sequential, stochastic screening problems
DESCRIPTION
My 2011 INFORMS Computing Society Conference presentation on risk-based passenger screening for aviation security.TRANSCRIPT
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
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
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
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
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
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
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
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
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
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
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
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
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.
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
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
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.
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
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
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
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.
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.
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.
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
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
*
Remaining capacity when assessedRemaining capacity when assessed threat value distribution and p inaccurate
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
Thank you!