I(i) v e (t)=0;8i6=s;d; 8t (25)

 Flowpropagationconstraints:

Ifplatoondispersionisnotallowedthen theincoming owat timet will

causeexactlyoutgoing owafter

e

(t)unittime, i.e.

u ep (t)=v ep (t+ e (t));8e2A;8p2A(p);8t (26)  Boundaryconditions: x e (0)=0;8e2A (27)  Nonnegativityconstraints: u e (t);x e (t);v e (t)0;8e2A; 8t (28)

X p2A(p) u ep (t)=u e (t); X p2A(p) x ep (t)=x e (t) ; X p2A(p) x ep (t)=x e (t); 8e2A; 8t (29)

Tohandlenonlinearityofthetraveltimeweuseagainatwoleveliterative

algorithm. Initiallytherstlevel(outerlevel)estimates thetraveltimesusing

free- owtraveltimes. Thetraveltimesarexedtemporallyinordertobeused

inthesecondlevel(innerlevel). Underxedtraveltime, themodelhasconvex

objectivefunction butlinearconstraints. Theinner levelusestheFrank-Wolfe

algorithm (see [40], [60]) to obtain the optimal ows. The resulting optimal

owsare used to recalculate traveltimes and then compare thenewest travel

times tothe previousones. Iftheresultis notsignicantlydierent,then the

algorithm stops. Otherwise, the second level is repeated using thenew travel

times as input. The reader is refered to Tjandra ([64]) for more details on

numericalproceduresandresults.

10 ContinuousTimeDynamic NetworkFlowModel

Wehaveseenintheprevoussections,thatdiscretizationplaysavitalroleinthe

modelingof evacuationusingdynamicnetwork ows. Toincreasetheaccuracy

of themodel onecanset the basictimeunit  (see Section 3) verysmall, but

thiswillenlargethesizeofthenetworkandthusthecomputationalcomplexity

of the solutions algorithm. Obviously, a tradeo is necessary between good

accuracyandcomputationaltracktabiliy. Independentofthis,thefactthatthe

choiceof discretization by choosinga specic basic time unit  predetermines

thepossibleset ofevacuationplansissomewhatunsatisfying.

We therefore discuss in this section acontinuous-time approach to ev

acu-ation modelling. Continous time dynamic network ow problems have been

considered byvarious authors includingTyndall [65] [66], Grinold[31], Perold

[54],Anderson,etal. [3],Philpott[55],Pullan[57],PhilpottandCraddock[56],

Pullan [59]. Most of existing works emphasize on the analysis of primal-dual

relationshipsandtheexistenceoftheoptimalsolution.

Inthecontinuousmodelweconsiderboundedmeasurablefunctionsc(t)and

b(t) whereeachofthemcomponentsassignsthecostof owandupperbound

ontherateof owin oneofthearcsattimet,respectively. Variablesx(t) and

y(t)deneratesof owineacharcandlevelsofstorageineachnodeattimet,

respectively. Thelevelofstorageisboundedabovebycontinuousfunctiona(t).

minZ= R T t=0 P (i;j)2A c ij (t)x ij (t)dt (30) s.t. y j (t)=y j (0)+ R t 0 [ P (i;j)2A x ij (  ij ) P (j;i)2A x ji ()]d; t2[0;T] (31) y j (t)a j (t);j2N ; t2[0;T]; (32) x ij (t)b ij (t);(i;j)2A;t2[0;T] (33) x ij (t);y j (t)0 (34)

Constraint(31)iscalledintegralconstraintandconstraints(32)-(33)arecalled

instantaneousconstraints.

Sincetheintegralandinstantaneousconstraintsareseparated,thisnetwork

ow problem is included into a specic class of continuous linear programs,

namely separated continuous linear programs (SCLP) proposed by Anderson,

Nash andPerold([3]). Thefollowingtheoremgivestheformatof theoptimal

ow function of SCLP under specic assumptions on the capacity and cost

functions.

Theorem10.1([58]) Suppose that a(t);b(t) and c(t) are piecewise analytic

on [0;T] with a(t) continuous. If the feasible region of SCLP is bounded and

nonempty,thenthereexistsanoptimalsolutionforSCLPinwhichx(t)is

piece-wise analyticon [0;T].

AnalogoustothethreeproblemsinTheorem7.1derivedinthediscrete

dy-namicnetworkmodelcontext,Philpott[55]formulatedthreecontinuousmodels

whichcanbeusedforevacuationplanning.

(a) Maximize owsintothesinknodein theinterval[0;T].

max R T 0 [ P (i;d)2A x id (  id ) P (d;i)2A x di ()]d subjectto (31)-(34)

(b) Minimizethetimetoclearthenetworkinitialoccupancies.

minT subjectto (31)-(34) y(t)=0;tT;x ij (t)=0;tT  ij

(c) Minimizingtotalegresstime.

min R T 0 P (i;j)2A c ij (t)x ij (t)dt+ R T 0 P j2N fdg h j (t)y j (t)dt subjectto (31)-(34) withh j

is holdingcostin nodej.

Problem (b) is known as the continuous version of the quickest ow problem

explained in Section 6. Bydening c

ij

(t) asturnstile cost,problem (c) solves

the problemof minimizing theaverage evacuation timeasin Section 3.

Rela-tionshipsamong thosethreeproblemsareexplainedbythenextresult.

Theorem10.2

 Any ow which solves (a)forany timeT

0

{ holding costfor eachnode isequal tooneunit,

{ arccostatany timetisdenedas

c ij (t)=   ij ;0t<T  ij T t ;T  ij t<T;

Problems (a)and(c) areequivalent.

This Theorem guarantees the existence of a universal maximum ow in the

continuoustimedomain.

Fromamodel accuracypointof view,modeling evacuationproblems using

dynamic networkproblems withcontinuous timeis preferabletodiscrete time

models. But continuous models can to date not be solved satisfactorily for

the large scale problems which need to be tackled in the evacuation context.

Solutionalgorithmsforcontinuousdynamicnetwork owalgorithmsaredueto

AndersonandPhilpott[5](continuous-timenetworksimplexalgorithm),Pullan

[57],PhilpottandCraddock[56](discretizationapproaches).Anderson,et.al[2]

and Philpott[55](continuous-timeversionof Ford-Fulkerson's maximum ow

labellingalgorithm).

Morework is certainlyneeded in this area. First additionalresultscanbe

foundinTjandra([64]). HeproposesanalgorithmtosolveUMFproblemwith

timedependentcapacityandutilizetheresulttosolvethequickest owproblem

underthesameassumption.

11 Microscopic Models

In microscopic models each evacuee is considered as a separate ow object.

An evacueewill be exposed to accidenteects depending onthe route he/she

followsand thelengthof time spent in dierentlocations. An evacueeselects

the route 'stepby step',which means that thechoice of the nextpiece of the

route is decided at every node along this route. The initially selected route

might be changeddue to some reasons, for instance, blockageby re orhigh

congestion. Figure 10 shows an example of the evacuation process as it can

be modeled for each person. Microscopic models emphasize the modeling of

human behavior during an emergency situation. The human model can be

provided with somepersonal attributes, for example, walking speed, personal

memoryandpsychologicalcondition. Theseattributewillbeusedtodetermine

themovementdecisions,forexample[21],toselectthenearestwalkway,moveon

thewalkwayonlywhenthereisnoblockageattheend,orchangethedestination

target before reaching it. Lvas [47] proposed some dierent probability laws

for personal movement relative to the route components (nodes and arcs) as

follows.

 Randomchoice. It is applied when the evacueeis not familiar with the

surrounding. LetsdeneX asthestateof theunderlyingsystem,for

in-stance, thenumberof evacuees oneach arc/nodeorthe hazard (smoke,

re,etc.) levelof each arc/node. Moreover,letp

k

(i;j;X)bethe

Initial response

Movement

Route strategy

Reach

the safety ?

NO

End

yes

Figure10: Individual EvacuationProcess[46].

nodej whenthewhole systemisin stateX. Ifthenumberof walkways

connectedtonodeiisdenotedasÆ

i ,then p k (i;j;X):= 1 Æ i

 Modied random choice. It includes the possibility that evacuees will

returntotheirpreviouswalkway. Denen

k

asthenode-numberofthelast

nodevisitedbyapersonkandintroduceanumber!,inwhich0!1.

Theparameter! isafactorshowinghowaperson iswillingto returnto

his/herpreviouswalkway. Usingthisparameter,theprobabilityp

k

(i;j;X)

canberedenedasfollows.

p k (i;j;X):= 8 > < > : ! Æi ;j=n k 1 ! Æ i Æi 1 ;j6=n k 0 ;otherwise

 Suppose thereis anevacuationrouteP determined apriori bythe

evac-uation planner (this is atypical situation, indicated by evacuation sign

attachedtothewallofroomsorcorridors).

P :=(s=origin;:::;i;j;:::;destination=d)

Supposethat Æ

i

>1forallnodesin thisroute,exceptpossiblytheorigin

anddestination. Whenanemergencysituationarisesthereisahigh

i

routeatanintersectionnode. Thenp

k

(i;j;X)canbewritten as

p k (i;j;X):=  1 i ;j=j i Æi 1 ;j6=n k

Pathchoice rules likethe ones presentedabovecanbe used together with

attributes of evacuees and building components to design a simulation of a

building evacuation (see [63]). Obviously, such simulation is a good tool to

model individual behaviour in evacuation planning and can also be used to

validateoptimizationmodelsaspresentedinthepreceding sections.

Inrecentyears,thereisanincreasingtrendtousesimulationbasedon

cellu-larautomata(CA).Eitherdeterministicorprobabilisticrulescanbeappliedto

modelthemovementpatternsbetweentimeperiodsrelativetothemovementof

other persons and/orphysical barriers. CA simulationoersthepossibilityto

emulate theessential,diverse movementsofevacueesasbehaviouralresponses

to varying and uncertainlocal conditions. A cellular automata isdened asa

regular n-dimensional latticepartitioned into discrete elements called cells or

siteswhichhasadicretesteptimeevolution. Usingcellularautomata,thespace

of theevacuation areais dividedinto accessible andnon accessiblecells, each

ofxedandequalsize. Thestateofeachcellisoneofnitelymanyvaluesand

has adynamical behaviour. It is updatedsimultaneously basedon thevalues

ofthestatesin itsneighborhoodattheprecedingtimestepandaccordingtoa

specic local rules . The set of local rulesis dened to control themovement

of evacuees, orthestate transition of each cell. Since therules are neededto

governonlylocalrelationshipsamongtheneighboringcells,CAisconsideredto

beveryeectiveforsimulatingphysicalphenomena, therelationshipsofwhich

overthewholedomainareunknown. Inthe1-dimensionallattice,acellhasonly

twoneighbourhoodcells,theleftandrightcelladjacenttoit. For2-dimensional

lattice, thereare someneighbourhoodclassicationsof acellasshownin

Fig-ure 11. The von Neumann neighbourhood consists of 4 cells, the cell above,

below, rightand left of the reference cell. The radius r of the von Neumann

neighborhood is 1,sinceit considers only thenextlayerof acell. TheMoore

neighborhood isan extensionof thevonNeumannneighborhood in which the

diagonalcellsare added. Itsradius isstill r=1. TheexpandedMoore

neigh-bourhoodfurtherextendstheMooreneighborhoodbyincludingtwolayers(i.e.,

r = 2). Anothertype of neigborhood is the Margolus neigborhood in which

22 cellsof latticesare considerd at once. Thetransition from onestateto

anotheris arrangedbyaset ofrules accordingtothe neighboringstate ofthe

preceding timestep. Ifx(t)isastateofcellxattimet thenthestateofcellx

attime t+1canbewritten as

x(t+1)=f(x(t);N

x (t))

whereN

x

(t)isstateoftheneighborhoodcellsofcellxattimetandwheref is

therule. Asimpleexampleofarulecanbegivenasfollows. Consideranarrow

corridor (assumedwide enough onlyfor one person) with a given lengththat

canbemodeledasaonedimensionallattice(seeFigure12). Acellrepresentsa

spacein thecorridorthatmaybeoccupiedby onepersonoritmaybeempty.

The walking speed of the i

th

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x x

x

x

x

x

x

x

x

x

x

x x x

x x

(a)

(b)

(c)

Figure 11: (a) von NeumannNeighborhood (b) Moore Neighborhood (c)

Ex-panded MooreNeighborhoodofaCell(ShadedCell).

thatapersonadvancesinoneiteration(onetimestep). Bydeningx(i)asthe

position of thei

th

person in thecorridor, thedistance betweenthe i

th person

andthepersonimmediatelyaheadisgivenby

g(i)=x(i+1) x(i) 1

Usingthese3variablesthetransitionrulecanbedenedasfollow:

 Accelerationof freeperson:

Ifv(i)<v max

andg(i)v(i)+1thenv(i)=v(i)+1,wherev

max isthe

maximumpossiblespeed.

 Slowingdowndueto otherperson :

Ifv(i)>g(i) 1thenv(i)=g(i)

 Movement: personisadvancedv(i)cells.

Asamethodofdiscretesimulation,CAisusedbysomeevacuationsoftwares

includingEGRESSandFlightSim.

EGRESSisaC++programdevelopedformodellingthebehaviourof

evac-ueesinemergencysituations,especiallyinoshoreenvironments. Thephysical

structure oftheoshoreinstallationis representedbyusing hexagonalcellular

grids (see, e.g., Doheny and Fraser [20]). It models evacuation using cellular

automata where the movementsand interactions of the automata on the

cel-lulargridsimulate themovementsandphysical interactionsofevacueesonthe

platform. EGRESSintegratesMOBEDIC(ModellingBehaviouralDecisionsIn