Fair Dynamic Resource Alloca2on in Transit-based Evacua2on Planning

SoheilaAalamiPhDCandidate,

&LinaKa.an,PEng,PhD

Professor,UrbanAllianceProfessorinTransporta;onSystemsOp;miza;on

DepartmentofCivilEngineeringSchulichSchoolofEngineering,UniversityofCalgary

Calgary,Canada

Outline -  Introduc;on-  ResourceAlloca;onandU;lity- WeightedPropor;onalFairness

-  Assump;onsandProblemFormula;on

-  DistributedSolu;onAlgorithm

-  NumericalExampleandDiscussionofResults

-  Conclusion

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Shelter

CommunityA

CommunityB

Factory

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Model Application: short notice evacuation such as forest fire, chemical leaks

from a factory, volcanic eruption, etc.

Introduction

What is a fair allocation of busses to these communities?

Resource Allocation and Utility

u1(r1) u2(r2) max F(u1(r1),..., un(rn)) … Σ r i ≤ ᚱ un(rn)

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Utility Function’s Properties:

1.  ∀r,Ui(ri)≥0,i.e.,u;lityisalwaysnon-nega;ve.

2.  Ui(0)=0,i.e.,inthecasenoevacueeismovedtoashelter,theu;lityiszero.

3.  Ui(·)isanon-decreasingfunc;onofri,i.e.,alloca;ngmoreresourcestoapickup

loca;ondoesnotreducetheu;lityofthepickuploca;on.

4.  ∃riM<∞,s.t.∀r≥riM,Ui(r)=Ui(∞)<∞,i.e.,thereisaop;mumrateri

M of evacua;on

forwhichalltheevacueescanbesafelymovedtosheltersbeforethedeadline.Having

ahigherevacua;onratethanriMisnothelpfulanddoesnotincreasetheu;lity.

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Sigmoidal u2lity func2on

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Different Objectives

•  Minimum network clearance time

•  Maximum social welfare

•  Fair resource allocation

•  ...

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Fairness

•  Fairness is a debatable topic

•  There are many different definitions for fairness •  Equal allocation •  Max-Min fairness •  Proportional fairness

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Weighted Proportional Fairness Proportional Fairness [Frank Kelly 1998]

Severitylevelwi>0foreachpickuploca;onPi9

Weighted Proportional Fairness: Equivalent Formulation

Severitylevelwi>0foreachpickuploca;onPi

Is equivalent to:

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Assumptions / Formulation

●  Severitylevelwi>0foreachpickuploca;onPi

●  AfleetFofpublictransitvehicleswithlimited

capacityisgiven

●  Capacityofeachshelterislimited

●  Totalcapacityofshelterscanaccommodate

thewholepopula;on

●  Roundtriptravel;mescanchangeover;me

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Shelter1 Shelter2 Shelter3

PickupP1 f11f12

f13

PickupP2f21 f22

f23

PickupP3f31

f32

f33

PickupP4f41

f42

f43

PickupP5f51

f52

f53

|S1| |S1| |S1|

?

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Flee

t siz

e C

onst

rain

t

Problem Formulation:

First constraint ensures that the number of evacuees moved to each shelter should be less than the capacity of the shelter.

Second constraint ensures that capacity of all transit vehicles allocated to move evacuees from pickup locations to shelters does not exceed the size of the fleet

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Lagrangian Dual of the Problem

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Net benefit of Pi

Weighted Utility (Benefit/Gain) associated with pick up location Pi

Cost associated with allocating fij between i and j

Cost associated with the use of Shelter j

16 We can split the dual problem into three smaller sub-problems and propose one algorithm for each part

Threeindependentagents:1.  Theshelters2.  Thetransitvehicledispatchingcenter3.  Thepickuploca;ons

•  Totalnumberofagents:n+m+1

•  AgentsinteractbyworkingcooperaBvelytoachievethesinglegoalofop;mizingpropor;onalfairnessduringtheevacua;onprocess

•  Agentscommunicatethroughpublishingprices/bids

PickupLocaBons(n)

Shelters(m)

BusDispatchcenter(1)

Resource Managers

Resource Consumers

Solution Algorithm: Proportionally Fair Dynamic Distributed Algorithm (PFD2A)

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The central idea of the suggested algorithm is to interpret the Lagrange multipliers µ as the virtual unit prices associated with resource consumption

Iterative Algorithm – Market Clearance Price:

PickupLoca;ons(n)

Shelters(m)BusDispatchcenter

(1)

Resource Consumers:

Resource Managers:

Shelters: Receive bids from pick-up locations and decide about the unit price of available shelters

Solution Algorithm (Cntd.) PFD2A

Dispatch Center: Receive bids from pick-up locations and decide about the unit price of available Resource

Pickup locations: Receive prices from resource managers. Then accept or offer new Bids

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Resource Consumers

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Resource Managers

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Numerical Example

5PickUpLoca;onsand3shelters

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Different Varia2ons of the Resource Alloca2on Problem

Ø MaximumevacuaBonrateresourceallocaBon(MR-RA):Theobjec;veisto

maximizethenumberofevacueeswhoreachsafetybyagivenevacua;ondeadline.

Ø MaximumsocialwelfareresourceallocaBon(MSW-RA):maximizingthesumma;on

oftheweightedu;lityfunc;onsofthepickuploca;onswhiletheseverityofthe

disasterineachpick-uploca;onandevacua;ondeadlinesareconsidered.

Ø ProporBonallyfairresourceallocaBon(PF-RA):Theobjec;veistoallocatetheresourcesamongdifferentpick-uploca;onsaccordingtothecriterionof

propor;onalfairness

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Results of MR-RA: Maximum evacuation rate resource allocation

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Results of MSW-RA: Maximum social welfare (i.e. summation of the weighted utility functions of the pickup locations)

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Results of PF-RA: Weighted Propor2onal Fairness: assigns a non-zero share to each pickup location

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•Atitera;on50,thepopulaBonofP5isreducedby10%anditsseveritylevelisdoubled.

•Atitera;on100,thepopulaBonofP2isincreasedby20%anditsseverityistripled.

•Atitera;on150,thefleetsizeischangedfrom500to700.

PFD2Aisabletoconvergequicklyandadapttothechangesintheparameters.

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Introducing changes over time

Summary of the Results:

• MR-RAisbiasedandfavorsthepickuploca4onsclosertothesheltersinordertomaximizethenumberofevacueesreachingsafety.

• MSW-RAisextremelyunfairinsomecasesandassignsnoresourcetosomeofthepickuploca;ons.

• PF-RAwasshowntohavethefollowingproper;es:1.  Isfair,whileittriesnottosacrificeefficiencyforfairness.2.  Canhandledifferentseveritylevelsanddeadlines.3.  Canadapttochangesintheevacua;onparameters(popula;on,deadlines,severity

andtravel;mes).4.  Canbeefficientlysolved.

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Contribution of the paper •  Introducingtheseman;cof“propor;onalfairness”totheemergencyevacua;on

problem:Canbeapplicabletomanyothertransporta;onproblemswherethefocusis

achievingFairness

•  Developingadynamicanddistributedalgorithm(PFD2A)basedontheLagrangiandual

methodtofindapropor;onalfairalloca;onofresourcestorespondtothedynamic

changesintheemergencysitua;on

•  Developingaunifiedmethodtoanalyze/comparedifferentvaria;onoftheproblem

(Max.evacua;onrate,Maximizingsocialwelfare,etc..)29

Limitations and Possible Extensions

•  Thispaperfocusespropor;onalfairnessforthepopula;onthatreliessolelyontransitforevacua;on.

•  Inreallifesitua;on,dependingonthetypeofdisaster,somepeoplemaychoosetoevacuateonfoot,otherswouldtakepublictransit,whiletherestoftheevacueeswouldtakepersonalvehicles

•  Consideringpersonalvehiclesaspartoftheevacua;onmayincreasetransittravel;me.Masspanicmayresultinmul;pleaccidentorextremelyoveru;lizedroutes,possiblyblockingsomeemergencyevacua;onroutesandthushighunreliabilityinthees;matesoftravel;me.

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Thank You!

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