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Sequential, St h ti                                    S      ti l Stochastic                                    Screening...
Passenger SP         Screening B k                i Background                           d   Passenger screening visible ...
Uniform S    U if    Screening vs. S l i S                  i       Selective Screening                                   ...
MotivationM ti ti   Static models for passenger screening   Passenger screening is dynamic      Staffing loads      Pa...
FrameworkF       k   Passengers/bags screened by series of devices      System response a function of device responses ...
Schematic  S h   ti          passenger 1              Passenger T            arrives                  arrivest=0        t=...
Schematic  S h   tipassenger 1                      passenger t                Passenger T  arrives                       ...
Schematic  S h   tipassenger 1             passenger t                           Passenger T  arrives                 arri...
Sequential Stochastic Multilevel PassengerScreening Problem (SSMPSP)   T stages in time interval when passenger can check...
SSPSP,SSPSP cont’d.         t’d   Goal: find policy that maximizes expected total security   Variables:      xS(t) = 1 ...
Markov decisionM k d i i process (MDP)   T+1    T 1 stages      Stage t describes system after t passengers assigned to ...
MDP V l F    Value Functions              ti   Vt(s(t)) = optimal expected security for assigning    passenger t and rema...
SSPSP properties            ti Optimal policy for SSPSP is deterministic and MarkovianFollows from the number of states b...
SSPSP Implications Based on DSKP   Theorem: Optimal policy for SSPSP is to classify passenger t as a    Th          O i l...
SSPSP Implications Based on DSKP   Theorem:    Th      Vt( c ) is a concave nondecreasing function of c , t=1,2,…,T     ...
SSPSP Implications Based on DSKP   Passengers with lower assessed threat values are more likely t b    P              ith...
SSPSP I li i        Implications B                     Based on SSA                         d     SSA determines “b k i t...
Sequential Stochastic Assignment    Heuristic (SSAH)   Note: SSPSP is the Generalized SSA (GSSA)      A passenger does n...
IllustrativeIll t ti example              l   T = 2000 stages with p = 0 5               t        ith  0.5   c = 50, 100...
Monotonicity resultsM   t i it       lt Value function and critical assessed th t value as a f V l f     ti     d iti l   ...
Monotonicity resultsM   t i it       lt Value function and critical assessed th t value as a f V l f     ti     d iti l   ...
Monotonicity resultsM   t i it       lt Value function and critical assessed th t value as a f V l f     ti     d iti l   ...
CriticalC i i l assessed threat values               d h        l                                      0.4                ...
Expected total security when assessedthreat value distribution inaccurateVar h pothesi ed distrib tion using : *  1/16Va...
Remaining capacity when assessedthreat value distribution and p inaccurate
Conclusions and F tC   l i       d Future W k                       Work   SSPSP formulated and optimal policy identified...
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Sequential, Stochastic Screening Problems

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

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Transcript of "Sequential, Stochastic Screening Problems"

  1. 1. 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
  2. 2. 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
  3. 3. 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
  4. 4. 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
  5. 5. 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
  6. 6. 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
  7. 7. 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
  8. 8. 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
  9. 9. 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
  10. 10. 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 
  11. 11. 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 )) ) ),
  12. 12. 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
  13. 13. 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.
  14. 14. 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.
  15. 15. 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
  16. 16. 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.
  17. 17. 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)
  18. 18. 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
  19. 19. 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
  20. 20. 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.
  21. 21. 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.
  22. 22. 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.
  23. 23. 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
  24. 24. Expected total security when assessedthreat value distribution inaccurateVar h pothesi ed distrib tion using : *  1/16Vary hypothesized distribution singTrue distribution  *
  25. 25. Remaining capacity when assessedthreat value distribution and p inaccurate
  26. 26. 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
  27. 27. Thank you!
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