sequential, stochastic screening problems

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S ti l St h ti Sequential, Stochastic Screening Problems Screening Problems Laura A. McLay Virginia Commonwealth University [email protected] Sheldon Jacobson Sheldon Jacobson The University of Illinois at Urbana-Champaign Alex Nikolaev The University of Buffalo The University of Buffalo This research was supported in part by the National Science Foundation (CBET-0735735) and the U.S. Department of Homeland Security under Grant Award Number 2008-DN-077-ARI001-02

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My 2011 INFORMS Computing Society Conference presentation on risk-based passenger screening for aviation security.

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Page 1: Sequential, Stochastic Screening Problems

S ti l St h tiSequential, Stochastic Screening ProblemsScreening Problems

Laura A. McLayVirginia Commonwealth [email protected]

Sheldon JacobsonSheldon JacobsonThe University of Illinois at Urbana-Champaign

Alex NikolaevThe University of BuffaloThe University of Buffalo

This research was supported in part by the National Science Foundation (CBET-0735735) and the U.S. Department of Homeland Security under Grant Award Number 2008-DN-077-ARI001-02

Page 2: Sequential, Stochastic Screening Problems

P S i B k dPassenger Screening Background Passenger screening visible aspect of aviation security Many changes in aviation security since 9/11

New technologies New screening strategies

Passenger prescreening CAPPS, selectees, nonselectees No fly list

TSA committed to a risk-based paradigm

Page 3: Sequential, Stochastic Screening Problems

U if S i S l i S iUniform Screening vs. Selective Screening All passengers treated the More security for passengers All passengers treated the

same More security scrutiny for all

More security for passengers perceived as higher-risk

Less security scrutiny for tpassengers

Simpler screening procedures & no privacy issues

most passengers System required for

determining who is higher-p y

Prohibitive cost to screen all ith ll it

g grisk

May be more cost-effectivepassengers with all security devices

Page 4: Sequential, Stochastic Screening Problems

M ti tiMotivation Static models for passenger screening Passenger screening is dynamic

Staffing loads Passenger arrival rates National risk level New procedures or technologies

What is the optimal way to screen passengers in a dynamic CAPPS-like environment?Markov decision process model Insights into optimal screening strategies

Page 5: Sequential, Stochastic Screening Problems

F kFramework Passengers/bags screened by series of devices

System response a function of device responses Passengers check-in sequentially Passengers assigned to one of two classes upon check-

iin Focus on one time interval during day at airport

f Day consists of several time intervals Resource availability constant over time interval Passenger arrival rate constant over time interval

Page 6: Sequential, Stochastic Screening Problems

S h tiSchematicpassenger 1

arrivesPassenger T

arrives

t=0initial state

t=1Assign passenger 1 to a class

t=T

to a class

Passenger 1 assigned to a classRemaining T-1 passengers have not arrived and their assessed threat values are unknown

Page 7: Sequential, Stochastic Screening Problems

S h tiSchematicpassenger 1

arrivespassenger t

arrivesPassenger T

arrives

t=0initial state

t=1Assign passenger 1

Assign passenger t to a class

t=T

to a class

Passenger t assigned to a class-Passengers 1,2,…,t-1 have arrived and have been assigned to classes-Remaining T-t passengers have not arrived

Page 8: Sequential, Stochastic Screening Problems

S h tiSchematicpassenger 1

arrivespassenger t

arrivesPassenger T

arrives

t=0initial state

t=1Assign passenger 1

Assign passenger t to a class

t=Tt=T-1

to a class

Passenger T assigned the most secure class with space remaining-Passengers 1,2,…,T-1 have arrived and were assigned to classes

Page 9: Sequential, Stochastic Screening Problems

Sequential Stochastic Multilevel PassengerSequential Stochastic Multilevel Passenger Screening Problem (SSMPSP)

T t i ti i t l h h k i T stages in time interval when passenger can check-in Passenger arrives in each stage with probability p,

resulting in N passengers checking in over T stagesresulting in N passengers checking in over T stages (t) = assessed threat value of passenger t (random

variable)) (t) = realized assessed threat value

(t) = 0 if no one checks in at t f() pdf of assessed threat values a capacity c associated with the selectee (S) class

( it f l t (NS) l d t b(capacity of nonselectee (NS) class assumed to be infinite)

the security level of each class LS and LNS the security level of each class LS and LNSwith 0 LS (and LNS) 1

Page 10: Sequential, Stochastic Screening Problems

SSPSP t’dSSPSP, cont’d. Goal: find policy that maximizes expected total security

Variables: xS(t) = 1 if passenger t classified as a selectee, 0 if

classified as a nonselectee

Objective: Find policy that determines passenger assignments Find policy that determines passenger assignments

xS(1), xS

(2),…, xS(T) such that the number of selectees

less than c andless than c and

T

SNS

T

SS txtLtxtLEz ))(1)(()()(sup tt 11

Page 11: Sequential, Stochastic Screening Problems

M k d i i (MDP)Markov decision process (MDP)T 1 T+1 stages Stage t describes system after t passengers assigned to classes Stage 0 is initial state Stage 0 is initial state

S denotes set of states, with s(t) capturing the remaining selectee capacity c

Variables: xS(t) = 1 if passenger t classified as a selectee, 0 if classified as

a nonselecteea nonselectee Transition probabilities

h i0

)()()1( if 1))(),(|)1((

txtststxtstsp S

S

Rewards

otherwise0

))(),(|)((p S

( ( ) ( ) ( )) ( ) ( ) ( )(1 ( ))r s t t x t L t x t L t x t ( ( ), ( ), ( )) ( ) ( ) ( )(1 ( ))S S S NS Sr s t t x t L t x t L t x t

Page 12: Sequential, Stochastic Screening Problems

MDP V l F tiMDP Value Functions Vt(s(t)) = optimal expected security for assigning

passenger t and remaining T – t passengers))(1)(()()(

))()(())(1)(()()(

max))((1

}1,0{)(

txtsV

txtLtxtLEtsV

St

SNSSS

txts

Ts(t)for and ,...,2,1for

Tt0))((1 tsV

Optimal policy found by dynamic programming

0))((1 tsVT

Page 13: Sequential, Stochastic Screening Problems

SSPSP tiSSPSP properties Optimal policy for SSPSP is deterministic and MarkovianFollows from the number of states being finite.

Relationship to Dynamic and Stochastic Knapsack Problem (DSKP)

Papastavrou et al. 1996, Management Science 42(12), 1706 – 1718.

Sequential Stochastic Assignment Problem (SSA) Sequential Stochastic Assignment Problem (SSA)Derman et al. 1972, Management Science 18(7), 349 – 355.

Page 14: Sequential, Stochastic Screening Problems

SSPSP Implications Based on DSKPTh O i l li f SSPSP i l if Theorem: Optimal policy for SSPSP is to classify passenger t as a selectee if (threshold policy)

1 1( ) ( 1)V c V c 1 1( ) ( 1)( ) ( ) t tt

S NS

V c V ct H cL L

Refer to as the critical assessed threat value.)(cHt

Page 15: Sequential, Stochastic Screening Problems

SSPSP Implications Based on DSKPTh Theorem: Vt( ) is a concave nondecreasing function of , t=1,2,…,T Vt( ) is a concave nonincreasing function of t =1 2 c

c cc c Vt( ) is a concave nonincreasing function of t, =1,2,…,c

Ht( ) is nonincreasing with , t=1,2,…,T Ht( ) is nonincreasing with t, =1,2,…,c

cc

cc

c

c

Proposition (new) Vt( ) is a concave nondecreasing function of p, =1,2,…,c,

t=1 2 Tc c

t=1,2,…,T Ht( ) is nondecreasing with p, =1,2,…,c, t=1,2,…,Tc c

Page 16: Sequential, Stochastic Screening Problems

SSPSP Implications Based on DSKPP ith l d th t l lik l t b Passengers with lower assessed threat values are more likely to be classified as selectees at the end of the time interval than at the beginningC it f V ( ) ith i li th t h i t it i Concavity of Vt( ) with implies that having extra capacity is more beneficial when the remaining capacity is lower.

The total security increases when more passengers are expected to

c c

arrive. Monotonicity of the critical assessed threat value with respect to p

implies that any given passenger is less likely to be classified as a p y g p g yselectee when more passengers are expected to arrive.

The concavity of the critical assessed threat value implies that increasing p when p is large has a more conservative effect on the g p p gpolicy than when p is small.

Page 17: Sequential, Stochastic Screening Problems

SSPSP I li i B d SSASSPSP Implications Based on SSASSA d t i “b k i t ” (J t 1 2 T t’ T t 1) i d th t SSA determines “breakpoints” (Jt’,t t=1,2,…,T, t’=T-t+1) in assessed threat value range that determines whether a passenger should be a selectee

Jt’ 0 J JJ JJt’,0 Jt’,1 … Jt’,cJt’,2 …Jt’,t’

Adapts main result from Derman et al. (1972) given f(), F(a(t))

Nonselectee class Selectee class

Compute intervals for passenger t (t’=T-t+1)',

' 1, ', 1 ', 1 ', ',( ) [1 ( )] ( )t jJ

t j t j t j t j t jJ J F J J F J y dF y E[t] = JT+1,t, t=1,2,…,T (expected value of tth smallest assessed threat

value)

', 1t jJ

Page 18: Sequential, Stochastic Screening Problems

Sequential Stochastic AssignmentSequential Stochastic Assignment Heuristic (SSAH)Note: SSPSP is the Generalized SSA (GSSA) Note: SSPSP is the Generalized SSA (GSSA) A passenger does not always arrive in each time period

Optimal policy for GSSA/SSPSP adapts SSA policy

P i i O i l li f SSPSP d d d

1,( )t T t c tH c J

Proposition: Optimal policy for SSPSP does not depend on security levelsC ti J d d th d th t lComputing Jt’,t depends on the assessed threat value

distributions, not the security values

Page 19: Sequential, Stochastic Screening Problems

Ill t ti lIllustrative exampleT 2000 t ith 0 5 T = 2000 stages with p = 0.5

c = 50, 100, 200 L = 0 9 L = 0 7 LS = 0.9, LNS = 0.7 Assessed threat values truncated exponential with

parameter 16 (mean 1/16)parameter 16 (mean 1/16)

Results over 100 replications with c = 50, 100, 200p , ,Optimal MDP solutions within 0.001 of optimal

screening when passenger set known a priori

Page 20: Sequential, Stochastic Screening Problems

M t i it ltMonotonicity results

V l f ti d iti l d th t l f ti f i l 10Value function and critical assessed threat value as a function of arrival, c = 10.

Page 21: Sequential, Stochastic Screening Problems

M t i it ltMonotonicity results

V l f ti d iti l d th t l f ti fValue function and critical assessed threat value as a function of remaining capacity in the selectee class, t = 1.

Page 22: Sequential, Stochastic Screening Problems

M t i it ltMonotonicity results

V l f ti d iti l d th t l f ti f 10 t 1Value function and critical assessed threat value as a function of p, c = 10, t = 1.

Page 23: Sequential, Stochastic Screening Problems

C i i l d h lCritical assessed threat values0 4

0.35

0.4al

ue

c = 50c = 100c=200

0.25

0.3

thre

at v

a c 200

0.15

0.2

asse

ssed

0.05

0.1

Crit

ical

a

0 500 1000 1500 20000

Stage

Stage

Page 24: Sequential, Stochastic Screening Problems

Expected total security when assessedExpected total security when assessed threat value distribution inaccurate

Var h pothesi ed distrib tion sing * 1/16Vary hypothesized distribution using : 1/16True distribution

*

Page 25: Sequential, Stochastic Screening Problems

Remaining capacity when assessedRemaining capacity when assessed threat value distribution and p inaccurate

Page 26: Sequential, Stochastic Screening Problems

C l i d F t W kConclusions and Future WorkSSPSP f l t d d ti l li id tifi d SSPSP formulated and optimal policy identified

A heuristic is identified that is retrospectively shown to always be optimal in an illustrative exampleE ample ill strates Example illustrates: Extremely high-risk passengers almost always classified as selectees Critical assessed threat values relatively constant Optimal policy sensitive to accuracy of assessed threat value

distribution Optimal policy less sensitive to accuracy of passenger arrival rate

Th d l i h d l b d d d d d i h The underlying methodology can be adapted and extended to examine how to optimally screen trucks entering the US at land border crossings

l t ti f ll l t t id f US t select proportion of small vessels to screen outside of US ports select the level of screening to assign to trucks or small vessels select which biosensor alarms to which to respond dispatch (heterogeneous) ambulances to prioritized 911 calls

Page 27: Sequential, Stochastic Screening Problems

Thank you!