Optimization and Engineering, 6, 339–359, 2005 c 2005 Springer Science + Business Media, Inc. Manufactured in The Netherlands
Integer Programming Models for Deployment of Airport Baggage Screening Security Devices SHELDON H. JACOBSON∗ LAURA A. MCLAY Simulation and Optimization Laboratory, Department of Mechanical and Industrial Engineering, University of Illinois, Urbana, Illinois 61801-2906 email:
[email protected], netfiles.uiuc.edu/shj/www/shj.html email:
[email protected] JULIE L. VIRTA 4406 W. Quill Lane, Waukegan, IL 60085 email:
[email protected] JOHN E. KOBZA Department of Industrial Engineering, Texas Tech University, Lubbock, Texas 79409-3061 email:
[email protected] Received May 13, 2002; Revised June 18, 2003
Abstract. Aviation security is an important problem of national interest and concern. Baggage screening security devices and operations at airports throughout the United States provide an important defense against terrorist actions targeted at commercial aircraft. Determining where to deploy such devices, and how to best use them can be quite challenging. This paper presents NP-complete decision problems concerning the deployment and utilization of baggage screening security devices. These problems incorporate three different deployment performance measures: uncovered baggage segments, uncovered flight segments, and uncovered passenger segments. Integer programming models are formulated to address optimization versions of these problems and to identify optimal baggage screening security device deployments (i.e., determine the number and type of baggage screening security devices that should be placed at different airports, and determining which baggage should be screened with such devices). The models are illustrated with an example that incorporates data extracted from the Official Airline Guide (OAG). Keywords: analysis
1.
aviation security, homeland security, NP-completeness, integer programming models, cost
Introduction
Terrorist activities, such as the explosion of a bomb on Pan Am Flight 103 in 1988, and more recently, the events of September 11th, 2001 that resulted in the destruction of the World Trade Center in New York City and significant damage to the Pentagon, have made aviation security a major concern of the general public. In 1996, the Commission on Aviation Safety and Security, headed by (then) Vice-President Albert Gore, outlined several recommendations to improve aviation security through an expansion of existing explosive detection ∗ Corresponding
author.
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technologies, automated passenger profiling, and positive passenger baggage matching. Unfortunately, concerns over the cost of security procedures resulted in the airline industry being hesitant to implement many of these recommendations. To objectively evaluate some of these concerns, Barnett et al. (2001) conducted a detailed study that showed that requiring positive passenger baggage matching on domestic flights would result in average delays of one minute per flight, far less than estimated by the airlines. On November 19, 2001, the Aviation and Transportation Security Act (ATSA) was signed into legislation, which required additional aviation security procedures to be implemented. One such procedure was the screening of all checked baggage for explosives (using federally certified screening devices and procedures) by the end of 2002. To meet this deadline, it was necessary to purchase and install security devices at all commercial airports throughout the United States, resulting in the deployment of more than 6,000 baggage screening security devices (Mead, 2003b). This deadline was later extended to the end of 2003, and currently, nearly all checked baggage is screened for explosives with such devices at every commercial airport in the United States (Mead, 2003b). Moreover, the ATSA mandates that all screening be performed by federal employees and an increased deployment of federal air marshals to monitor high security risk flights (Mead, 2002, 2003a). The primary objective of all these efforts is to improve security at airports throughout the nation. The ATSA established the Transportation Security Administration (TSA), now a part of the Department of Homeland Security (DHS), to provide a government unit devoted to the security of the entire transportation system within the United States. The TSA is committed to developing new security system paradigms that can optimally use and simultaneously coordinate next generation security technologies and procedures. Given the complexity of the airspace system and the many threats that can impact its safe operation, it is a challenge to determine the type and number of security devices to deploy and to determine how to use them to have the greatest impact on security. The objective of this paper is to demonstrate how integer programming models can be used to obtain optimal deployment of baggage screening security devices for a set of flights traveling between a given set of airports. The paper is organized as follows. Section 2 reviews performance measures for assessing the effectiveness of different airport baggage screening security device deployments. This section also contains definitions and notation needed to describe the integer programming models. Section 3 formulates NP-complete decision problems based on these performance measures. Section 4 describes the integer programming models for the problems described in Section 3. Section 5 outlines a Greedy algorithm for addressing these problems. Section 6 provides an illustrative example for the integer programming models for a particular set of airports and flights, based on data extracted from the Official Airline Guide (OAG). Section 7 presents concluding comments and directions for future research. 2.
Definitions and notation
Determining how to optimally deploy different types of baggage screening security devices throughout airports in the United States is of critical importance to national security. Limited available baggage screening security device capacities and time restrictions initially lead
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to the implementation of procedures that required the screening of only a fraction of all passenger checked baggage (Mead, 2003b). However, congressional mandates now require that all checked bags must be screened by federally approved baggage screening security devices (Mead, 2003a). Currently, explosive detection systems (EDS) are the preferred method for checked baggage screening (Mead, 2003a). However, in airports with smaller volumes, the cost of such devices makes it difficult to justify their deployment. Therefore, less expensive explosive trace detection (ETD) machines are often the only devices that are used in such airports. In the future, new, more effective (and potentially more expensive) baggage screening technologies will be developed and reach the market. How and where this new technology should be deployed is a problem of critical importance. Until sufficient baggage screening capacity is deployed at airports throughout the nation, only a targeted fraction of all checked baggage will be screened with this new technology. One method of identifying passengers warranting this increased scrutiny is to use a computer-aided passenger prescreening system (CAPPS) (Mead, 2003b). The TSA uses CAPPS to provide a binary risk assessment of all passengers. Passengers are termed selectees if CAPPS is unable to clear them from being a potential risk to the system. The selectee rate is the fraction of passengers (between zero and one) on a given flight who are selectees (i.e., number of selectees on a given flight divided by the total number of passengers on the flight). Selectee baggage (i.e., checked baggage belonging to selectees) is classified as either screened or unscreened, based on whether or not it has been processed through a next-generation airport baggage screening security device. It is assumed that all checked baggage is subject to screening by current-generation EDS and ETD deployments. The issue is how to best deploy a limited number of next generation security devices to have the biggest impact on system security. Selectee baggage considered at the point at which it first enters the system (i.e., origin) is termed originating baggage. Any non-originating baggage at an airport enroute to another airport is termed baggage in transit. A baggage screening security device deployment for a set of airports is an allocation of baggage screening security devices to these airports and an assignment of selectee baggage that should be screened for the set of flight segments between these airports, where a flight segment (or flight) is a takeoff and landing of an aircraft from one airport to another. From this definition, the takeoff and landing of an aircraft from Airport A to Airport B with no intermediate stops counts as a single flight. A selectee is said to be on a direct route if their flight path is composed of one flight segment, or on a connecting route if their flight path is composed of two or more flight segments. A flight is covered if all selectee baggage on it has been screened. Lastly, the number of covered passengers on a covered flight is the number of passengers on the flight. Note that unless otherwise stated, all baggage screening occurs at origin (i.e., no baggage is screened in transit along a connecting route). Three security performance measures have been identified (Jacobson et al., 2003) for a given set of flights carrying both selectee and non-selectee passengers and baggage. (1) Uncovered Baggage Segments (UBS)—Number of unscreened selectee bags on uncovered flights. (2) Uncovered Flight Segments (UFS)—Number of flights carrying one or more unscreened selectee bags.
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(3) Uncovered Passenger Segments (UPS)—Number of passengers on uncovered flights. This paper formulates a set of integer programming models to determine the optimal baggage screening security device deployment for a set of airports, where optimality is measured using one of these three security performance measures. Note that the focus of this paper is to demonstrate the application of integer programming models rather than the development of new theory (i.e., solving an applied problem rather than improving solution time through algorithms). Therefore, all the models are solved using CPLEX, a commercial software package. The problem of determining an optimal deployment of baggage screening security devices is similar to the location analysis problem, in that facilities (devices) are assigned to predetermined locations in the network (airports) in order to cover flow (originating selectee baggage). However, these problems differ in some key aspects. The security deployment problem considers multiple baggage screening security device types with differing capacities and costs. The flow capturing models used by Berman et al. (1992, 1995) and Hodgson (1990) consider the placement of identical facilities in the network in order to cover customers’ paths (flow) before the customers leave the network. In security deployment problems, there is an enormous distinction between screening a bag at its origin rather than screening it at its destination, and hence, when a bag is screened is of critical importance. This can be contrasted with the flow capturing models, in which a customer is covered if there is a facility along or near the customer’s path. Moreover, the flow covering models do not consider a budget constraint since the number of facilities to be located in the network is a parameter and the facilities are identical. The objective of the flow capturing models is either to maximize the number of potential customers (Berman et al., 1992, 1995) or to minimize the expected inconvenience, the distance that a customer is expected to deviate from their path (Berman et al., 1995). These objectives are not comparable to the UFS and UPS measures since the passengers in the security deployment problem are partitioned into flights, and both UFS and UPS objectives seek to cover flights, which is achieved indirectly by screening (covering) originating selectee baggage. To describe the integer programming models, several additional definitions and assumptions are needed. First, airport activity is typically stochastic, varying both across days as well as within a given day. Therefore, the models consider a worst case scenario based on activity during an airport’s peak period, defined as the sixty minute period during the day in which the largest number of originating passengers enters the airport. This assumption ensures that the number of baggage screening devices deployed is sufficient to optimally address checked baggage screening requirements during any time period during a day. Since the “hub and spoke” system used by many airlines results in carefully sequenced and timed flights into and from the hub airport, assume that during an airport’s peak period, there is at most one flight scheduled to each of the other airports in the system of airports under study. Note that this assumption is reasonable only if a single airline is being considered. If such an assumption is relaxed to allow for multiple flights to a particular airport, then this can be modeled by treating this airport as multiple airports, one for each of the flights scheduled
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to fly into it. Also, assume that all peak periods coincide so that any selectee bags on a flight into a hub airport, regardless of arrival time, can transfer on any flight departing from the hub airport during the hub’s peak period (i.e., once the peak hours are determined, the actual arrival and departure time of the flights are not considered). Though this assumption is unlikely to hold in practice, it once again allows for the system of airports to be studied under a worst case scenario. Therefore, all averages discussed in the paper correspond to averages during each airport’s peak period. Let N denote the total number of airports in a system, where An , n = 1, 2, . . . , N denotes the nth airport. The average number of selectee bags on flights from Airport Ai to Airport A j is denoted by f i, j , i, j = 1, 2, . . . , N , i = j. These values can be obtained using four parameters: the average number of bags per passenger (b), the capacity of flights from Airport Ai to Airport A j (ci, j ), the selectee rate from Airport Ai to Airport A j (ρi, j ), and the enplanement rate from Airport Ai to Airport A j (σi, j ). Airline reservation systems have the capacity to collect the information needed to provide values for these parameters. However, given that schedules are constantly changing and being updated, this information will also change accordingly. Therefore, updated information may need to be regularly considered to provide the most current parameter values. Moreover, since peak period averages and information are being used, the baggage screening security device deployments that arise from the integer programming models should be reasonable for a variety of schedules, provided that the peak period flight schedules are similar. Each passenger is assumed to have no more than two connecting flights. This assumption is reasonable, since the “hub and spoke” system used by most major commercial airlines in the United States facilitates such routing situations. The flow of selectee baggage can be further described to capture connections. This information is needed to account for selectee baggage in transit. For a given set of flights out of Airport Ai , let gi, j denote the average number of selectee bags on direct flights from Airport Ai to Airport A j , and let m i, j,k denote the average number of selectee bags originating at Airport Ai with final destination Airport Ak , connecting through Airport A j . Therefore,
f i, j = (b)(ci, j )(ρi, j )(σi, j ) = gi, j +
N k=1 k=i k= j
m i, j,k +
N
m h,i, j
i, j = 1, 2, . . . , N , i = j.
h=1 h=i h= j
Note that the values for gi, j and m i, j,k can also be computed using information extracted from an airlines reservation system. However, the data collection requirement for such detailed, passenger-by-passenger information, may be prohibitively expensive to secure, given the volume of traffic that must be accounted for. Lastly, let R denote the number of baggage screening security device types available, each with a capacity (λr ) per sixty minute period, a purchase cost ( pr ), an average annual maintenance and operation cost (including airport rental space fees for the device) (or ), and a number of available devices of type r (sr ), r = 1, 2, . . . , R. Lastly, let B denote the budget available to cover the purchase of the baggage screening security devices and the first year of operating cost, for a given set of airports.
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Decision problem formulations
This section defines decision problems that model the baggage screening security device deployment problem, using the three security performance measures described in Section 2. The difficulties associated with finding an optimal baggage screening security device deployment that satisfies physical and operational constraints (such as not exceeding budget levels) are also discussed. The decision problems formulated are all NP-complete (Garey and Johnson, 1979). The Multiple Airport Baggage Problem (MABP) determines whether a feasible allocation of baggage screening security devices exists such that all selectee bags can be screened at their origin. Therefore, if the answer to MABP is yes, then all selectee bags can be screened, hence all flights and passengers can be covered. The MABP is formally stated. Multiple Airport Baggage Problem (MABP) Instance: – A set of N airports A = {A1 , A2 , . . . , A N }, – a set of M baggage screening devices, D = {d1 , d2 , . . . , d M }, – a cost for each element of D, C(d) ∈ Z + , d ∈ D, – a baggage screening capacity for each element of D, λ(d) ∈ Z + , d ∈ D, – a baggage screening security device allocation budget B ∈ Z + , – the number of originating selectee bags to be screened at Airport Ai , qi ∈ Z + , i = 1,2, . . . ,N . Question: Are there N subsets of baggage screening security devices Di ⊆ D, i = N 1, 2, . . . , N , where Di ∩ D j = ∅, i, j = 1, 2, . . . , N , i = j, such that i=1 d∈Di C(d) ≤ B and d∈Di λ(d) ≥ qi , i = 1, 2, . . . , N ? Theorem 1.
The MABP is NP-complete.
Proof: See Appendix 1. N The inequality i=1 d∈Di C(d) ≤ B in the statement of the MABP (and for all following decision problems in this section) corresponds to the budget constraint, while the constraint λ(d) ≥ qi , i = 1, 2, . . . , N , determines if all selectee bags can be screened at d∈Di origination. When the answer to the MABP is no, then it is not possible to screen (cover) all the baggage segments, flight segments, and passenger segments. This leads to a new set of problems, namely, determining the maximum number of baggage segments (MADBP), flight segments (MADFP), or passenger segments (MADPP) that can be covered; each of these problems is NP-complete using transformations from MABP or the knapsack problem (Martello and Toth, 1990). The Multiple Airport Direct and Connecting Baggage Problem (MADCBP) is a decision problem for determining whether a feasible allocation of baggage screening security devices exists such that a minimum number of baggage segments are screened.
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The Multiple Airport Direct and Connecting Baggage Problem (MADCBP) Instance: – A set of N airports A = {A1 , A2 , . . . , A N }, – a set of M baggage screening devices, D = {d1 , d2 , . . . , d M }, – a cost for each element of D, C(d) ∈ Z + , d ∈ D, – a baggage screening capacity for each element of D, λ(d) ∈ Z + , d ∈ D, – a baggage screening security device allocation budget B ∈ Z + , – the number of originating selectee bags on direct flights from each Airport Ai available to be screened, qi1 ∈ Z + , i = 1, 2, . . . , N , – the number of originating selectee bags on connecting flights from each Airport Ai available to be screened, qi2 ∈ Z + , i = 1, 2, . . . , N , – the minimum number of selectee bags to be screened, α ∈ Z + . Question: Is there a set of N subsets of baggage screening security devices Di ⊆ D, i = 1, 2, . . . , N , where Di ∩ D j = ∅, for all i, j = 1, 2, . . . , N , i = j, and a number N ˆ i1 + qˆ i2 ≤ of bags to screen qˆ i1 ≤ qi1 and qˆ i2 ≤ qi2 such that i=1 d∈Di C(d) ≤ B, q N ˆ i1 + 2qˆ i2 ) ≥ α? d∈Di λ(d), i = 1, 2, . . . , N , and i=1 (q Theorem 2.
The MADCBP is NP-complete.
Proof: See Appendix 2. The term qˆ i1 + qˆ i2 ≤ d∈Di λ(d) captures whether the capacity of the baggage screening security deployed exceeds the number of selectee bag segments screened, while the Ndevices term i=1 (qˆ i1 + 2qˆ i2 ) ≥ α captures whether the minimum number of selectee bags can be screened. Note that each selectee bag on a connecting flight, if screened at origination, will be covered for two flight segments, thus impacting the number of covered baggage segments twice (hence the coefficient two in this expression). The Multiple Airport Direct and Connecting Flight Problem (MADCFP) is a decision problem for determining whether a feasible allocation of baggage screening security devices exists such that a minimum number of flight segments, χ , can be screened. Therefore, the maximum value of χ that satisfies the conditions of the MADFP will minimize UFS. More specifically, the MADCFP determines if the baggage screening security device deployment is within budget, the number of flights screened is greater than the minimum required, and the capacity of the baggage screening security devices located at each airport is greater than the number of selectee bags on the flights screened. The Multiple Airport Direct and Connecting Flight Problem (MADCFP) Instance: – A set of N airports A = {A1 , A2 , . . . , A N }, – a set of M baggage screening devices, D = {d1 , d2 , . . . , d M }, – a cost for each element of D, C(d) ∈ Z + , d ∈ D, – a baggage screening capacity for each element of D, λ(d) ∈ Z + , d ∈ D, – a set of flights departing Airport Ai , Fi , i = 1, 2, . . . , N , – the number of originating direct selectee bags on flight f ∈ Fi , i = 1, 2, . . . , N , Q d ( f ),
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– the number of originating connecting selectee bags on flight f ∈ Fi , i = 1, 2, . . . , N , Q o ( f ), – a baggage screening security device allocation budget B ∈ Z + , – the minimum number of flight segments to be screened, χ ∈ Z + . Question: Is there a set of N subsets of baggage screening security devices Di ⊆ D, i = 1, 2, . . . , N , where Di ∩ D j = ∅, for all i, j = 1, 2, . . . , N , i = j, and a subset of flights that are covered Fi ⊆ Fi at each Airport Ai , i = 1, 2, . . . , N (i.e., all selectee baggage N on each flight f ∈ Fi , i = 1, 2, . . . , N , are screened) such that i=1 d∈D C(d) ≤ N i d o B, f ∈Fi Q ( f ) + Q ( f ) ≤ d∈Di λ(d), for each i = 1, 2, . . . , N , and i=1 |Fi | ≥ χ ? Theorem 3.
The MADCFP is NP-complete.
Proof: Since MADCFP includes MADFP as a special case, then MADCFP is NPcomplete. In particular, any arbitrary instance of MADFP is a particular instance of MADCFP with Q c ( f ) = 0, f ∈ Fi , i = 1, 2, . . . , N (i.e., no baggage in transit hence no connecting routes). The Multiple Airport Direct Passenger Problem (MADCPP) is a decision problem for determining whether a feasible allocation of baggage screening security devices exists such that a minimum number of passenger segments can be covered. Therefore, the maximum value of δ which satisfies the conditions of the MADPP will minimize UPS. The Multiple Airport Direct and Connecting Passenger Problem (MADCPP) Instance: – A set of N airports A = {A1 , A2 , . . . , A N }, – a set of M baggage screening devices, D = {d1 , d2 , . . . , d M }, – a cost for each element of D, C(d) ∈ Z + , d ∈ D, – a baggage screening capacity for each element of D, λ(d) ∈ Z + , d ∈ D, – a set of flights departing Airport Ai , Fi , i = 1, 2, . . . , N , – the number of originating direct selectee bags on flight f ∈ Fi , i = 1, 2, . . . , N , Q d ( f ), – the number of originating connecting selectee bags on flight f ∈ Fi , i = 1, 2, . . . , N , Q o ( f ), – the total number of passengers on flight f ∈ Fi , i =, 1, 2, . . . , N , P( f ) ∈ Z +, – a baggage screening security device allocation budget B ∈ Z + , – the minimum number of passenger segments to be screened, δ ∈ Z + . Question: Is there a set of N subsets of baggage screening security devices Di ⊆ D, i = 1, 2, . . . , N , where Di ∩ D j = ∅, for all i, j = 1, 2, . . . , N , i = j, and a subset of flights that N are covered Fi ⊆ Fi at each Airport Ai , i = 1, 2, . . . , N , such that i=1 d∈D C(d) ≤ B, N i d o f ∈Fi Q ( f ) + Q ( f ) ≤ d∈Di λ(d), for each i = 1, 2, . . . , N , and f ∈Fi P( f ) ≥ i=1 δ? Theorem 4. The MADCPP is NP-complete.
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Proof: Since MADCPP includes MADPP as a special case, then MADCPP is NPcomplete. In particular, any arbitrary instance of MADPP is a particular instance of MADCPP with Q c ( f ) = 0, f ∈ Fi , i = 1, 2, . . . , N (i.e., no baggage in transit hence no connecting routes). Note that MADCBP, MADCFP, and MADCPP all remain NP-complete even if the flights are restricted to only direct flights. 4.
Integer programming models
This section describes the integer programming models for obtaining optimal baggage screening security device deployments. The resulting models are designed to optimally determine where to assign the different baggage screening security devices and which selectee baggage to screen, where optimality is based on minimizing either the number of uncovered baggage segments, the number of uncovered flight segments, or the number of uncovered passenger segments, (hence resulting in three different integer programming models). In these three models, all decision variables are either non-negative integers or binary. The following is a list of decision variables for these models. X i,r
Ui, j Ti, j,k
Vi, j,k
Fi, j
= the number of units of baggage screening security device type r = 1, 2, . . . , R, deployed at Airport Ai , i = 1, 2, . . . , N , where R denotes the number of different types of baggage screening security devices, = the number of uncovered originating direct selectee bags leaving Airport Ai for Airport A j , i, j = 1, 2, . . . , N , i = j, = the number of uncovered originating connecting selectee bags leaving Airport Ai enroute to connecting Airport A j with final destination Airport Ak , i, j, k = 1, 2, . . . , N , i = j = k, = the number of uncovered non-originating connecting selectee bags originating at Airport Ai , leaving Airport A j with final destination Airport Ak , i, j, k = 1, 2, . . . , N , i = j = k, = 1(0) if the flight from Airport Ai to Airport A j is (un)covered, i, j = 1, 2, . . . , N , i = j.
The objective functions, corresponding to the three security performance measures, are defined. The objective function for minimizing the number of uncovered bag segments is Minimize
N N i=1 j=1 j=i
Ui, j +
N N N i=1
j=1 k=1 i= j=k
Ti, j,k +
N N N i=1
Vi, j,k.
j=1 k=1 i= j=k
The objective function for minimizing the number of uncovered flight segments is Minimize
N N i=1 j=1 j=i
(1 − Fi, j ).
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The objective function for minimizing the number of uncovered passenger segments is Minimize
N N
(ci, j • σi, j )(1 − Fi, j ).
i=1 j=1 j=i
Since the number of uncovered flight segments and the number of uncovered passenger segments are affected only by covered flights, the formulations presented here do not consider partial screening. That is, if there is not enough capacity available to cover a flight, then a baggage screening security device is not used to screen additional bags. Therefore, it is unlikely that all the available baggage screening capacity will be used. The constraints for minimizing the number of uncovered flight segments and uncovered passenger segments include (note that the terms in parenthesis in (ii) represent a relaxation to allow screening in transit, if permitted): (i) The purchase cost plus the first year annual operating and maintenance budget of the baggage screening devices must not exceed the given budget N R
( pr + or )X i,r ≤ B.
i=1 r =1
(ii) The baggage screening security device capacity plus the number of uncovered originating direct and connecting selectee bags leaving Airport Ai (plus the number of uncovered non-originating selectee bags leaving Airport Ai ) must equal or exceed the average number of originating selectee bags leaving Airport Ai (plus any uncovered connecting selectee bags)
R
λr X i,r +
r =1
≥
N
Ui, j +
j=1 j=i
N j=1 j=i
gi, j +
N N j=1 k=1 j=i k=i k= j
N N j=1 k=1 j=i k=i k= j
m i, j,k
Ti, j,k
N N + Vh,i, j h=1 j=1 h=i
j=i j=h
N N + Th,i, j , h=1 j=1 h=i
i = 1, 2, . . . , N .
j=i j=h
(iii) The number of uncovered originating selectee bags on a route cannot exceed the average number of originating selectee bags for that route. Ui, j ≤ gi, j i, j = 1, 2, . . . , N , i = j, Ti, j,k ≤ m i, j,k i, j, k = 1, 2, . . . , N , i = j = k.
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(iv) When baggage screening in transit is not permitted (as in most cases), then all unscreened non-originating connecting selectee bags leaving Airport A j remain unscreened, and Vi, j,k is equal to Ti, j,k, hence (a) is used. If baggage screening in transit is permitted, then Vi, j,k is not fixed, but rather bounded by the number of uncovered connecting selectee bags on the route, hence (b) is used. (a) Vi, j,k − Ti, j,k = 0 i, j, k = 1, 2, . . . , N , i = j = k, (b) Vi, j,k − Ti, j,k ≤ 0 i, j, k = 1, 2, . . . , N , i = j= k. (v) A flight segment is uncovered if any unscreened baggage exists on that flight segment from either unscreened baggage in transit or unscreened originating baggage, hence, Fi, j is forced to zero by the minimizing function if any selectee bags on a flight from Airport Ai to Airport A j are left uncovered, and forced to one otherwise. Note this constraint is only necessary when minimizing the uncovered flight segment or uncovered passenger segment performance measures. f i, j (1 − Fi, j ) − Ui, j −
N k=1 k=i k= j
Ti, j,k −
N
Vh,i, j ≥ 0 i, j = 1, 2, . . . , N , i = j.
h=1 h=i h= j
The size of the resulting integer programming models is polynomial in the number of airports (N ) and the number of baggage screening security device types (R). In the worst case, with flights between each airport and baggage in transit at each airport, there are N (2N 2 −5N + R +3) integer variables and N (N −1) binary variables. In addition, there are N (N − 1)2 simple bound constraints (i.e., variables constrained by an integer value). Lastly, the number of constraints (excluding simple bounds) is N 3 −3N 2 +3N +1 when minimizing the uncovered baggage segment performance measure, and N 3 − 2N 2 + 2N + 1 when minimizing the uncovered flight segment or uncovered passenger segment performance measures. 5.
Greedy algorithm heuristics
This section describes a Greedy algorithm heuristic for the problems presented in Section 3. The knapsack-like structures of these problems suggest that a Greedy algorithm is wellsuited to provide reasonable solutions for them. The Greedy algorithm heuristic is presented for the MADCBP, MADCFP, and MADCPP. One heuristic is presented for the three performance measures, with slight differences in its execution based on the performance measure being considered. The Greedy algorithm heuristic executes for all performance measures as follows. First, the airport with the largest number of unscreened originating selectee bags is selected. The baggage screening device type with the lowest screening capacity to screen the unscreened originating baggage is selected, if such a device type exists. In this case, all of the unscreened
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originating selectee baggage is screened. If such a device type does not exist, then the screening device type with the largest ratio of the screening capacity to cost is selected. A portion of the unscreened originating selectee bags is then screened according to a Greedy rule that varies according to the performance measure considered. For the UFS measure, the flights are sorted in nondecreasing order of the total number of unscreened selectee bags on each flight. For the UPS measure, the flights are sorted in nonincreasing order of the ratio of the number of passengers to the total number of unscreened selectee bags. The flights are unsorted for the UBS measure. Based on the order of the flights, the maximum number of unscreened selectee bags is then screened. The time complexity of the Greedy algorithm heuristic for the three performance measures is O(M(R + |F|2 )). The heuristic is now presented in pseudo-code form. Greedy Algorithm Heuristic: Set the remaining capacity B = B Set the array of devices used µ(d, a) = 0, a ∈ A, d ∈ D Set the airport screening capacities cap(i) = 0, i ∈ A While B − min {C(d): d ∈ D} ≥ 0 a = airport with the largest number of unscreened originating bags (ba ) If ba ≤ maxd {λ(d)}, then k = mind {λ(d) ≥ ba } Else k = maxd {λ(d)/C(d)} µ(k, a) = µ(k, a) + 1 B = B − C(k) cap(a) = cap(a) + λ(k) If UFS, sort f ∈ Fa in nondecreasing order of the total number of unscreened selectee bags If UPS, sort f ∈ Fa in nonincreasing order of the ratio of the number of passengers to unscreened selectee bag Greedily assign cap(a) to originating connecting bags in f ∈ Fa Greedily assign cap(a) to originating direct bags in f ∈ Fa Determine whether each f ∈ F is covered and adjust ba Determine the values of UBS, UFS, or UPS 6.
Illustrative example
This section illustrates the integer programming models (in Section 4) and the Greedy algorithm heuristic (in Section 5) using data extracted from the Official Airline Guide (OAG). Optimal baggage screening deployments are obtained based on the three security measures described in Section 2. The data extracted from the OAG is for a single airline carrier and the flights by that carrier between a set of ten airports in the United States. See Table 1 for a list of these airports. While the OAG provides some explicit data for the integer programming models (e.g., the number of peak bags per hour at an airport), other data must be created to support these models (e.g., the number of bags in transit). Table 2 lists the nineteen flights that exist between the ten airports under study, as well as the total number of available seats between the city pairs, the number of total (selectee
DEPLOYMENT OF AIRPORT BAGGAGE SCREENING SECURITY DEVICES
Table 1.
Airports under study. Airport
Table 2.
Airport code
Airport location
A1
ATL
Atlanta, GA
A2
CLE
Cleveland, OH
A3
CLT
Charlotte, NC
A4
DTW
Detroit, MI
A5
ERI
Erie, PA
A6
FAY
Fayetteville, NC
A7
GSO
Greensboro, NC
A8
ITH
Ithaca, NY
A9
ORF
Norfolk, VA
A10
PIT
Pittsburgh, PA
Flight information. Flight route Depart ATL
Arrive
Number of seats
Number of originating bags
Selectee rate
CLT
100
17
0.30
ATL
PIT
126
21
0.30
CLE
CLT
112
49
0.22
CLT
ATL
112
33
0.15
CLT
FAY
85
25
0.15
CLT
GSO
112
33
0.15
CLT
ORF
37
11
0.15
CLT
PIT
85
25
0.25
DTW
PIT
100
31
0.22
ERI
PIT
85
55
0.09
FAY
CLT
85
56
0.16
GSO
CLT
126
39
0.13
ITH
PIT
100
66
0.09
ORF
CLT
85
27
0.25
ORF
PIT
85
27
0.20
PIT
ATL
112
37
0.10
PIT
CLE
30
10
0.10
PIT
CLT
85
28
0.25
PIT
ERI
37
12
0.15
351
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and non-selectee) originating bags, and the selectee rate on each flight. Note that the true selectee rates (available from the TSA) are security sensitive information, and hence, cannot be reported here. Therefore, a broad range of selectee rates are used so that the models can be solved and the analysis procedure can be illustrated. Let the peak hour at each airport be the hour of the day in which the largest number of flights is recorded in the OAG (e.g., 12 PM). To approximate the number of bags on a flight, an enplanement rate is used to predict how many people board the flight (e.g., 80% of the seats are occupied). The number of bags is then determined by multiplying the number of passengers by the average number of bags per passenger (e.g., 1.5 bags per passenger). The number of selectee bags on each flight is the number of bags on the flight multiplied by the selectee rate of that flight, and rounded to the nearest integer. To determine the number of selectee bags in transit on a flight, first assume that all peak hours coincide so that any selectee bags on a flight into a hub airport can transfer on any flight departing from the hub airport (i.e., once the peak hours are determined disregard the actual arrival and departure time of the flights). In practice, a transferring selectee bag at a hub airport can originate from any airport and transfer to any another airport. Specifically, a bag can transfer from an airport under study to an airport not under study, or from an airport not under study to an airport under study. For this example, assume that a selectee bag in transit must have originated from an airport under study and must transfer to an airport under study. This effectively assumes that there is a degree of interchangeability between various non-hub airports. To approximate the number of transferring selectee bags on a flight, for each flight into the hub airport, distribute the selectee bags on the incoming flight among the flights departing the hub airport based on the number of seats on each departing flight, and round to the nearest integer. If, due to rounding, the sum of the transferring selectee bags is larger (smaller) than the total number of selectee bags from the originating airport to the hub airport, then round down (up) the value(s) previously rounded up (down) with a fractional component nearest .5, as needed. For example, 2.6 should be rounded down to 2 before 2.9, since it is closer to 2.5. Also, any value of transferring selectee bags greater than one should be rounded down before any value less than one is rounded down. For example, 2.7 should be rounded down to 2 before 0.6 is rounded down to 0. If the origin of the flight into the hub airport is the same as the destination of the departing flight out of the hub airport, then that portion of selectee bags are direct route bags. To illustrate this rounding rule, consider a flight from ERI to PIT (a hub airport) that contains five selectee bags. Four flights depart PIT during the peak hour to the following airports: ATL (112 seats), CLE (30 seats), CLT (85 seats), and ERI (37 seats). Distributing the five bags would yield: two (2.12 rounded down) selectee bags transfer to ATL, one (0.57 rounded up) selectee bag transfers to CLE, one (1.61, rounded down) selectee bag transfers to CLT, and one (0.70 rounded up) selectee bag exits the system. Lastly, the number of passengers on a flight is the enplanement rate times the number of seats, and rounded up to the next integer. Assume an enplanement rate of 80% for each flight. Assume there are four types of baggage screening security devices with associated capacity and total cost given in Table 3. Note that the actual rates and costs are security sensitive data and could not be reported here.
DEPLOYMENT OF AIRPORT BAGGAGE SCREENING SECURITY DEVICES Table 3.
353
Baggage screening security device information. Device type
Capacity bags/hour
Total cost
1
5
$550,000
2
10
$600,000
3
15
$750,000
4
25
$1,100,000
By inspection, the minimum budget required for 100% selectee coverage is $7 million. That is, for each airport, deploy those devices with sufficient screening capacity to screen all originating selectee bags at the lowest cost. As the budget decreases from $7 million to $0, deployment choices must be made to minimize UBS, UFS, or UPS. These results are illustrated in Figures 1–4 and were found by solving the integer programming models formulated in Section 4 using CPLEX 7.0. The approximate run-times for minimizing UBS, UFS, and UPS were 1800, 700, and 800 CPU seconds, respectively, using an Intel Pentium III XeonTM processor (approximately 550 MHz). Figure 1 depicts the optimal UBS solution, the UBS value for the optimal UFS solution, the UBS value for the optimal UPS solution, and the UBS value of the solutions obtained with the UBS Greedy algorithm heuristic, all as a function of budget. The ratios of the UBS value to the optimal UFS value, the UPS solution to the optimal UBS value, and the UBS Greedy heuristic value to the optimal UBS value are also given in Figure 2. From these figures, the optimal UPS and the optimal UFS solution rarely minimize the number of uncovered bag segments. However, the optimal UPS solution tends to result in smaller UBS values than the optimal UFS solution. This suggests a stronger relationship between UPS and UBS than UFS and UBS.
Figure 1.
Uncovered baggage segments.
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Figure 2.
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Ratios for uncovered baggage segments.
Figure 3 depicts the UFS value as a function of budget for the optimal UPS solution and the optimal UFS solution. The ratio of these two values is also given on the secondary (right-hand side) axis. For this example, the optimal UPS solution provides a good UFS value compared to the optimal UFS solution value. Figure 3 also depicts the UFS value of the Greedy algorithm heuristic as well as the ratio of this value to the optimal UFS solution as a function of the budget. Figure 4 depicts the UPS value as a function of the budget for the optimal UFS solution and the optimal UPS solution. The ratio of these two values is also given on the
Figure 3.
Uncovered flight segments and ratio.
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Figure 4.
355
Uncovered passenger segments and ratio.
secondary (right-hand side) axis. Figure 4 also depicts the UPS value of the Greedy algorithm heuristic and the ratio of this value to the optimal UPS value as a function of the budget. It is clear that the UFS and the UPS solutions are significantly different, since UFS favors the screening of smaller flights, resulting in fewer passengers covered, while UPS favors the screening of larger flights, resulting in a greater number of passengers covered. This example illustrates the effectiveness of and the relationship between each of the three performance measures. From Figures 2–4, the optimal UPS solution results in reasonable UFS values, while the UBS values are significantly lower than or similar to the UBS values for the optimal UFS solution. Also, at higher budgets, the optimal UFS solution did not yield acceptable UPS values, which were more than 1.4 times the value of UFS. While UBS ensures that the greatest number of selectee bags will be screened, it provides limited insight into the other performance measures (i.e., there is no guarantee that any flight segment or any passenger segment will be covered). Therefore, these results suggest that the UPS solution provides a reasonable baggage screening strategy for this airline carrier and these ten airports. This example illustrates the effectiveness of the Greedy algorithm heuristic for each of the three performance measures. The UBS value obtained using the Greedy heuristic is always within 91 bags of the optimal solution (of 170 total bags). The Greedy heuristic consistently finds UBS values that are within two times the optimal UBS value when B ≤ $3.45 million. The UFS value of the Greedy algorithm heuristic is within two times the optimal UFS value for all budget values, and when B ≤ $4.40 million, the Greedy algorithm heuristic finds values that are no more than 1.3 times the optimal UFS value. The UPS value of the Greedy algorithm heuristic is within three times the optimal UFS value for all budget values, and when B ≤ $4.95 million, the Greedy algorithm heuristic finds UPS values that are no more than 1.5 times the optimal UPS value. These results suggest that the Greedy algorithm heuristic provides good UFS and UPS values for budget values that are not sufficient to cover most of the flights or passengers.
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Conclusions
The critical need for high-level security for checked baggage has driven the development of new and improved baggage screening security devices. However, the optimal deployment of these devices is critical to take full advantage of their capabilities. The integer programming models introduced can be used to determine the optimal deployment of new generations of baggage screening security devices across a set of airports, based on three different security performance measures. The relationship between these three performance measures is an area of current research, along with the possibility of simultaneously optimizing all three of these measures, using multi-criteria optimization tools. In the proposed models, as well as with the illustrative example, each airport is considered as a single unit; no distinction is made between different airlines at an airport. To account for individual airlines, the integer programming models can be modified by creating a separate station for each airline at each of the N airports. Note that more detailed data would be required to use such models, (e.g., specific flight and passenger routing information). Consequently, the solutions obtained would determine the optimal deployment of baggage screening security devices among airlines at the airports, rather than just among airports. To effectively use and implement the proposed models, a large amount of data may be needed. Since some of this data, such as the flight routings of passengers, may be unavailable or difficult to obtain from the airlines, work is in progress to modify the models to include only readily available or easily estimated data. Moreover, while the models can provide useful deployment strategy information, the data to support it must be estimated and is constantly changing. This may require deployment decisions to be continually updated based on such new information. The analysis provided in this paper focuses on the optimal deployment and utilization of a limited number of next-generation checked baggage screening devices at a set of airports. The analysis assumes that a system such as CAPPS is in place to determine which passengers require additional security screening for their checked baggage. A frequently mentioned criticism of using a system such as CAPPS is that it can be defeated through extensive trial and error sampling by a variety of passengers through the system (Barnett, 2004; Chakrabarti and Strauss, 2002). Moreover, a potential threat to the system intent on gaining access to a flight and causing damage may “game the system” and board a plane at an airport where the new baggage screening security devices have not been deployed. However, all passengers are subject to a minimum level of security specified by the United States Congressional mandate of 100% checked baggage screening (Mead, 2002, 2003a). The intent here is to deploy new technologies as quickly as possible. To wait until adequate capacity and resources are available to deploy the new devices without leaving such holes delays the benefit that these future technologies will provide. The models presented provide initial insight into the optimal deployment of baggage screening security devices across a set of flights between a set of airports. This formulation models such a system as closed, which it clearly is not. However, given runway capacities and space limitations, peak period analysis overcomes some of the error caused by this restriction. Work is in progress to extend these models to cover a broader scope of baggage screening deployment issues, as well as to study such models using multi-criteria tools, to better understand how to design more effective aviation security systems.
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Appendix 1 Proof of Theorem 1: First, the MABP is in NP since if the subsets Di , i = 1, 2, . . . , N , N are guessed, then i=1 d∈Di C(d) ≤ B and d∈Di λ(d) ≥ qi , i = 1, 2, . . . , N , can be checked in linear time in the size of the problem instance. To complete the proof, a polynomial transformation from an arbitrary instance of the knapsack problem to the particular instance of the MABP must be constructed. The knapsack problem (Martello and Toth, 1990) is first formally stated. Knapsack Problem – A finite set U = {u 1 , u 2 , . . . , u n }, – a size S(u) ∈ Z + , u ∈ U , – a value V (u) ∈ Z + , u ∈ U , – a size constraint B ∈ Z + , – a value goal K ∈ Z + . Question: Is there a subset U ⊆ U such that u∈U S(u) ≤ B and u∈U V (u) ≥ K ? Instance:
To show that the MABP is NP-complete, transform the knapsack problem stated above. To obtain the desired transformation, let N = 1 and M = n. Let di = u i , i = 1, 2, . . . , n, C(di ) = S(u i ), i = 1, 2, . . . , n, and λ(di ) = V (u i ), i = 1, 2, . . . , n, B = B , and q1 = K . This transformation is obtained in polynomial time in the size of the arbitrary instance of the knapsack problem. Finally, it remains to be shown that the answer to the arbitrary instance of the knapsack problem is yes if and only if the answer to the particular instance of the MABP is yes. First, if the answer to the particular instance of the MABP is yes, then there exists a subset of baggage screening security devices D1 ⊆ D such that d∈D1 C(d) ≤ B and d∈D1 λ(d) ≥ q1 . By the definition of the transformation, D1 maps one-to-one onto a subset U ⊆ U for the knapsack problem. The sum of the sizes for the elements in this subset is u∈U S(u) = d∈D1 C(d) ≤ B = B . Similarly, the sum of the values for the elements in this subset is u∈U V (u) = d∈D1 λ(d) ≥ q1 = K . Therefore, the answer to the arbitrary instance of the knapsack problem is yes. Conversely, if the answer to the arbitrary instance of the knapsack problem is yes, then there exists a subset U ⊆ U such that u∈U S(u) ≤ B and u∈U V (u) ≥ K . Once again, by the definition of the transformation, there is a one-to-one mapping ontoa set D1 ⊆ D for the MABP. The sum of the sizes for the elements in this subset is d∈D1 C(d) = B = B, and the sum of the values for the elements in this subset is u∈U S(u) ≤ d∈D1 λ(d) = u∈U V (u) ≥ K = q1 . Therefore, the answer to the particular instance of the MABP is yes. Appendix 2 Proof of Theorem 2: First, the MADCBP is in NP since if the subsets Di , i = 1, 2, . . . , N , N ˆ i1 + qˆ i2 ≤ qˆ i1 and qˆ i2 are guessed, then i=1 d∈Di C(d) ≤ B, q d∈Di λ(d), i =
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N 1, 2, . . . , N , and i=1 (qˆ i1 + 2qˆ i2 ) ≥ α can be checked in linear time in the size of the problem instance. To complete the proof, a polynomial transformation from an arbitrary instance of the MABP to a particular instance of the MADCBP must be constructed. To show the MADCBP is NP-complete, transform the MABP. To obtain the desired transformation, let N = 1 and set q11 = α = q1 and q12 = 0. All other elements, A1 , D, C(d), λ(d), M, and B, are exactly the same. This transformation can be executed in linear time in the size of the arbitrary instance of the MABP. Finally, it remains to be shown that the answer to the arbitrary instance of the MABP is yes if and only if the answer to the particular instance of the MADBP is yes. First, if the answer to the particular instance of the MADCBP is yes, then there exists a subset of baggage screening security devices D1 ⊆ D such that d∈Di C(d) ≤ B, qˆ 11 ≤ d∈Di λ(d), and qˆ 11 ≥ α, i.e., all selectee bags are selected to be screened and the allocation is within budget. Hence, d∈Di λ(d) ≥ qˆ 11 = q11 = q1 , and the answer to the arbitrary instance of the MABP is yes. Conversely, if the answer to the arbitrary instance of the MABP isyes, then there exists a subset of baggage screening security devices D1 ⊆ D such that d∈Di C(d) ≤ B and within d∈Di λ(d) ≥ q1 (i.e., all selectee bags are able to be screened and the allocation is budget). By definition of the particular instance of the MADCBP, it is implied that d∈Di λ(d) ≥ q1 = q11 = qˆ 11 and qˆ 11 ≥ α = q11 is satisfied at equality. Therefore, the answer to the particular instance of the MADCBP is yes.
Acknowledgment This research has been supported in part by the National Science Foundation (DMI-0114046, DMI-0114499). The first author has also been supported in part by the Air Force Office of Scientific Research (FA9550-04-1-0110). The computational work was done in the Simulation and Optimization Laboratory housed within the Department of Mechanical and Industrial Engineering at the University of Illinois. The authors would also like to thank the Editor-in-Chief, Dr. Tam´as Terlaky, the associate editor, and the two anonymous reviewers for their helpful comments and suggestions that have resulted in a significantly improved manuscript.
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