Sensors Threshold in Port-of-Entry Inspection Systems1
Hao Zhang, Christina Schroepfer and Elsayed A. Elsayed
Department of Industrial and Systems Engineering, Rutgers University,
96 Frelinghuysen Road, Piscataway, NJ 08854-8138, USA
elsayed@rci.rutgers.edu
ABSTRACT
This paper considers the problem of container inspection at the port-of-entry. Containers are
inspected at inspection stations using a predetermined sequence. The threshold levels of sensors
might result in accepting undesired containers or subjecting “good” containers to unnecessary
additional inspections. In this paper we investigate an approach for determining the optimum
threshold levels of the sensors at inspection stations in order to minimize the overall cost of
inspection. Numerical examples are presented to demonstrate the use of the proposed approach.
Keywords: sensors threshold levels, probability of false positive, probability of false negative,
Boolean function.
1. Introduction
Seaports are critical gateways for the movement of international commerce. More than 95 percent
of our non–North American foreign trade arrives by ship. With “just-in-time” deliveries of goods,
the expeditious flow of commerce through these ports is so essential. Containers are inspected in
order to detect the smuggling of nuclear materials, hazardous and illegal shipments. Slowing the
flow long enough to inspect either all or a statistically significant random selection of imports
would be economically intolerable (Loy and Ross 2002). There are only two techniques of noninvasively “seeing” into a container, both involving projection and the resultant transmission or
reflection of waves: 1) Electromagnetic (EM) waves (radio waves, light, x-rays, gamma rays, etc.)
and 2) Material vibration waves (ultrasound). A variety of means exist for converting these waves
into images suitable for human inspectors to interpret (James et al. 2002). The interpretation may
lead to accepting a container that contains undesirable material (Type II error) or may lead to
further inspection of a container that is acceptable (Type I error) which results in delays and added
cost. Clearly, the threshold levels of these techniques have a direct effect on these types of errors
and the inspection section.
In this paper, we decompose the port-of-entry inspection problem into two sub-problems. The first
problem deals with the determination of the optimum sequence of inspection or the structure of the
1
This research is conducted with partial support from ONR grant number N00014-05-1-0237 and NSF grant number
NSFSES 05-18543 to Rutgers University.
inspection decision tree in order to achieve the minimum expected inspection cost. The second
problem deals with the determination of the optimum thresholds of the sensors at inspection stations
so as to minimize the cost associated with false positive (false alarm, which results in additional
manual inspection) and false negative (failure to identify illicit materials or weapons). The first
problem can be formulated and investigated using approaches parallel to those used in the optimal
sequential inspection procedure for reliability systems as described by Butterworth (1972), Halpern
(1974, 1977) , Ben-Dov (1981), Cox et al. (1989), Cox et al. (1996), and Azaiez et al. (2004). After
the sequences of inspection and the structure of the inspection decision tree are determined, we
determine the optimum thresholds of the sensors at inspection stations. We intend to address the
first problem in section 2 and the second problem in section 3.
2. Optimum Sequence Problem for Port-of-Entry Inspection
2.1. Optimum Sequence Determination
In the port-of-entry inspection application, these are containers (entities) being off loaded from
ships. Containers are inspected and classified according to observations we make regarding their
attributes. There are several categories into which we seek to classify entities. In the simplest case,
these are positive and negative, 1 or 0, with “0” designating entities that are considered “acceptable”
and “1” designating entities that raise suspicion and require special treatment. After each
observation, we either classify the entity or subject it to another inspection process.
The
classification will be thought of as a decision function F that assigns to each binary string of
attributes a category. In this paper, we first focus on the case where there are two categories. Thus,
F is a Boolean function. For instance, consider the Boolean function defined by F(111) = 1 and
F(abc) = 0 otherwise. This is the function that classifies an entity as positive if and only if it has all
of the attributes. Boolean functions provide the selection logic for an inspection scheme. If F is
known, we seek to determine its value by testing the attributes one by one. In a typical case, the
attributes are assumed to be independent. We shall return to the question of whether the distribution
of the attribute states is known. In particular, in the binary case where the probability that the ith
attribute takes on the value 0, which is defined as pi , and the probability that it takes on the value 1,
which is defined as qi  1  pi . The inspection policy determines the order of testing the container’s
attributes. At any point, the inspection scheme might tell us to stop inspecting and output the value
of F based on the outcome of inspections so far. We must make enough observations to allow us to
classify the entity and decide whether to designate it as an entity that needs further inspection or
accept it. The problem is to find an optimum inspection sequence that minimizes the total expected
inspection cost.
This inspection problem is analogous to the inspection problem of a reliability system. Consider a
reliability system, the components (analogous to attributes of a container) of which are to be tested
sequentially in order to identify the state of the system (operating or failed, analogous to the
classification of a container, “acceptable” or “suspicious”). A cost is incurred for testing each
component of the system. The initial failure probability of each component (before testing) is
known, which is analogous to the initial probability that the ith attribute takes on the value 1, which
is defined as qi , as well as the system configuration, which is analogous to the Boolean function F .
The inspection problem of a reliability system is to determine the optimum inspection procedure for
identifying the system state at minimal expected inspection cost. So the optimum inspection
sequence problem for port-of-entry can be formulated and investigated using an approach similar to
that used in the optimal sequential inspection procedure for reliability systems.
The assumptions of port-of-entry inspection system are:
1. There are n inspection stations in the system; each station is used to identify one attribute of the
container being inspected. Let xi be the state of the i th attribute.
2. The classification of each container is thought of as a decision function F that assigns to each
string of attributes a class. In this section, we focus on the case where there are only two classes,
0 and 1, i.e. F ( x1, x2 ,
, xn )  0 means negative class and that there is no suspicion with the
container and F ( x1, x2 ,
, xn )  1 means positive class and that additional manual inspection of
the container is required.
3. Let ci be the unit cost associated with the inspection at inspection station i, i  1, 2,
,n .
4. From inspection history, we might have information about the probability that the attribute i is
present or absent for a group of containers, which are imported by the same company or from
the same origin at the same time. Let pi be the probability that the attribute i is absent and qi
be the probability that the attribute i is present, i.e. pi  P( xi  0), qi  P( xi  1)  1  pi , for
i  1, 2,
,n .
5. The inspection stations are perfect, which means that the true attributes can always be identified
at the corresponding inspection stations without any error.
We now state the results for optimum inspection of containers with series and parallel Boolean
function. By definition, a series Boolean function is a decision function F that assigns the
container class “1” if any of the attributes is present, i.e. xi  1 for any i  {1, 2,
, n} , and a
parallel Boolean function is a decision function F that assigns the container the class “1” if all of
the attributes are present, i.e. xi  1 for all i  {1, 2,
, n} .
Now, consider a group of containers with a series Boolean function and n independent attributes.
Let the inspection procedure be such that attribute i  1 is inspected only if attribute i is found
absent, for all attributes i  1, 2,
, n  1 . The following result holds:
Theorem 1: For a series Boolean decision function, inspecting attributes i  1, 2,
, n in sequential
order is optimum, minimizes expected inspection cost, if and only if:
c1 / q1  c2 / q2 
cn / qn .
(1)
In this case, the expected inspection cost is given by
n
i 1
i 2
j 1
C  c1  [ p j ] ci .
(2)
Now, consider a parallel Boolean decision function, and assume that the inspection procedure is
such that attribute i  1 is inspected only if attribute i is found present, for all attributes
i  1, 2,
, n  1 . Then, using the same notation as above, the following result also holds:
Theorem 2: For a parallel Boolean decision function, inspecting attributes i  1, 2,
, n in
sequential order is optimum (in the sense that it minimizes expected inspection cost) if and only if:
c1 / p1  c2 / p2 
cn / pn .
(3)
In this case, the expected inspection cost is given by
n
i 1
i 2
j 1
C  c1  [ q j ] ci .
The proofs and examples for series/parallel systems are given in Ben-Dov (1981).
(4)
We now generalize the results given above to containers with more general combined series/parallel
Boolean decision functions. Here, we focus on systems of independent attributes that can be
represented without replications, that is Boolean decision functions that can be represented using
only AND/OR logic in such a way that each attribute appears only once.
We first introduce the following definitions:
1. A subsystem S is called a series (parallel) subsystem with constituents S1 ,
obtained by placing S1 ,
, Sn if S can be
, Sn in series (parallel).
2. A series (parallel) subsystem S is called a maximal series (parallel) subsystem if no other
subsystems of the entire system can be obtained by placing additional attributes or subsystems
in series (parallel) with S .
3. The constituents S1 ,
, Sn of a series (parallel) subsystem S are called the basic constituents of
S if none of them is itself a series (parallel) subsystem. That is, each basic constituent Si of a
series subsystem must be either a simple attribute or a parallel subsystem, and conversely for the
basic constituents of a parallel subsystem.
For instance, the system represented in Figure 1 is a maximal series subsystem whose basic
constituents are subsystem S 3 and attribute 5. In turn, S 3 is a maximal parallel subsystem whose
basic constituents are subsystem S 2 and attribute 4, and so on.
Consider a combined series/parallel system that can be represented as discussed above. The
following initialization algorithm is used to order the basic constituents of all subsystems of such a
Boolean decision function, prior to identifying the optimal inspection policy.
Step 1:
Consider any maximal series subsystem S for which all the basic constituents S1 ,
, Sn are simple
attributes. Let ci be the inspection cost of attribute Si , and let pi and qi be the prior absent and
present probabilities of attribute Si , respectively, for all i  1,
, n . Then, do the following:
Figure 1. Sample system for a Boolean decision function
1. Reorder and re-label the attributes (if necessary) so that condition (1) above holds. We say
that S  ( S1 ,
, Sn ) is now ordered.
2. Since the expected cost of inspection the series subsystem S  ( S1 ,
n
i 1
i 2
j 1
, Sn ) is given by
equation (2), set C ( S )  c1  [ p j ] ci .
n
3. Set P( S )   pi and Q ( S )  1  P( S ) to be the absent and present probabilities of
i 1
subsystem S , respectively.
Similarly, for any maximal parallel subsystem S for which all the basic constituents S1 ,
, Sn are
simple attributes, and using the same notation as above, do the following:
4. Reorder and re-label the attributes (if necessary) so that condition (3) above holds. We say
that S  ( S1 ,
, Sn ) is now ordered. Whenever S is to be inspected, it should be inspected
sequentially according to the established order, such that attribute Si 1 is inspected only if
attribute Si if found present.
5. Since the expected cost of inspecting the parallel subsystem S  ( S1 ,
n
i 1
i 2
j 1
equation (4), set C ( S )  c1  [ q j ] ci .
, Sn ) is given by
n
6. Set Q ( S )   qi and P( S )  1  Q ( S ) to be the present and absent probabilities of
i 1
subsystem S , respectively.
If the inspection sequence is now determined, i.e, if all maximal series or parallel subsystems are
now ordered according to steps 1-3 or 4-6, then stop, otherwise go to step 2.
Step 2:
Consider each non-ordered maximal series subsystems S1 ,
, Sn in which all basic constituents are
either ordered subsystems or simple attributes. If any basic constituents Si is a simple attribute, then
let C ( Si ) be the inspection cost of Si , and P( Si ) and Q( Si ) be the absent and present probabilities
of Si , respectively. For each maximal series subsystem S1 ,
, Sn in turn, do the following:
7. Reorder and re-label the basic constituents (if necessary) so that the following condition
holds:
C ( S1 ) / Q( S1 )  C ( S2 ) / Q( S2 ) 
n
i 1
i 2
j 1
C ( Sn ) / Q( Sn ) ,
(5)
8. Set C ( S )  C ( S1 )  [ P( S j )] C ( Si ) .
n
9. Set P ( S )   P ( Si ) and Q ( S )  1  P( S ) to be the absent and present probabilities of
i 1
subsystem S , respectively.
Similarly, for each non-ordered maximal parallel subsystems S1 ,
, Sn , do the following:
10. Reorder and re-label the basic constituents (if necessary) so that the following condition
holds:
C ( S1 ) / P( S1 )  C ( S2 ) / P( S2 ) 
n
i 1
i 2
j 1
C ( S n ) / P( S n ) ,
(5)
11. Set C ( S )  C ( S1 )  [ Q( S j )] C ( Si ) .
n
12. Set Q ( S )   Q ( Si ) and P( S )  1  Q ( S ) to be the present and absent probabilities of
i 1
subsystem S , respectively.
Repeat step 2 as until all subsystems have been ordered.
We are now ready to establish the main result of this section.
Theorem 3: Consider a general system S for a Boolean decision function F , ordered according to
the initialization algorithm. Then, the optimal inspection policy that minimizes the expected
inspection cost follows the orderings specified in the initialization algorithm. Moreover, if a basic
constituent Si j of subsystem S j  ( S1j ,
, Smj j ) is to be inspected, then it should be inspected to
completion before moving on to the inspection of basic constituent Si j 1 of that subsystem (or
inspection of some other subsystem), if needed. In this case, the optimal expected testing cost of the
system is C ( S ) .
2.2. Numerical Example
We now apply the above algorithm to the system given in Figure 1. Let the inspection costs be
c1  10 , c2  12 , c3  7 , c4  6 , and c5  10 . Also, let the success probabilities be p1  0.7 ,
p2  0.8 , p3  0.67 , p4  0.6 , and p5  0.9 .
In step 1, we consider the maximal parallel subsystem S 1 formed by attributes 2 and 3. The ratios of
cost to absent probability are 15 and 10.45 for components 2 and 3, respectively. Based on step 1 of
the initialization algorithm, we renumber the attributes so that constituent S11 of subsystem S 1 will
be attribute 3, and constituent S 21 of subsystem S 1 will be attribute 2. To distinguish the subsystem
before and after ordering, in this example we will use S 1 to denote the subsystem after ordering.
Thus, the ordered subsystem is S 1 = (3, 2). The corresponding expected cost is
C( S 1 )  c3  q3c2  10.96 . Also, the present probability is given by Q( S 1 )  q2q3  0.066 and the
absent probability is therefore P( S 1 )  1  Q ( S 1 )  0.934 .
In step 2, we begin by considering the series subsystem S 2 with attribute 1 and ordered subsystem
S 1 , i.e., S 2  (1, S 1 ) . Set C(1)  c1  10 and P(1)  p1  0.7 . The ratios of cost to present
probability are 33.3 and 166.1 for attribute 1 and ordered subsystem S 1 , respectively. Therefore, the
ordered series subsystem S 2 is given by S 2  (1, S 1 ) . The corresponding expected cost of the
ordered subsystem S 2 is given by C ( S 2 )  C (1)  P(1)C ( S 1 )  17.67 . Also, the absent and present
probabilities of S 2 are 0.65 and 0.35 respectively.
We next consider the parallel subsystem S 3  ( S 2 , 4) . The ratios of cost to absent probability are
28.4 and 10.0 for ordered subsystem S 2 and attribute 4 respectively. This yields the ordered
subsystem S 3  (4, S 2 ) . The present probability of S 3 is Q ( S 3 )  q4Q ( S 2 )  0.14 , leading to
P ( S 3 )  0.86 .
The
corresponding
expected
cost
of
the
order
subsystem
S3
is
C ( S 3 )  C (4)  Q (4)C ( S 2 )  13.07 .
Finally, we consider the entire system S  ( S 3 , 5) . Set C (5)  c5  10 and P(5)  p5  0.9 . The
ratios of cost to present probability are 94.7 and 100.0 for ordered subsystem S 3 and attribute 5,
respectively. Therefore, S is already ordered. However, to distinguish between the initial version of
system S and the ordered one, we denote the latter by S . Moreover, the expected inspection cost
for the ordered system is given by C ( S )  C ( S 3 )  P( S 3 )C (5)  13.07  0.86(10)  21.67 . The
system absent probability is P( S )  P( S 3 ) P(5)  0.78 .
The optimum inspection sequence of a container with a Boolean decision function as represented by
system S is determined by Theorem 3 as follows:
 Inspect S 3 first. If the combined attribute of S 3 is present, then conclude that the container
is suspicious and needs additional manual inspection and inspection of attribute 5 is skipped.
Otherwise, inspect attribute 5. If attribute 5 is present, then we perform additional manual
inspection of the container. Otherwise, accept the container.
 To inspect S 3 , start by inspecting attribute 4. If it is absent, then conclude that the combined
attribute of S 3 is absent. Otherwise, inspect S 2 .
 To inspect S 2 , first inspect attribute 1. If it is present, then conclude that the attribute of
S 2 (and therefore S 3 and the entire S ) is present. Otherwise, inspect S 1 .
 To inspect S 1 , inspect attribute 3 first. If it is absent, then S 1 , S 2 , and S 3 are also absent.
Otherwise, inspect attribute 2. If it is present, then the combined attributes of S 1 , S 2 , and
S 3 are also present; otherwise, the combined attributes of S 1 , S 2 , and S 3 are absent.
Finally, the binary decision tree is determined as shown in Figure 2.
a4
0
1
a5
a1
0
0
1
“0”
1
a3
“1”
0
“1”
1
a2
a5
0
“0”
0
1
“1”
0
“0”
a5
1
1
“1”
“1”
Figure 2. Binary Decision Tree for the Example
3. Optimum Thresholds Problem for Port-of-Entry Inspection
3.1. Problem Description
The inspection of a container at the port-of-entry is performed sequentially at different inspection
stations. There are n inspection stations in the system; each station is used to identify one attribute
of the container being inspected. Let xi be the true state of the i th attribute, and Di be the decision
of i th station after inspection. For simplicity we assume that there are only two states, 1 and 0, for
presence or absence of the attribute respectively. The classification of each container is thought of
as a decision function F that assigns a class to each string of attributes. In this section, we focus on
the case where there are only two classes, 0 and 1, i.e. F ( x1, x2 ,
that the container is not a suspect and F ( x1, x2 ,
, xn )  0 means negative class and
, xn )  1 means positive class and that additional
manual inspection is required. The decision of classification can be made before all attribute values
are available as in Figure 3.
Figure 3. Binary Decision Tree
For each inspection station i , a measurement X i is taken by the sensor at the station and compared
against a given threshold value Ti which has a certain range of possible discrete values. The
decision Di of station i is made based on the following criterion:
1,
Di  
0,
if X i  Ti
if X i  Ti
.
There are potential errors in making a decision at each station. There is a conditional probability
that given an attribute with xi  1 the decision Di will be 0. This is known as a false negative, and
denoted by i 0  P  D  0 | xi  1 . The opposite case is making a decision Di  1 given the true
state xi  0 , and the conditional probability of this false positive or false alarm is denoted by
i1  P  D  1| xi  0 .
A normal distribution for attribute i with true state of 1 or 0 might be assumed or estimated from
previous inspections as X i1
Norm(1,i , 1,i 2 ) or X i 0
Norm(0,i ,  0,i 2 ) .
In this case, the
conditional probabilities of error for a decision from station i can be written as follows:
 Ti  0,i 

  0,i 
 i1  P  Di  1| xi  0   P  X i  Ti | X i  X i 0   P  X i 0  Ti   1   
and
 Ti  1,i 
 .
  1,i 
 i 0  P  Di  0 | xi  1  P  X i  Ti | X i  X i1   P  X i1  Ti    
The final decision in the classification of a container is based on all or partial decisions at inspection
stations and the Boolean function F ( x1, x2 ,
, xn ) , i.e. D  F ( D1, D2 ,
, Dn ) , which means that we
may not need all the values of Di ’s to make the final decision of classification. The conditional
probability of a false positive in the final classification of a container is  1  P( D  1| F  0) and
the probability of false negative in the final classification is  0  P( D  0 | F  1) .
From inspection history, we have information about the probability of a container’s true
classification being 1 or 0: P( F  1) and P( F  0). Let cFP be the unit cost associated with a
false positive (the cost of additional inspection), and cFN be the unit cost associated with a false
negative.
The expected total cost associated with false positive and false negative is
CF  cFP 1P  F  0  cFN 0 P  F  1 .
The objective of the problem is to minimize CF with the decision variables being the sensor
threshold levels Ti at inspection stations, i  1, 2,
, n . This is achieved by expressing the false
negative and false positive probabilities in terms of the threshold levels. The calculation of  1 and
 0 depends on the Boolean function F ( x1, x2 , , xn ) . We assume that the inspections and decisions
made based on the sensor measurements at different stations are independent.
3.2. System Probabilities of Error
In this section, conditional probabilities of false positive and false negative for the overall
inspection system are estimated for series and parallel Boolean functions.
3.2.1. Series Boolean Function
A simple series system with n inspection stations, where any attribute with a state xi  1 causes the
overall
classification
to
F ( x1 , x2 ,..., xn )  1  (1  x1 )(1  x2 )
F 1
be
,
has
the
Boolean
function
1  xn  .
The conditional probability of false positive  1 is found by:
 1  P  D  1| F  0 
 P  ( Di  1) | ( xi  0)   P  ( Di  1|
   i1    i1 j1  ...   1
i j
r 1

i1 i2 ...ir
xi  0) 
 i 1 i 1... i 1  ...   1  i 1 i 1... i 1
n 1
1
2
r
1
2
n
For a system with 3 attributes,  1  11   21   31  11 21  11 31   21 31  11 21 31 .
The conditional probability of false negative is found by considering possible combinations of
attributes. A value for  0  P  D  0 | F  1  P  ( Di  0) | ( xi  1)  is estimated by conditioning
on these ( x1 , x2 ,
, xn ) arrangements and considering how likely each one is to occur. For a certain
sequence with r attributes of type 1,  0 is found by multiplying  i 0 for these r attributes and 1   i1
for all remaining n-r attributes. Therefore if the attribute sequence is reordered with the first r
attributes having type 1, the conditional probability of false negative for that sequence can be
 0( r ) 1   1( r 1) 
written as:  0 (k )   0 (1) 0(2)
1    where k is an indicator for a possible
1
( n)
attribute sequence. The k indicator associates an  0 (k ) with a k , which is an estimate of the
likelihood of a specific arrangement of attribute values. Then the overall conditional probability of
false negative in the system is
 
k
0
(k ) .
Consider n  3 , assume equal prior probabilities for the seven possible sequences with F  1 , we
obtain  0 (k ) for all possible sequences as shown in Table 1.
Table 1.  0 (k ) for All Possible Sequences
 0 (k )
# possible sequences
One “1” attribute
3
 0 (1) 1   1(2) 1   1(3) 
Two “1” attributes
3
 0(1) 0(2) 1   1(3) 
All “1” attributes
1
 0(1) 0(2) 0(3)
Combining
   k
0
the
0
conditional

k 
probability




of


false

negative


in
1
1
0
1
1
0
1
1
 0
1 1 1   2 1   3   2 1  1 1   3   3 1  1 1   2
 
7 10 20 1   31  10 30 1   21   20 30 1  11  10 20 30





the
system
is

.

3.2.2. Parallel Boolean Function
A simple parallel system with n inspection stations, where all attributes must have the state xi  1
for the overall classification to be F  1 , has the Boolean function F ( x1 , x2 ,..., xn )  x1 x2
xn .
The conditional probability of false positive  1  P  D  1| F  0  P  Di  1| xi  0 is
conditioned on possible attribute sequences and their relative likelihood. For a certain sequence
with r attributes of type 0,  1 is found by multiplying  i1 for these r attributes and 1   i 0 for all
remaining n-r attributes.
Therefore if the attribute sequence is reordered so that the first r attributes have type 0, the
conditional
probability
 1 (k )   1(1) 1(2)
of
false
 1( r ) 1   0( r 1) 
positive
for
1    where
0
(n)
that
sequence
can
be
written
as:
k is an indicator for a possible attribute
sequence. The k indicator associates an  1 (k ) with a k , which is an estimate of the prior percent
observed of that specific arrangement of attribute values. Then the overall conditional probability
of false positive in the system is
   (k ) .
1
k
Again consider the case when n  3 , assume equal prior probabilities for the seven possible
sequences with F  0 , then  1 (k ) is estimated using the values shown in Table 2.
Table 2.  1 (k ) for All Possible Combinations
# possible sequences
 1 (k )
One “0” attribute
3
 1(1) 1   0 (2) 1   0(3) 
Two “0” attributes
3
 1(1) 1(2) 1   0(3) 
All “0” attributes
1
 1(1) 1(2) 1(3)
Combining the above lines gives the conditional probability of false positive in the system,
   k
1
1

k 









0
0
1
0
0
1
0
0
 1
1 1 1   2 1   3   2 1  1 1   3   3 1  1 1   2
 
7 11 21 1   30  11 31 1   20   21 31 1  10  11 21 31






.

The conditional probability of false negative in this system is found by:
 0  P  D  0 | F  1
 P  ( Di  0) | ( xi  1)   P  ( Di  0 |
xi  1) 
   i 0    i 0 j 0  ...   1
2
i j
r 1

i1 i2 ...ir
 i 0 i 0 ... i 0  ...   1
1
r
n 1
 i 0 i 0 ... i 0
1
2
n
For n  3 ,  0  10   20   30  10 20  10 30   20 30  10 20 30 .
3.3. Numerical Example
This section provides an example of the procedure used to determine the optimum threshold values
for a system with three inspection stations whose overall classification is based on a series Boolean
function. The optimum thresholds Ti , i  1, 2, 3 are the solutions to the minimization problem with
an objective function of the total expected cost associated with the possibility of false positive and
false negative in the final decision, CF  cFP 1P  F  0  cFN 0 P  F  1 .
Assuming equal probabilities for the seven possible sequences with F=1, the total expected cost of
errors is:


1   1      1   1   

1       1        
CF  cFP P  F  0  11   21   31  11 21  11 31   21 31  11 21 31



10 1   21 1   31   20
cFN


P  F  1 
0
0
1
0
0
7
1  2 1   3  1  3


1
1
1
2
1
3
0
2
0
1
1
3
0
1
1
3
0
1
1
2
0
2
0
3
which is a function of the threshold levels and system parameters. To carry out the optimization a
search loop is implemented which tests all combinations of possible threshold level values for each
station, calculating overall α0 and α1 values for each combination.
Given
the
following
system
information:
CFP  $200 , P  F  0  0.99 ,
CFN  $200,000 , P  F  1  0.01 and the distribution parameters for attribute ‘i’ which is
measured by station ‘i’ are shown in Table 3.
The optimum threshold levels determined for each station are T1  1.76 , T2  3.33 , and T3  3.93 .
These threshold values determine the system’s probabilities of false negative and false positive
which are  0  0.0117 and  1  0.0753 . These values return a minimum expected cost associated
with false negative and false positive of CF= $38.35 .
Table 3. Distribution Parameters for Attribute ‘i’
i
1
2
3
m0
s0
m1
s1
0
0
0
1.0
1.5
2.0
5
8
10
2.0
2.0
3.0
4. Conclusions
In this paper we investigate an approach for determining the optimum threshold levels of sensors for
containers subject to inspection at the port-of-entry. Methods for estimating the probabilities of
false negative and false positive we developed for two commonly used inspection systems: series
and parallel. The approach is general and it requires prior knowledge of “misclassifications” as a
function of the sensors’ threshold levels. The optimization approach searches within discrete values
of the threshold levels to determine the levels that minimize the overall cost of inspection.
References
1. Azaiez, N., Bier, V. M., (2004), Optimal resource allocation for security in reliability
systems, CREATE Report.
2. Ben-Dov, Y., (1981), Optimal testing procedures for special structures of coherent systems,
Management Science, 27(12):1410-1420.
3. Butterworth, R. W., (1972), Some reliability fault-testing models, Operations Research, 20,
335-343.
4. Cox L., Qiu, Y., Kuehner, W., (1989), Heuristic least-cost computation of discrete
classification functions with uncertain argument values, Annals of Operations Research, 21
1-30.
5. Cox, L., Chiu, S., Sun, X., (1996), Least-cost failure diagnosis in uncertain reliability
systems, Reliability Engineering and System Safety, 54, 203-216.
6. Elsayed, E. A., (1997), Reliability Engineering, Addison Wesley.
7. Elsayed, E. A., (2003), Distributed sensing for quality and productivity improvement,
(Panelist) INFORMS, Atlanta, GA.
8. Halpern, J., (1974), Fault-testing of a k-out-n system, Operations Research, 22, 1267-1271.
9. Halpern, J., (1974), A sequential testing procedure for a system's state identification, IEEE
Trans. Reliab., Vol R-23, No 4, 267-272.
10. Halpern, J., (1977), The sequential covering problem under uncertainty, INFOR, Vol. 15,
76-93.
11. James, K. A., de Sulima-Przyborowski, J. C. and Haydn, N. T. A., (2002), Non-invasive
means of investigating container contents for customs agents at port, University Southern
California.
12. Liao, H., Elsayed, E. A. and Chan, L-Y, (2005), Maintenance of continuously monitored
degrading systems, (in press) , European Journal of Operations Research.
13. Loy, J. M. and Ross, R. G., (2002), Global Trade: America’s Achilles’ Heel, Defense
Horizon, Center for Technology and National Security University