Transcript of "Sequential, Stochastic Screening Problems"
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Sequential, St h ti S ti l Stochastic Screening Problems Laura A. McLay Virginia Commonwealth University lamclay@vcu.edu Sheldon Jacobson The University of Illinois at Urbana-Champaign Alex Nikolaev The University of BuffaloThis 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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Passenger SP Screening B k i Background d 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
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Uniform S U if Screening vs. S l i S i Selective Screening i All passengers treated the More security for passengers same perceived as higher-risk More security scrutiny for all Less security scrutiny for passengers most passengers t Simpler screening procedures System required for & no privacy issues p y determining who is higher- g g risk Prohibitive cost to screen all May be more cost-effective passengers with all security ith ll it devices
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MotivationM ti ti 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
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FrameworkF k 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- in Focus on one time interval during day at airport Day consists of several time intervals f Resource availability constant over time interval Passenger arrival rate constant over time interval
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Schematic S h ti passenger 1 Passenger T arrives arrivest=0 t=1 t=Tinitial Assignstate passenger 1 to a class Passenger 1 assigned to a class Remaining T-1 passengers have not arrived and their assessed threat values are unknown
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Schematic S h tipassenger 1 passenger t Passenger T arrives arrives arrivest=0 t=1 Assign t=Tinitial Assign passenger tstate passenger 1 to a class 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
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Schematic S h tipassenger 1 passenger t Passenger T arrives arrives arrivest=0 t=1 Assign t=T-1 t=Tinitial Assign passenger tstate passenger 1 to a class 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
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Sequential Stochastic Multilevel PassengerScreening Problem (SSMPSP) T stages in time interval when passenger can check-in t i ti i t l h h ki Passenger arrives in each stage with probability p, resulting 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 (capacity of nonselectee (NS) class assumed t b ( it f l t l d to be infinite) the security level of each class LS and LNS with 0 LS (and LNS) 1
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SSPSP,SSPSP cont’d. t’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 xS(1), xS(2),…, xS(T) such that the number of selectees less than c and T T z sup E LS (t ) xS (t ) LNS (t )(1 xS (t )) t 1 t 1
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Markov decisionM k d i i process (MDP) T+1 T 1 stages Stage t describes system after t passengers assigned to classes 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 nonselectee Transition probabilities 1 if s (t 1) s (t ) xS (t ) p ( s (t 1) | s (t ), xS (t )) 0 otherwise h i Rewards r ( s (t ), (t ) xS (t )) LS (t ) xS (t ) LNS (t )(1 xS (t )) ) ),
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MDP V l F Value Functions ti Vt(s(t)) = optimal expected security for assigning passenger t and remaining T – t passengers LS (t ) xS (t ) LNS (t )(1 xS (t )) Vt ( s (t )) max E xs ( t ){0 ,1} Vt 1 ( s (t ) xS (t )) for t 1,2,..., T and for s(t) TVT 1 ( s (t )) 0 Optimal policy found by dynamic programming
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SSPSP properties ti 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) Derman et al. 1972, Management Science 18(7), 349 – 355.
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SSPSP Implications Based on DSKP Theorem: Optimal policy for SSPSP is to classify passenger t as a Th O i l li f i l if selectee if (threshold policy) Vt 1 (c ) Vt 1 (c 1) (t ) H t (c ) LS LNSRefer to H t (c ) as the critical assessed threat value.
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SSPSP Implications Based on DSKP Theorem: Th Vt( c ) is a concave nondecreasing function of c , t=1,2,…,T Vt( c ) is a concave nonincreasing function of t c =1 2 t, =1,2,…,cc Ht( c ) is nonincreasing with c, t=1,2,…,T Ht( c ) is nonincreasing with t, c =1,2,…,c Proposition (new) Vt( c ) is a concave nondecreasing function of p, c =1,2,…,c, t=1,2,…,T t=1 2 T Ht( c ) is nondecreasing with p, c =1,2,…,c, t=1,2,…,T
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SSPSP Implications Based on DSKP Passengers with lower assessed threat values are more likely t b P ith l d th t l lik l to be classified as selectees at the end of the time interval than at the beginning Concavity of Vt( c ) withc implies th t having extra capacity i more C it f ith i li that h i t it is beneficial when the remaining capacity is lower. The total security increases when more passengers are expected to arrive. Monotonicity of the critical assessed threat value with respect to p implies that any g p y given p passenger is less likely to be classified as a g y selectee 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 g policy than when p is small.
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SSPSP I li i Implications B Based on SSA d SSA determines “b k i t ” (Jt’,t t 1 2 d t i “breakpoints” t=1,2,…,T, t’ T t 1) i assessed th t T t’=T-t+1) in d threat value range that determines whether a passenger should be a selecteeJt’ 0 Jt’,1 Jt’,2 t’,0 … Jt’,c …Jt’,t’ J Nonselectee class Selectee class Adapts main result from Derman et al. (1972) given f(), F(a(t)) Compute intervals for passenger t (t’=T-t+1) J t , j J t 1, j J t , j 1 F ( J t , j 1 ) J t , j [1 F ( J t , j )] J t , j 1 y dF ( y ) E[t] = JT+1,t, t=1,2,…,T (expected value of tth smallest assessed threat value)
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Sequential Stochastic Assignment Heuristic (SSAH) 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 H t (c ) J T t c 1,t Proposition: O i l policy f SSPSP d P ii Optimal li for does not d depend on d security levels CComputing Jt’,t d ti depends on th assessed th t value d the d threat l distributions, not the security values
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IllustrativeIll t ti example l T = 2000 stages with p = 0 5 t ith 0.5 c = 50, 100, 200 LS = 0.9, LNS = 0.7 09 07 Assessed threat values truncated exponential with parameter 16 (mean 1/16) Results over 100 replications with c = 50, 100, 200 p , , Optimal MDP solutions within 0.001 of optimal screening when passenger set known a priori
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Monotonicity resultsM t i it lt Value function and critical assessed th t value as a f V l f ti d iti l d threat l function of arrival, c = 10 ti f i l 10.
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Monotonicity resultsM t i it lt Value function and critical assessed th t value as a f V l f ti d iti l d threat l function of ti f remaining capacity in the selectee class, t = 1.
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Monotonicity resultsM t i it lt Value function and critical assessed th t value as a f V l f ti d iti l d threat l function of p, c = 10 t = 1 ti f 10, 1.
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CriticalC i i l assessed threat values d h l 0.4 04 c = 50 0.35 c = 100 alue c 200 c=200 Critical assessed threat va 0.3 0.25 0.2 0.15 a 0.1 0.05 0 0 500 1000 1500 2000 Stage
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Expected total security when assessedthreat value distribution inaccurateVar h pothesi ed distrib tion using : * 1/16Vary hypothesized distribution singTrue distribution *
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Remaining capacity when assessedthreat value distribution and p inaccurate
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Conclusions and F tC l i d Future W k Work SSPSP formulated and optimal policy identified f l t d d ti l li id tifi d A heuristic is identified that is retrospectively shown to always be optimal in an illustrative example Example illustrates: E ample ill strates 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 The d l i Th underlying methodology can b adapted and extended to examine h h d l be d d d d d i how to optimally screen trucks entering the US at land border crossings select proportion of small vessels t screen outside of US ports l t ti f ll l to t id f t 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
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