Algorithms for Port of Entry Inspection for WMDs

1
Algorithms for Port of Entry Inspection for WMDs
Fred S. Roberts DIMACS Center, Rutgers University
Philip D. Stroud Los Alamos National Laboratory
2
Port of Entry Inspection Algorithms

• Goal Find ways to intercept illicit
• nuclear materials and weapons
• destined for the U.S. via the
• maritime transportation system
• Currently inspecting only small
• of containers arriving at ports
• Even inspecting 8 of containers in Port of NY/NJ
  might bring international trade to a halt
  (Larrabbee 2002)

3
Port of Entry Inspection Algorithms

• Aim Develop decision support algorithms that
  will help us to optimally intercept illicit
  materials and weapons subject to limits on
  delays, manpower, and equipment
• Find inspection schemes that minimize total
  cost including cost of false positives and
  false negatives

Mobile Vacis truck-mounted gamma ray imaging
system
4
Sequential Decision Making Problem

• Stream of containers arrives at a port
• The Decision Makers Problem
• Which to inspect?
• Which inspections next based on previous results?
• Approach
• decision logics
• combinatorial optimization methods
• Builds on ideas of Stroud
• and Saeger at Los Alamos
• National Laboratory
• Need for new models
• and methods

5
Sequential Diagnosis Problem

• Such sequential diagnosis problems arise in many
  areas
• Communication networks (testing connectivity,
  paging cellular customers, sequencing tasks, )
• Manufacturing (testing machines, fault diagnosis,
  routing customer service calls, )
• Artificial intelligence/CS (optimal derivation
  strategies in knowledge bases, best-value
  satisficing search, coding decision trees, )
• Medicine (diagnosing patients, sequencing
  treatments, )

6
Sequential Decision Making Problem

• Containers arriving to be classified into
  categories.
• Simple case 0 ok, 1 suspicious
• Inspection scheme specifies which inspections
  are to be made based on previous observations

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Sequential Decision Making Problem

• Containers have attributes, each
• in a number of states
• Sample attributes
• Levels of certain kinds of chemicals or
  biological materials
• Whether or not there are items of a certain kind
  in the cargo list
• Whether cargo was picked up in a certain port

8
Sequential Decision Making Problem

• Currently used attributes
• Does ships manifest set off an alarm?
• What is the neutron or Gamma emission count? Is
  it above threshold?
• Does a radiograph image come up positive?
• Does an induced fission test come up positive?

Gamma ray detector
9
Sequential Decision Making Problem

• We can imagine many other attributes
• This project is concerned with general
  algorithmic approaches.
• We seek a methodology not tied to todays
  technology.
• Detectors are evolving quickly.

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Sequential Decision Making Problem

• Simplest Case Attributes are in state 0 or 1
• Then Container is a binary string like 011001
• So Classification is a decision function F that
  assigns each binary string to a category.

011001
F(011001)
If attributes 2, 3, and 6 are present and others
are not, assign container to category F(011001).
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Sequential Decision Making Problem

• If there are two categories, 0 and 1, decision
  function F is a boolean function.
• Example
• F(000) F(111) 1, F(abc) 0 otherwise
• This classifies a container as positive iff it
  has none of the attributes or all of them.

1
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Sequential Decision Making Problem

• Given a container, test its attributes until know
  enough to calculate the value of F.
• An inspection scheme tells us in which order to
  test the attributes to minimize cost.
• Even this simplified problem is hard
  computationally.

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Sequential Decision Making Problem

• This assumes F is known.
• Simplifying assumption Attributes are
  independent.
• At any point we can stop inspecting and output
  the value of F based on outcomes of inspections
  so far.
• Complications May be precedence relations in the
  components (e.g., cant test attribute a4 before
  testing a6.
• Or cost may depend on attributes tested before.
• F may depend on variables that cannot be
  directly tested or for which tests are too
  costly.

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Sequential Decision Making Problem

• Such problems are hard computationally.
• There are many possible boolean functions F.
• Even if F is fixed, problem of finding a good
  classification scheme (to be defined precisely
  below) is NP-complete.
• Several classes of functions F allow for
  efficient inspection schemes
• k-out-of-n systems
• Certain series-parallel systems
• Read-once systems
• regular systems
• Horn systems

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Sensors and Inspection Lanes

• n types of sensors measure presence or absence
  of the n attributes.
• Many copies of each sensor.
• Complication different characteristics of
  sensors.
• Entities come for inspection.
• Which sensor of a given type to
• use?
• Think of inspection lanes and
• queues.
• Besides efficient inspection
• schemes, could decrease costs by
• Buying more sensors
• Change allocation of containers to sensor lanes.

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Binary Decision Tree Approach

• Sensors measure presence/absence of attributes.
• Binary Decision Tree (BDT)
• Nodes are sensors or categories (0 or 1)
• Two arcs exit from each sensor node, labeled left
  and right.
• Take the right arc when sensor says the attribute
  is present, left arc otherwise

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Binary Decision Tree Approach

• Reach category 1 from the root only through the
  path a0 to a1 to 1.
• Container is classified in category 1 iff it has
  both attributes a0 and a1 .
• Corresponding boolean function F(11) 1, F(10)
  F(01) F(00) 0.

Figure 1
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Binary Decision Tree Approach

• Reach category 1 from the root by
• a0 L to a1 R a2 R 1 or
• a0 R a2 R1
• Container classified in category 1 iff it has
• a1 and a2 and not a0 or
• a0 and a2 and possibly a1.
• Corresponding boolean function F(111) F(101)
  F(011) 1, F(abc) 0 otherwise.

Figure 2
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Binary Decision Tree Approach

• This binary decision tree corresponds to the same
  boolean function
• F(111) F(101) F(011) 1, F(abc) 0
  otherwise.
• However, it has one less observation node ai. So,
  it is more efficient if all observations are
  equally costly and equally likely.

Figure 3
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Binary Decision Tree Approach

• Even if the boolean function F is fixed, the
  problem of finding the optimal binary decision
  tree for it is very hard (NP-complete).
• For small n number of attributes, can try to
  solve it by brute force enumeration.
• Even for n 4, not practical.
• (n 4 at Port of Long Beach-Los Angeles)

Port of Long Beach
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Binary Decision Tree Approach

• Promising Approaches
• Heuristic algorithms, approximations to optimal.
• Special assumptions about the boolean function F.
  
• For monotone boolean functions, integer
  programming formulations give promising
  heuristics.
• Stroud and Saeger enumerate all
• complete, monotone boolean functions
• and calculate the least expensive
• corresponding binary decision trees.
• Their method practical for n up to 4,
• not n 5.

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Binary Decision Tree Approach

• Monotone Boolean Functions
• Given two strings x1x2xn, y1y2yn
• Suppose that xi ? yi for all i implies that
  F(x1x2xn) ? F(y1,y2yn).
• Then we say that F is monotone.
• Then 111 has highest probability of being in
  category 1.

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Binary Decision Tree Approach

• Incomplete Boolean Functions
• Boolean function F is incomplete if F can be
  calculated by finding at most n-1 attributes
  and knowing the value of the input string on
  those attributes
• Example F(111) F(110) F(101) F(100) 1,
  F(000) F(001) F(010) F(011) 0.
• F(abc) is determined without knowing b (or
  c).
• F is incomplete.

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Binary Decision Tree Approach

• Complete, Monotone Boolean Functions
• Stroud and Saeger brute force algorithm for
  enumerating binary decision trees implementing
  complete, monotone boolean functions and choosing
  least cost BDT.
• Feasible to implement up to n 4.
• n 2
• There are 6 monotone boolean functions.
• Only 2 of them are complete, monotone
• There are 4 binary decision trees for calculating
  these 2 complete, monotone boolean functions.

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Binary Decision Tree Approach

• Complete, Monotone Boolean Functions
• n 3
• 9 complete, monotone boolean functions.
• 60 distinct binary trees for calculating them
• n 4
• 114 complete, monotone boolean functions.
• 11,808 distinct binary decision trees for
  calculating them.
• (Compare 1,079,779,602 BDTs for all boolean
  functions)

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Binary Decision Tree Approach

• Complete, Monotone Boolean Functions
• n 5
• 6894 complete, monotone boolean functions
• 263,515,920 corresponding binary decision trees.
• Combinatorial explosion!
• Need alternative approaches enumeration not
  feasible!
• (Even worse compare 5 x 1018 BDTs corresponding
  to all boolean functions)

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Cost Functions

• Stroud-Saeger method applies to more
  sophisticated cost models, not just cost number
  of sensors in the BDT.
• Using a sensor has a cost
• Unit cost of inspecting one item with it
• Fixed cost of purchasing and deploying it
• Delay cost from queuing up at the sensor station
• Preliminary problem disregard fixed and delay
  costs. Minimize unit costs.

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Cost Functions Delay Costs

• Tradeoff between fixed costs and delay costs Add
  more sensors cuts down on delays.
• Stochastic process of containers arriving
• Distribution of delay times for inspections
• Use queuing theory to find average delay times
  under different models

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Cost FunctionsUnit CostsTree Utilization

• Complication Assume cost depends on how many
  nodes of BDT are actually visited during an
  average containers inspection. (This is sum of
  unit costs.)
• Depends on characteristics of population of
  entities being inspected.
• I.e., depends on distribution of containers.
• In our early models, we assume we are given
  probability of sensor errors and probability of
  bomb in a container.
• This allows us to calculate expected cost of
  utilization of the tree Cutil.

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Cost Functions

• Cost of false positive Cost of additional tests.
  
• If it means opening the container, its very
  expensive.
• Cost of false negative
• Complex issue.
• What is cost of a bomb going off in Manhattan?

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Cost Functions Sensor Errors

• One Approach to False Positives/Negatives
• Assume there can be Sensor Errors
• Simplest model assume that all sensors checking
  for attribute ai have same fixed probability of
  saying ai is 0 if in fact it is 1, and
  similarly saying it is 1 if in fact it is 0.
• More sophisticated analysis later describes a
  model for determining probabilities of sensor
  errors.
• Notation X state of nature (bomb or no bomb)
• Y outcome (of sensor or entire inspection
  process).

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Probability of Error for The Entire Tree

• State of nature is zero (X 0), absence of a bomb

State of nature is one (X 1), presence of a bomb
Probability of false positive (P(Y1X0)) for
this tree is given by
Probability of false negative (P(Y0X1)) for
this tree is given by
P(Y1X0) P(YA1X0) P(YB1X0)
P(YA1X0) P(YB0X0) P(YC1X0)
Pfalsepositive
P(Y0X1) P(YA0X1) P(YA1X1)
P(YB0X1)P(YC0X1) Pfalsenegative
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Cost Function used for Evaluating the Decision
Trees.

• CTot CFalsePositive PFalsePositive
  CFalseNegative PFalseNegative Cutil

CFalsePositive is the cost of false positive
(Type I error) CFalseNegative is the cost of
false negative (Type II error) PFalsePositive is
the probability of a false positive
occurring PFalseNegative is the probability of a
false negative occurring Cutil is the expected
cost of utilization of the tree.
34
Stroud Saeger Experiments

• Stroud-Saeger ranked all trees formed
• from 3 or 4 sensors A, B, C and D
• according to increasing tree costs.
• Used cost function defined above.
• Values used in their experiments
• CA .25 P(YA1X1) .90 P(YA1X0) .10
• CB 10 P(YC1X1) .99 P(YB1X0) .01
• CC 30 P(YD1X1) .999 P(YC1X0) .001
• CD 1 P(YD1X1) .95 P(YD1X0) .05
• Here, Ci unit cost of utilization of sensor i.
  
• Also fixed were CFalseNegative, CFalsePositive,
  P(X1)

35
Stroud Saeger Experiments Our Sensitivity
Analysis

• We have explored sensitivity of the Stroud-Saeger
  conclusions to variations in values of these
  three parameters.
• We estimated high and low values for these
  parameters.
• n 3 (use sensors A, B, C)
• We chose one of the values from the interval of
  values and then explored the highest ranked tree
  as the other two were chosen at random in the
  interval of values. 10,000 experiments for each
  pair of fixed values.
• We looked for the variation in the top-ranked
  tree and how the top-rank related to choice of
  parameter values.
• Very surprising results.

36
Stroud Saeger Experiments Our Sensitivity
Analysis

• CFalseNegative was varied between 25 million and
  10 billion dollars
• Low and high estimates of direct and indirect
  costs incurred due to a false negative.
• CFalsePositive was varied between 180 and 720
• Cost incurred due to false positive
• (4 men (3 -6 hrs) (15 30 /hr)
• P(X1) was varied between 1/10,000,000 and
  1/100,000

37
Frequency of Top-ranked Trees when CFalseNegative
and CFalsePositive are Varied

• 10,000 randomized experiments (randomly selected
  values of CFalseNegative and CFalsePositive from
  the specified range of values) for the median
  value of P(X1).
• The above graph has frequency counts of the
  number of experiments when a particular tree was
  ranked first or second, or third and so on.
• Only three trees (7, 55 and 1) ever came first. 6
  trees came second, 10 came third, 13 came fourth.

38
Frequency of Top-ranked Trees when CFalseNegative
and P(X1) are Varied

• 10,000 randomized experiments for the median
  value of CFalsePositive.
• Only 2 trees (7 and 55) ever came first. 4 trees
  came second. 7 trees came third. 10 and 13 trees
  came 4th and 5th respectively.

39
Frequency of Top-ranked Trees when P(X1) and
CFalsePositive are Varied

• 10,000 randomized experiments for the median
  value of CFalseNegative.
• Only 3 trees (7, 55 and 1) ever came first. 6
  trees came second. 10 trees came third. 13 and 16
  trees came 4th and 5th respectively.

40
Values of CFalseNegative and CFalsePositive when
Tree 7 was Ranked First

• This is a graph of CFalsePositive against
  CFalseNegative values obtained from the
  randomized experiments. The black dots represent
  points at which tree 7 scored first rank.

41
Values of CFalseNegative and CFalsePositive when
Tree 55 was Ranked First

• Tree 55 fills up the lower area in the range of
  CFalseNegative and CFalsePositive values.

42
Values of CFalseNegative and CFalsePositive when
Tree 1 was Ranked First

• Tree 1 fills up the upper area in the range of
  CFalseNegative and CFalsePositive.

43
Values of CFalseNegative and CFalsePositive for
the Three First Ranked Trees

• Trees 7, 55 and 1 fill up the entire area in the
  range of CFalseNegative and CFalsePositive among
  themselves.

44
Values of CFalseNegative and P(X1) when Tree 7
was Ranked First

• Tree 7 again fills up the major area in the range
  of CFalseNegative and P(X1).

45
Values of CFalseNegative and P(X1) when Tree 55
was Ranked First

• Tree 55 fills up the rest of the area in the
  range of CFalseNegative and P(X1).

46
Values of CFalseNegative and P(X1) for First
Ranked Trees

• Together trees 7 and 55 fill up the entire region
  of CFalseNegative and P(X1).

47
Values of CFalsePositive and P(X1) When Tree 7
was Ranked First

• Tree 7 fills up the major area in the range of
  CFalsePositive and P(X1).

48
Values of CFalsePositive and P(X1) when Tree 55
was Ranked First

• Tree 55 fills up the upper area in the range of
  CFalsePositive and P(X1).

49
Values of CFalsePositive and P(X1) when Tree 1
was Ranked First

• Tree 1 fills up the lower area in the range of
  CFalsePositive and P(X1).

50
Values of CFalsePositive and P(X1) for First
Ranked Trees

• Trees 7, 55 and 1 fill up the entire area in the
  range of CFalsePositive and P(X1) among
  themselves.

51
Stroud Saeger Experiments Our Sensitivity
Analysis 4 Sensors

• Second set of computer experiments n 4
• (use sensors, A, B, C, D).
• Same values as before.
• Experiment 1 Fix values of two of
  CFalseNegative, CFalsePositive, P(X1) and vary
  the third through their interval of possible
  values.
• Experiment 2 Fix a value of one of
  CFalseNegative, CFalsePositive, P(X1) and vary
  the other two.
• Do 10,000 experiments each time.
• Look for the variation in the highest ranked
  tree.

52
Stroud Saeger Experiments Our Sensitivity
Analysis 4 Sensors

• Experiment 1 Fix values of two of
  CFalseNegative, CFalsePositive, P(X1) and vary
  the third.

53
CTot vs CFalseNegative for Ranked 1 Trees (Trees
11485(9651) and 10129(349))
Only two trees ever were ranked first, and one,
tree 11485, was ranked first in 9651 out of
10,000 runs.
54
CTot vs CFalsePositive for Ranked 1 Trees (Tree
no. 11485 (10000))
One tree, number 11485, was ranked first every
time.
55
CTot vs P(X1) for Ranked 1 Trees (Tree no.
11485(8372), 10129(488), 11521(1056))
Three trees dominated first place. Trees
10201(60), 10225(17) and 10153(7) also achieved
first rank but with relatively low frequency.
56
Stroud Saeger Experiments Our Sensitivity
Analysis 4 Sensors

• Experiment 2 Fix the values of one of
  CFalseNegative, CFalsePositive, P(X1) and vary
  the others.

57
Frequency of First Ranked Trees when Two
Parameters (CFalseNegative and CFalsePositive)
were Varied Keeping P(X1) Constant at Randomly
Selected Values.

• 10,000 randomized experiments with randomly
  selected values of P(X1)
• The experiments were repeated for 20 different
  randomly selected values of P(X1)

58
Frequency of First Ranked Trees when Two
Parameters (CFalseNegative and P(X1)) were
Varied Keeping CFalsePositive Constant at
Randomly Selected Values.

• 10,000 randomized experiments with randomly
  selected values of CFalsePositive
• The experiments were repeated for 20 different
  randomly selected values of CFalsePositive

59
Frequency of First Ranked Trees when Two
Parameters (P(X1) and CFalsePositive) were
Varied Keeping CFalseNegative Constant at
Randomly Selected Values.

• 10,000 randomized experiments with randomly
  selected values of CFalseNegative
• The experiments were repeated for 20 different
  randomly selected values of CFalseNegative

60
Modeling Sensor Errors

• One Approach to Sensor Errors Modeling Sensor
  Operation
• Threshold Model
• Sensors have different discriminating power
• Many use counts (e.g., Gamma radiation counts)
• See if count exceeds
• threshold
• If so, say attribute is present.

61
Modeling Sensor Errors

• Threshold Model
• Sensor i has discriminating power Ki,
  threshold Ti
• Attribute present if counts exceed Ti
• Calculate fraction of objects in each category
  whose readings exceed T
• Seek threshold values that minimize all costs
  inspection, false positive/negative
• Assume readings of category 0 containers follow a
  Gaussian distribution and similarly category 1
  containers
• Simulation approach

62
Probability of Error for Individual Sensors

• For ith sensor, the type 1 (P(Yi1X0)) and type
  2 (P(Yi0X1)) errors are modeled using Gaussian
  distributions.
• State of nature X0 represents absence of a bomb.
• State of nature X1 represents presence of a
  bomb.
• ?i represents the outcome (count) of sensor i.
• Si is variance of the distributions
• PD prob. of detection, PF prob. of false pos.

Ki
?i
63
Modeling Sensor Errors
The probability of false positive for the ith
sensor is computed as P(Yi1X0) 0.5
erfcTi/v2 The probability of detection for the
ith sensor is computed as P(Yi1X1) 0.5
erfc(Ti-Ki)/(Sv2) erfc complementary error
function erfc(x) ?(1/2,x2)/sqrt(?)
The following experiments have been done using
sensors A, B, C and using KA 4.37 SA 1 KB
2.9 SB 1 KC 4.6 SC 1 We then varied
the individual sensor thresholds TA, TB and TC
from -4.0 to 4.0 in steps of 0.4. These values
were chosen since they gave us an ROC curve
for the individual sensors over a complete range
P(Yi1X0) and P(Yi1X1)
64
Frequency of First Ranked Trees for Variations in
Sensor Thresholds

• 68,921 experiments were conducted, as each Ti was
  varied through its entire range.
• The above graph has frequency counts of the
  number of experiments when a particular tree was
  ranked first. There are 15 such trees. Tree 37
  had the highest frequency of attaining rank one.

65
Conclusions from Sensitivity Analysis

• Considerable lack of sensitivity to modification
  in parameters for trees using 3 or 4 sensors.
• Very few optimal trees.
• Very few boolean functions arise among optimal
  and near-optimal trees.

66
Some Complications

• More complicated cost models bringing in costs
  of delays
• More than two values of an attribute
• (present, absent, present with probability gt 75,
  absent with probability at least 75)
• (ok, not ok, ok with probability gt 99, ok with
  probability between 95 and 99)
• Inferring the boolean function from observations
  (partially defined boolean functions)

67
Some Research Challenges

• Explain why conclusions are so insensitive to
  variation in parameter values.
• Explore the structure of the optimal trees and
  compare the different optimal trees.
• Develop less brute force methods for finding
  optimal trees that might work if there are more
  than 4 attributes.
• Develop methods for
• approximating the optimal tree.

Pallet vacis
68
Closing Remark

• Recall that the cost of inspection includes the
  cost of failure, including failure to foil a
  terrorist plot.
• There are many ways to lower the total cost of
  inspection
• Use more efficient
• orders of inspection.
• Find ways to inspect
• more containers.
• Find ways to cut down
• on delays at inspection lanes.

69
Research Team

• Saket Anand, Rutgers, ECE graduate student
• Endre Boros, Rutgers, Operations Research
• Elsayed Elsayed, Rutgers, Ind. Systems
  Engineering
• Liliya Fedzhora, Rutgers, Operations Res. grad.
  student
• Paul Kantor, Rutgers, Schl. of Infor. Library
  Studies
• Abdullah Karaman, Rutgers Ind. Syst. Eng. grad.
  student
• Alex Kogan, Rutgers, Business School
• Paul Lioy, Rutgers/UMDNJ, Environmental and
  Occupational Health and Sciences Institute
• David Madigan, Rutgers, Statistics
• Richard Mammone, Rutgers, Center for Advanced
  Information Processing
• S. Muthukrishnan, Rutgers, Computer Science
• Saumitr Pathek, Rutgers ECE graduate student
• Richard Picard, Los Alamos, Statistical Sciences
  Group
• Fred Roberts, Rutgers, DIMACS Center
• Kevin Saeger, Los Alamos, Homeland Security
• Phillip Stroud, Los Alamos, Systems Engineering
  and Integration Group
• Hao Zhang, Rutgers Ind. Systems Eng., graduate
  student

70

• Collaborators on Sensitivity Analysis
• Saket Anand
• David Madigan
• Richard Mammone
• Saumitr Pathak
• Research Support
• Office of Naval Research
• National Science Foundation
• Los Alamos National Laboratory
• Rick Picard
• Kevin Saeger