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Models for nuclear smuggling interdictionDavid P. Morton a , Feng Pan b & Kevin J. Saeger ba Graduate Program in Operations Research , 1 University Station, C2200, The University ofTexas at Austin , Austin, TX, 78712-0292, USAb D-6, Risk Analysis & Decision Support Systems, Los Alamos National Laboratory, Mail StopF603 , Los Alamos, NM, 87545, USAPublished online: 08 Apr 2011.

To cite this article: David P. Morton , Feng Pan & Kevin J. Saeger (2007) Models for nuclear smuggling interdiction, IIETransactions, 39:1, 3-14, DOI: 10.1080/07408170500488956

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IIE Transactions (2007) 39, 3–14Copyright C© “IIE”ISSN: 0740-817X print / 1545-8830 onlineDOI: 10.1080/07408170500488956

Models for nuclear smuggling interdiction

DAVID P. MORTON1,∗, FENG PAN2 and KEVIN J. SAEGER2

1Graduate Program in Operations Research, 1 University Station, C2200, The University of Texas at Austin,Austin, TX 78712-0292, USAE-mail: morton@mail.utexas.edu2D-6, Risk Analysis & Decision Support Systems, Los Alamos National Laboratory, Mail Stop F603, Los Alamos, NM 87545, USAE-mail: fpan@lanl.gov, saeger@lanl.gov

Received April 2005 and accepted November 2005

We describe two stochastic network interdiction models for thwarting nuclear smuggling. In the first model, the smuggler travelsthrough a transportation network on a path that maximizes the probability of evading detection, and the interdictor installs radiationsensors to minimize that evasion probability. The problem is stochastic because the smuggler’s origin-destination pair is known onlythrough a probability distribution at the time when the sensors are installed. In this model, the smuggler knows the locations ofall sensors and the interdictor and the smuggler “agree” on key network parameters, namely the probabilities the smuggler will bedetected while traversing the arcs of the transportation network. Our second model differs in that the interdictor and smuggler canhave differing perceptions of these network parameters. This model captures the case in which the smuggler is aware of only a subsetof the sensor locations. For both models, we develop the important special case in which the sensors can only be installed at bordercrossings of a single country so that the resulting model is defined on a bipartite network. In this special case, a class of valid inequalitiesreduces the computation time for the identical-perceptions model.

Keywords: Network interdiction, integer programming, stochastic programming

1. Introduction

We develop stochastic network interdiction models de-signed to locate radiation sensors, which detect gamma andneutron emissions from nuclear material, at critical bordercrossings in the Former Soviet Union (FSU). The goal isto locate the sensors on an underlying transportation net-work to minimize the probability of a successful smugglingattempt. Our work supports the Second Line of Defense(SLD) program of the US Department of Energy.

The smuggling of nuclear material, equipment, and tech-nology has become a greater threat to international securitysince the dissolution of the Soviet Union. In the early 1990s,Russia inherited roughly 600–850 metric tons of highly en-riched uranium and plutonium, enough material to makeover 50 000 explosive devices (Jones, 2002; Cobb, 2002).The “first line of defense” concerns nuclear Material Pro-tection, Control and Accountability (MPC&A). In short,this involves securing and inventorying nuclear material atits storage sites in both civilian and defense facilities.

An International Atomic Energy Agency databaseincludes 540 incidents of trafficking of nuclear and radioac-

∗Corresponding author.

tive material from 1993–2003 that have been confirmedby a country’s government (IAEA, 2004). Of these 205involved nuclear material and 17 involved weapons-gradeuranium or plutonium. With sophisticated technology,non-weapons-grade nuclear material can be processed toobtain weapons-grade material. With minimal technologyit, or more widely available radioactive material, can beused with conventional explosives to build a radiologicaldispersal device, i.e., a dirty bomb. The majority of the inci-dents involved smugglers seeking to sell the illicit material.Weapons-grade material has been seized by authorities inRussia, Germany, the Czech Republic, Lithuania, Bulgaria,Kyrgyzstan, Georgia, Greece and France, and in the major-ity of the cases the material was traced to have originatedin Russia or other parts of the FSU (IAEA, 2004). Someincidents involved kilograms of material. Some othersinvolving smaller quantities actually represented samplesof stolen material or material at risk of being stolen. Thisclearly points to the vulnerability of Russia’s first line ofdefense. US efforts to assist the FSU in improving Russia’sfirst line of defense are ongoing (National Nuclear SecurityAdministration, 2006). These MPC&A efforts are criticallyimportant but by themselves, insufficient. An accurateinventory of the nuclear material that existed in Russia atthe beginning of the 1990s seems impossible.

0740-817X C© 2007 “IIE”

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4 Morton et al.

The SLD program seeks to reduce the risk of illicit traf-ficking of nuclear material through airports, seaports andborder crossings in Russia and other key transit states, withthe program’s initial efforts in the FSU (Ball, 1998). The firstSLD sensor installation was at Moscow’s Sheremetyevo In-ternational Airport in September of 1998. Such sensor in-stallations have two purposes: (i) to deter potential theftand smuggling of nuclear material, and (ii) to detect andtherefore prevent actual smuggling attempts.

In this paper, we describe two types of stochastic networkinterdiction models that can be used to select the sites toinstall sensors to minimize the probability a smuggler cantravel through a transportation network undetected. Ourtwo basic models are distinguished with respect to whetherthe interdictor and smuggler have the same or differing per-ceptions of key network parameters. Our first model, inwhich the smuggler and interdictor have identical percep-tions of the network, has been developed in collaborationwith the Los Alamos National Laboratory SLD team andhas been implemented for decision support for the SLDprogram. Our second model in which the interdictor andsmuggler can have differing perceptions is an importantextension. The primary emphasis in this paper is on mod-eling, as opposed to solution techniques and computation.Of course, modeling choices affect our ability to solve theseproblems, and so important parts of the development aredevoted to precisely these issues. Furthermore, we describe,and motivate from a modeling perspective, a class of validinequalities that strengthen our simplest model. Finally, indeveloping our basic model, we provide an outline of someof the techniques that have been successfully employed toobtain tractable network interdiction models in settings be-yond the specific models of this paper.

While there are earlier references (e.g., Wollmer, 1964),the study of network interdiction in operations researchbegan in earnest in the 1970s. During the Vietnam War,deterministic mathematical programs to disrupt flow of en-emy troops and supplies were developed (McMasters andMustin, 1970; Ghare et al., 1971). The problem of maximiz-ing an adversary’s shortest path is considered in Fulkersonand Harding (1977) and Golden (1978). A closely relatedproblem concerns maximizing the longest path in an ad-versary’s PERT network (Reed, 1994; Brown et al., 2004).When these are linear programs (LPs), the interdictor cancontinuously increase the length of an arc, subject to a bud-get constraint. A discrete version of maximizing the shortestpath removes an interdicted arc from the network, and whenthe budget constraint is simply a cardinality constraint, thisis called the k-most-vital-arcs problem (Corley and Sha,1982; Ball et al., 1989; Malik et al., 1989). Generalizationsof the k-most-vital arcs problem are considered in Isrealiand Wood (2002). The interdiction problem of removingarcs to minimize flow in an adversary’s maximum-flow net-work is considered in Wollmer (1964) and Wood (1993). SeeWashburn and Wood (1994) for game-theoretic approachesto related network interdiction problems, Chern and Lin

(1995) for an interdiction model on a minimum-cost-flownetwork, and Israeli and Wood (2001) for interdiction mod-els of more general systems.

The above interdiction models are deterministic in thefollowing senses. First, the arc lengths in the shortest pathand PERT problems, and the arc capacities in the maximumflow problem, are known with certainty. Second, when in-creasing the length of an arc in the former problems or whenremoving or decreasing the capacity of an arc in the latterproblem, these modifications are deterministic, i.e., withcertainty. The work of Wood (1993) on maximum flow net-work interdiction is generalized in Cormican et al. (1998)to allow for both random arc capacities and interdictionsuccesses. An interdiction model with uncertain networktopology is developed in Hemmecke et al. (2003). A stochas-tic interdiction model in which the adversary’s response ismodeled via a Markov decision process is considered inBailey et al. (2004).

The remainder of this paper is organized as follows.In Section 2, we formulate our basic model, which welabel SNIP, for stochastic network interdiction problem,as a bilevel stochastic mixed-integer program (MIP). Thismodel exhibits a “min-max” structure, which does not lenditself to computation and so we formulate an equivalentstochastic linear MIP that can be solved, e.g., by commer-cial branch-and-bound solvers for integer programming.We then turn our attention in Section 3 to an importantspecial case of SNIP that arose in our work on the SLD pro-gram, in which sensors can only be installed at border cross-ings of a single country, namely Russia. We show the associ-ated MIP can be simplified in this special case. The resultingmodel is called BiSNIP, for bipartite stochastic network in-terdiction problem, because it may be viewed as an interdic-tion problem on a bipartite network. Section 4 generalizesSNIP and BiSNIP to models we call PSNIP and BiPSNIP,respectively. Here, the addition of “P” to the SNIP andBiSNIP labels indicates that these are models in which theinterdictor and the evader differ in their perceptions of thenetwork. Our emphasis here is on the simpler BiPSNIP case.In Section 5, we describe a class of valid inequalities, thatwe call step inequalities, to tighten the MIP formulation ofBiSNIP, and we present computational results when usingthese inequalities. We conclude the paper in Section 6.

2. SNIP on a general network

We model two adversaries, an interdictor and an evader(we will use the terms “evader” and “smuggler” inter-changeably), and an underlying directed network G(N, A)on which the evader travels. In the deterministic version ofour model, the evader starts at a source node s ∈ N andwishes to reach a terminal node t ∈ N. The model is deter-ministic in that this origin-destination pair is known. Theprobability that the evader can traverse arc (i, j) ∈ A unde-tected is pij if the interdictor has not installed a sensor on

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Models for nuclear smuggling interdiction 5

arc (i, j), and this probability is qij < pij if the interdictorhas installed a sensor. An evader can be caught by indige-nous law enforcement without radiation detection equip-ment, and so pij < 1. The events of the evader being detectedon distinct arcs are assumed to be mutually independent.The evader chooses an s−t path to maximize the prob-ability of traversing the network without being detected.With limited resources, the interdictor must select arcs onwhich to install sensors in order to minimize this evasionprobability.

Our stochastic network interdiction problem (SNIP) dif-fers from the above description only in that the identityof the evader is unknown when the interdictor installs thesensors. In our basic SNIP model, an evader’s identityis uniquely specified by an origin-destination pair, (s, t),which is assumed to be governed by a known probabilitymass function, pω = P{(s, t) = (sω, tω)}, ω ∈ �. The proba-bility that evader ω traverses the network undetected is thena sum of (conditional) evasion probabilities, each weightedby pω, over the population of possible evaders indexed byω ∈ �. While our basic model equates evader ω’s identitywith an origin-destination pair, straightforward extensionswe describe below allow this identity to involve further dis-tinguishing characteristics.

The timing of decisions and realizations is as follows.First, the interdictor installs sensors on a subset of the net-work’s arcs. Then, a random origin-destination pair is re-vealed and evader ω selects an sω−tω path to maximize theprobability of avoiding detection. The evader selects thispath with full knowledge of the sensor locations and eva-sion probabilities.

As indicated above, the model represents sensor instal-lations as occurring on arcs. In reality, detection equip-ment (e.g., in the form of radiation sensor portals) is in-stalled at transportation choke points such as airports,seaports, and international border crossings for automo-biles, railroads, and pedestrians. We capture this in ourmodel by splitting the node associated with such a locationinto two nodes with an arc representing travel through thecheckpoint.

In SNIP, we assume smugglers are aware of the sen-sor locations. The initial installation of SLD equipmentin Moscow in 1998 was accompanied by a ribbon-cuttingceremony (Richardson, 1998), and subsequent installationswere also reported in the news. The reason for this publicityis that completely sealing Russia’s 12 500 miles of bordersis impractical in today’s environment. As a result, in addi-tion to catching nuclear smugglers, the SLD program seeksto deter would-be smugglers. That said, as the program ex-pands beyond Russia these policies may change. So, in Sec-tion 4 we consider variants of SNIP in which the smugglerhas a perception of the network parameters pij and qij thatmay differ from that of the interdictor and in which thesmuggler is aware of only a subset of the sensor locations.The notation used in formulating SNIP is summarized be-low, followed by the formulation.

Network and sets:

G(N, A) = directed network with nodes N and arcs A;FS(i) = the set of arcs leaving node i;RS(i) = the set of arcs entering node i;AD ⊂ A = the arcs on which sensors may be placed.

Data:

b = the total budget for installing sensors;cij = the cost of installing a sensor on arc (i, j) ∈ AD;pij = the probability evader can traverse (i, j) undetected

with no sensor installed;qij = the probability evader can traverse (i, j) undetected

with a sensor installed; qij < pij.

Random elements:

ω ∈ � = a sample point and a sample space, indexing theevader’s identity;

(sω, tω) = the realization of the evader’s origin–destination pair;

pω = the probability mass function.

Interdictor’s decision variables:

xij ={

1 if a sensor is installed on arc (i, j),0 otherwise.

Evader’s decision variables:

yij is positive only if evader traverses (i, j) and no sensor isinstalled;

zij is positive only if evader traverses (i, j) and a sensor isinstalled.

Boundary conditions:

xij, zij ≡ 0 (i, j) /∈ AD

Formulation:

minx∈X

∑ω∈�

pωh(x, (sω, tω)), (1)

where X = {x :∑

(i,j) ∈ AD cijxij ≤ b, xij ∈ {0, 1}, (i, j) ∈ AD},and where h(x, (sω, tω)) is the optimal value of:

maxy,z

ytω , (2a)

subject to∑

(sω,j)∈FS(sω)

(ysω j + zsω j) = 1 : πsω , (2b)

∑(i,j)∈FS(i)

(yij + zij) =∑

(j,i)∈RS(i)

(pjiyji + qjizji),

i ∈ N \ {sω, tω} : πi, (2c)

ytω =∑

(j,tω)∈RS(tω)

(pjtω yjtω + qjtω zjtω ) : πtω , (2d)

0 ≤ yij ≤ 1 − xij, (i, j) ∈ A : λij, (2e)0 ≤ zij, (i, j) ∈ AD. (2f)

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6 Morton et al.

The optimal value of Model (2) is the conditionalprobability that the smuggler avoids detection, given(sω, tω). The objective function in Model (1) is theexpected value of this conditional evasion probability,where the expectation is taken over all possible origin-destination pairs. Dual variables πi, i ∈ N, and λij, (i, j) ∈A, are shown in model (2) with their correspondingconstraints.

Each network link on which a sensor can be placed ismodeled as two arcs in parallel. If a sensor is installed,i.e., xij = 1, then flow may occur only on the “sensor” arc,through zij. Otherwise flow can occur on the “no sensor”arc, via yij. Flow on arc (i, j) is multiplied by that arc’sgain (either pij or qij). So, if Psω,tω is a path from sω to tω

then:

ytω =∏

(i,j)∈Psω,tω

[pij(1 − xij) + qijxij], (3)

is the probability that evader ω can travel Psω,tω withoutbeing detected. The evader’s subproblem finds a path Psω,tω

maximizing ytω by forcing one unit of flow out of sω in Con-straint (2b), enforcing flow conservation at all intermediatenodes in Constraint (2c), defining the flow that reaches tω asytω in Constraint (2d) and maximizing that value in the ob-jective function (2a). Flow is forced on the appropriate arc,and incurs the associated gain (actually, loss), by the inter-dictor’s decision variable xij in Constraint (2e). Note thatan upper bound of xij in Constraint (2f) is not necessarysince qij < pij implies that the smuggler will not use zij inplace of yij when xij = 0.

When locating sensors, the interdictor knows: (i) the net-work topology; (ii) the indigenous detection probability oneach arc; (iii) the detection probability given the presenceof a sensor; (iv) the budget constraint; (v) the probabilitydistribution governing the random (s, t) pair; and (vi) themethod by which the evader will select a path. After (sω, tω)is revealed, evader ω selects an sω−tω path that maximizesthe evasion probability, knowing (i), (ii), and (iii) as well asthe sensor locations.

In the model’s current form, pij and qij are identical forall possible smugglers, and scenario ω simply specifies thesmuggler’s origin-destination pair, (sω, tω). However, theabove formulation can directly accommodate the case inwhich these are replaced by pω

ij and qωij . In this setting, the

smuggler’s identity, ω, specifies the origin-destination pair,(sω, tω), and the evasion probabilities, pω

ij and qωij . Here,

the evasion probabilities could differ for smugglers havingthe same origin-destination pair because of differences inthe nature of the material being smuggled and how the ma-terial is packaged (these affect a sensor’s ability to detectthe material), and could also include other traits of thesmuggler, which could affect the likelihood of detection viaindigenous law enforcement. For notational simplicity, wewill not label pij and qij with ω.

SNIP with h defined in model (2) is a bilevel stochasticMIP. In bilevel programs (e.g., Ben-Ayed (1993), Ishizuka

et al. (1997) and Bard (1998)) each player has an objectivefunction, and these can differ because the players’ motivesdiffer. In our case, the objective function is the same forboth players, but the interdictor is minimizing that func-tion and the evader is maximizing it. The problem is for-mulated with a nested “min-max” structure, which is notamenable to solution through standard optimization algo-rithms. There are at least three approaches one might taketo obtain a tractable optimization model from a problemsuch as SNIP, and we label these decomposition, dualityand reformulation.

2.1. Decomposition

The nested min-max structure is not a difficulty if one canemploy an outer-approximation cutting-plane algorithmsuch as Kelley’s cutting-plane method (Kelley, 1960), Ben-ders’ decomposition (Benders, 1962; Geoffrion, 1972), orthe L-shaped method (Van Slyke and Wets, 1969; Wollmer,1980). Unfortunately, h(x, (sω, tω)), defined by model (2),is a maximization linear program with x appearing onthe right-hand side of Constraint (2e). This implies thath(x, (sω, tω)), and hence Eh(x, (sω, tω)), is a concave func-tion over the convex hull of X . This does not bode wellfor employing the above cutting-plane schemes. We notethat Laporte and Louveaux (1993) have developed vari-ants of the L-shaped method that are valid for nonconvexEh(x, (sω, tω)) but we regard this approach as one of lastresort if our other approaches fail because they require cut-ting planes that are tight at a specific (binary) value of x = x̂and drop to an a priori lower bound at all other (binary)values of x ∈ X .

2.2. Duality

Another general approach is to take the dual of the innermaximization problem and then to formulate a single modelin which we simultaneously optimize over the interdictor’sdecision variables and the smuggler’s (dual) decision vari-ables. We assume the network G is such that an sω−tω pathexists for all ω ∈ �. So, model (2) is feasible and has a fi-nite optimal solution for all ω ∈ �. Hence, an equivalentexpression of h(x, (sω, tω)) is available via linear program-ming duality, i.e., using the dual variables as indicated inmodel (2) we have:

h(x, (sω, tω)) = minπ,λ

πsω +∑

(i,j)∈AD

(1 − xij)λij,

subject to πi − pijπj ≥ 0, (i, j) ∈ A \ AD,

πi − pijπj + λij ≥ 0, (i, j) ∈ AD,

πi − qijπj ≥ 0, (i, j) ∈ AD,

λij ≥ 0, (i, j) ∈ AD,

πtω = 1. (4)

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Models for nuclear smuggling interdiction 7

With the expression of model h(x, (sω, tω)) in model (4), wehave a nested “min-min” formulation and can now forma single optimization problem in which we simultaneouslyminimize over x and (πω, λω), ω ∈ �. However, the associ-ated objective function includes nonlinear terms between xand λω. The multipliers λij are bounded above by one, andso these terms are linearized via:

h(x, (sω, tω)) = minπ,λ,β

πsω +∑

(i,j)∈AD

βij,

subject to πi − pijπj ≥ 0, (i, j) ∈ A \ AD,

πi − pijπj + λij ≥ 0, (i, j) ∈ AD,

(5a)πi − qijπj ≥ 0, (i, j) ∈ AD,

βij − λij ≥ −xij, (i, j) ∈ AD, (5b)λij, βij ≥ 0, (i, j) ∈ AD,

πtω = 1.

Constraints (5a) and (5b) can be written as πi − pijπj ≥−λij ≥ −βij − xij, (i, j) ∈ AD. Variable λij appears in thisexpression and no where else in the model. Hence, we canreplace these constraints by πi − pijπj + βij ≥ −xij, (i, j) ∈AD, eliminating λij, (i, j) ∈ AD, in the process. Next, wecan take βij = 0 in an optimal solution. To see this, we ar-gue by contradiction. Assume we have an optimal solution(π∗, β∗) with β∗ �= 0. Using the fact that pij, qij ∈ [0, 1], it isstraightforward to construct a feasible solution with β = 0whose objective function value is no larger. So, model (5)simplifies to:

h(x, (sω, tω)) = minπ

πsω

subject to πi − pijπj ≥ 0, (i, j) ∈ A\AD,

πi − pijπj ≥ −xij, (i, j) ∈ AD,

πi − qijπj ≥ 0, (i, j) ∈ AD,

πtω = 1. (6)

Model (6) leads us to a single large-scale linear MIP in whichwe minimize over x and πω, ω ∈ �. Our above reasoningin linearizing the nonlinear terms in the objective functionof model (4) mirrors that of Wood’s application of the du-ality approach to the interdiction problem on a maximumflow network (Wood, 1993). Below, we arrive at model (6)through another, arguably simpler approach. However, theduality approach is important in interdiction modeling be-cause it is one way to arrive at a tractable model, and insome cases, such as the shortest-path interdiction model ofFulkerson and Harding (1977), the duality approach is thesimplest in that it does not give rise to the types of nonlinearterms we have seen above.

2.3. Reformulation

In Pan et al. (2003), we showed a minor variant of the fol-lowing result: h(x, (sω, tω)) as defined in model (2) may be

equivalently formulated as:

h(x, (sω, tω)) = maxy≥0,z≥0

ytω −∑

(i,j)∈AD

xijyij,

(7)subject to Constraints (2b)–(2d).

Model (7) differs from model (2) in that the upper boundin Constraint (2e) has been removed and a penalty term isnow included in the objective function. The intuition be-hind this reformulation is that while we now allow flow onyij even if a sensor is installed on (i, j), i.e., xij = 1, in thatcase, we subtract the associated flow in the objective func-tion. The validity of the reformulation is based on an exactpenalty result from Morton and Wood (1999). A proof isalso given above in our application of the duality approach,where we argued h(x, (sω, tω)) is given by model (6) becausemodel (7) is the dual of model (6). As mentioned above,the duality approach was applied to maximum flow net-work interdiction in Wood (1993). The reformulation ap-proach we consider here was applied to (stochastic versionsof) the same maximum-flow problem in Cormican et al.(1998). The following theorem tightens the reformulation inmodel (7).

Theorem 1. Assume that G has an sω−tω path ∀ω ∈ �, 0 ≤pij ≤ 1, (i, j) ∈ A, and 0 ≤ qij < pij, (i, j) ∈ AD. Then, forall x ∈ X and ω ∈ �, h(x, (sω, tω)) is the optimal value ofthe following LP:

minπ

πsω

subject to πi − pijπj ≥ 0, (i, j) ∈ A \ AD,

πi − pijπj ≥ −(pij − qij)xij, (i, j) ∈ AD, (8a)πi − qijπj ≥ 0, (i, j) ∈ AD, (8b)πtω = 1.

Proof. Model (7) has as its dual model (8) except that con-straints (8a) are instead:

πi − pijπj ≥ −xij, (i, j) ∈ AD. (9)

If xij = 0 then Constraints (8a) and (9) are equivalent. Onthe other hand, if xij = 1 then:

πi − qijπj = πi − pijπj + (pij − qij)πjxij

≤ πi − pijπj + (pij − qij)xij,

where the inequality holds since pij > qij and 0 ≤πj ≤ 1 in any optimal solution of model (7)’s dual.Thus, when xij = 1 constraint (8a) is redundant givenConstraint (8b). �

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8 Morton et al.

The value of Theorem 1 is that SNIP can be expressed asthe following large-scale MIP:

minx,π

∑ω∈�

pωπωsω ,

subject to x ∈ X,

πωi − pijπ

ωj ≥ 0, (i, j) ∈ A \ AD, ω ∈ �,

πωi − pijπ

ωj

+ (pij − qij)xij ≥ 0, (i, j) ∈ AD, ω ∈ �,

πωi − qijπ

ωj ≥ 0, (i, j) ∈ AD, ω ∈ �,

πtω = 1, ω ∈ �.

(10)

In summary, the goal of all three techniques (decompo-sition, duality and reformulation) is to achieve a computa-tionally tractable optimization model. In various settingseach technique has proven useful for attaining this goal. Inour case, the duality and reformulation approaches lead tothe same model. That said, the reformulation approach maybe more attractive in that it provides an intuitive modeling-based argument for the formulation and avoids the needto carry-out the types of transformations we performedfrom model (4) to model (6), which may not always be sostraightforward.

We can now attempt to solve the large-scale MIP (10) viacommercially available integer programming software. In-terestingly, the reformulations of models (6), (7) and (8) alsomake our problem amenable to solution by the L-shapeddecomposition method. As noted earlier, h(x, (sω, tω)) asdefined by model (2) is concave over the convex hull of X .However, h(x, (sω, tω)) as defined by model (8) is convexover the convex hull of X because it is an LP with x inthe right-hand side. (Having h be both convex and concaveover the convex hull of X is possible because the optimalvalues of these LPs are only ensured to be equal when xtakes on binary values.) The L-shaped method with a MIPmaster program is an attractive solution approach when thenumber of scenarios |�| is large.

To close this section, we comment on one other model-ing approach that could prove valuable in certain settings.In model (2), the smuggler is solving what is called themaximum reliability path problem, and it is well knownthat by taking logarithms of the objective function as ex-pressed in Equation (3) the model can be solved as ashortest-path problem (see, e.g., Ahuja et al., 1993, ex-ercise 4.39). When |�| = 1 this approach leads to a lin-ear MIP formulation by first taking logarithms and thenusing the type of “duality” approach outlined above.However, when |�| > 1 one cannot simply take loga-rithms of each h term in

∑ω∈� pωh(x, (sω, tω)) and obtain

an equivalent objective function. Instead, one can write∑ω∈� pω exp {ln [h(x, (sω, tω))]}. This leads to a nonlinear

MIP, and so we prefer the above approach which beginswith the generalized network flow formulation in model (2)and leads to a linear MIP. Next, we turn to an importantspecial case of SNIP on a bipartite network.

3. BiSNIP: SNIP on a bipartite network

The most pressing problem in our initial SLD work con-cerned smuggling of nuclear material out of Russia. Inthis special case of SNIP, potential sensor locations arerestricted to customs checkpoints. With origins (sω) lo-cated inside Russia and destinations (tω) located outsideRussia, the key to simplifying the formulation is that oneach sω − tω path there is exactly one customs checkpointarc on which the smuggler could encounter a sensor.

Let Pω denote the set of all paths for origin-destinationpair (sω, tω). In BiSNIP we assume that each path in Pω

contains exactly one arc in AD, i.e., each path has exactlyone arc that can receive a sensor. Let ADω = {(i, j) : (i, j) ∈AD, (i, j) ∈ Pω} be all such checkpoint arcs for ω ∈ �. Foreach ω, we compute the value of the maximum reliabil-ity path from sω to the tail of each checkpoint arc, andthe value of the maximum reliability path from the headof each checkpoint arc to tω. (As indicated in the previ-ous section, this can be done efficiently.) Let the product ofthese two probabilities be γ ω

k , k = (i, j) ∈ ADω. The prob-ability that the smuggler can travel from sω to tω via kundetected is γ ω

k pk if no sensor is installed at k, and thisprobability is γ ω

k qk if a sensor is installed at k. Smuggler ω’spath-selection decision is now reduced to the decision ofchoosing a checkpoint k through which to travel, and thatcheckpoint is found by solving:

h(x, (sω, tω)) = maxk∈ADω

{γ ω

k pk(1 − xk), γ ωk qkxk

}.

As a result, we can express BiSNIP as the following stochas-tic MIP:

minx,θ

∑ω∈�

pωθω,

subject to x ∈ X,

θω ≥ γ ωk pk(1 − xk), k ∈ ADω, ω ∈ �, (11a)

θω ≥ γ ωk qkxk, k ∈ ADω, ω ∈ �. (11b)

The bipartite network on which BiSNIP is defined hasnode sets � and AD = ∪ω∈� ADω, and arcs (ω, k) link eachorigin-destination pair (sω, tω), ω ∈ �, with its possible in-termediate checkpoints, k ∈ ADω. Figure 1(a and b) illus-trates the transformation from the underlying transporta-tion network to the corresponding bipartite network.

Since pk > qk and xk is binary, we can replace the right-hand side of Constraint (11b) with γ ω

k qk, and hence, replaceConstraint (11b) with the simple lower bound θω ≥ q̄ω ≡maxk∈ADω γ ω

k qk, ω ∈ �. Defining θ̄ω = θω − q̄ω, we trans-form model (11) to a model in which θ̄ω has simple lowerbounds of zero:

minx,θ̄

∑ω∈�

pωθ̄ω,

subject to x ∈ X, (12)θ̄ω ≥ rω

k (1 − xk), k ∈ ADω, ω ∈ �,

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Models for nuclear smuggling interdiction 9

Fig. 1. (a) An example transportation network for nuclear-material smuggling in which one set of arcs connects sources with the tailsof border-crossing arcs and another set of arcs connects the heads of border-crossing arcs with destinations; (b) the correspondingbipartite network in which arcs connect each scenario’s (sω, tω) pair with its possible border crossings, k ∈ ADω.

where rωk = (γ ω

k pk − q̄ω)+ and where (·)+ = max (·, 0). Thesimplified model (12) is equivalent to model (11) in thatthey have the same set of optimal solutions for locating thesensors, but their objective functions differ by a constant.We can view this as a transformation to a model in whichthe radiation sensors are perfectly reliable, i.e., model (12)has the form of model (11) with qk = 0.

4. (Bi)PSNIP: (Bi)SNIP with differing perceptions

In the previous two sections we developed SNIP on a gen-eral network and then specialized it to a bipartite network.In SNIP the interdictor and the evader agree upon, i.e., havethe same perception of, the network topology G(N, A), theindigenous detection probability on each arc, pij, (i, j) ∈ A,and the detection probability given the presence of a sensor,qij, (i, j) ∈ AD. This section generalizes SNIP to a model wecall PSNIP to handle the case in which the evader and theinterdictor differ in their perceptions of these network pa-rameters (the network topology remains agreed upon). Wethen focus on the special case of PSNIP in which we are re-stricted to locating sensors on arcs leaving a single country,i.e., model BiPSNIP.

Static Stackelberg programs and other bilevel programsdeal with asymmetries in players’ objective functions andinformation. In a sense, stochastic network interdiction cap-tures an asymmetry in that the interdictor does not knowwhich evader will appear from a population of possibleevaders. That said, there has been little work in the interdic-tion literature on handling other types of asymmetries. Onenotable exception is recent work on shortest-path networkinterdiction by Bailey and Bayrak (2005).

We redefine and extend the notation of Section 2 to nowaccommodate two sets of perceptions.

Data:

pij = interdictor’s perception of the probability that theevader can traverse arc (i, j) undetected when no sen-sor is installed;

qij = interdictor’s perception of the probability that theevader can traverse arc (i, j) undetected when a sensoris installed; qij < pij;

p2ij = evader’s perception of the probability that the evader

can traverse arc (i, j) undetected when no sensor isinstalled;

q2ij = evader’s perception of the probability that the evader

can traverse arc (i, j) undetected when a sensor isinstalled; q2

ij ≤ p2ij.

Evader’s decision variables:

yij, zij are analogous to y2ij and z2

ij below, except that theyare used to compute the interdictor’s perception ofthe probability the evader avoids detection;

y2ij is positive only if evader traverses (i, j) and no sensor

is installed;z2

ij is positive only if evader traverses (i, j) and a sensoris installed.

Boundary conditions:

xij, zij, z2ij ≡ 0 (i, j) /∈ AD.

In this setting, the interdictor knows both (pij, qij) and(p2

ij, q2ij) for all arcs. While the interdictor views the latter as

inferior estimates of these network parameters, the modelimplicitly assumes the interdictor knows all the informationthat the evader will use to select what the evader perceivesto be a maximum reliability sω − tω path for all ω ∈ �.

PSNIP captures the case in which the evader is only awareof a subset of the sensor locations: If p2

ij > q2ij then the evader

knows whether (i, j) has a sensor, but if p2ij = q2

ij the evader isunaware that (i, j) can receive a sensor. The case in which theevader is unaware of the sensors on all arcs (call it USNIP,for an uninformed evader) is easily handled because theevader’s path selection has nothing to do with the inter-dictor’s sensor-installation decisions. This is captured by arelatively straightforward modification of the SNIP modelthat we will not detail here. PSNIP generalizes SNIP in thatthe original model is recovered if p2

ij = pij and q2ij = qij for

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10 Morton et al.

all arcs, and also includes USNIP as a special case whenp2

ij = q2ij = pij for all arcs.

PSNIP is formulated as the SNIP of Model (1), exceptthat the definition of h from model (2) is replaced by:

h(x, (sω, tω)) = maxy,z,y2,z2

ytω , (13a)

subject to Constraints (2b), (2c), (2d), (2e), (2f),yij ≤ My2

ij, (i, j) ∈ A,

(13b)zij ≤ My2

ij + Mz2ij, (i, j) ∈ AD,

(13c)(y2, z2) ∈ Y 2(x, ω). (13d)

Model (13) differs from model (2) in that it has two sets ofdecision variables: the variables (y2, z2) select the evader’spath under (sω, tω) and are constrained to be in Y 2(x, ω),which denotes the argmax of the evader’s subproblem, i.e.,model (2) where p and q are replaced by the evader’s percep-tions p2 and q2. Variables (y, z) are used for “accounting”purposes, i.e., to compute the interdictor’s perception of theprobability that the smuggler evades detection. Constraints(2e) and (13b) allow yij to take positive flow only if (i, j) hasno sensor and (i, j) is on the evader’s optimal path. Simi-larly, pij > qij and constraints (13c) allow zij to be positivein an optimal solution to model (13) only if xij = 1 and ei-ther y2

ij > 0 or z2ij > 0. We state the latter condition in this

form because the evader may be unaware of a potentialsensor (p2

ij = q2ij) and could traverse (i, j) as if it has no sen-

sor. Of course, in implementation we simply remove z2ij and

q2ij from the smuggler’s network but the above approach is

notationally simpler.It is typical that constraints of the form of Con-

straint (13d) arise in a bilevel program. We avoided suchconstraints in SNIP because there the interdictor’s andevader’s objective functions were essentially identical, eventhough the former sought to minimize it and the lattersought to maximize it. Now, due to differing perceptionsthe objective functions differ.

The next task in the development is to achieve PSNIP’sanalog of model (10), i.e., to reformulate PSNIP as a large-scale MIP. We only outline how to accomplish this andrefer to Pan (2005) where this is carried out in detail. First,we state constraints (13d) in explicit form by: (i) writingthe primal constraints of the evader’s LP subproblem, i.e.,constraints of model (2) with (p, q) replaced by (p2, q2)and (y, z) replaced by (y2, z2); (ii) writing the dual con-straints of this evader’s LP subproblem; and (iii) enforcingstrong duality of these primal and dual LPs. As a result,model (13) can be stated as an LP (again, x ∈ X is fixedin model (13)). Second, to reformulate the resulting nested“min-max” problem, we use the kind of equivalent penalty-based reformulation employed in Section 2 and then takethe dual of this inner maximizing LP for each ω ∈ �. Theresult is a single large-scale linear MIP for PSNIP definedon a general network.

Instead of detailing the above process for PSNIP we turnto BiPSNIP, the special case of PSNIP defined on a bipartitenetwork, and we carry out the type of steps outlined aboveto obtain a single large-scale MIP formulation for BiPSNIP.In Section 3, we denoted evader ω’s probability of travelingfrom sω to tω via checkpoint k undetected as γ ω

k pk (γ ωk qk)

without (with) a sensor installed at k, and for BiPSNIP thesenow denote the interdictor’s perception of these probabili-ties. We add to this notation γ 2ω

k , p2k and q2

k , which representthe evader’s perception. Let q̄2ω = maxk∈ADω γ 2ω

k q2k , and de-

fine AD2ω = {k ∈ ADω : γ 2ωk p2

k ≥ q̄2ω}. Smuggler ω selectsa checkpoint from AD2ω by solving the following model:

minθ2ω

θ2ω,

subject to θ2ω ≥ r2ωk (1 − xk), k ∈ AD2ω : vk,

(14)

where r2ωk = γ 2ω

k p2k − q̄2ω and where we have carried out the

transformation to a problem with perfectly reliable detec-tors (from the smuggler’s perspective) as done in Section 3for BiSNIP. Decision variable θ2ω plays the role that θ̄ω

played in model (12), but here we suppress the “bar” no-tation for simplicity. The analog of constraints (13d) underscenario ω for BiPSNIP are represented by:

θ2ω ≥ r2ωk (1 − xk), k ∈ AD2ω, (15a)∑

k∈AD2ω

vωk = 1, vω

k ≥ 0, k ∈ AD2ω, (15b)

θ2ω =∑

k∈AD2ω

r2ωk (1 − xk)vω

k . (15c)

Here, the optimality conditions for model (14) are repre-sented by the primal feasibility of Constraint (15a), dualfeasibility of Constraint (15b), and strong duality of Con-straint (15c). For fixed x ∈ X , model (15) is a linear system.We assume that (15) has a unique solution for each x ∈ X ,i.e., that the smuggler is not indifferent between selectingtwo or more checkpoints. Note that vω

k , k ∈ AD2ω, takesvalue one for the checkpoint selected by smuggler ω and iszero otherwise.

We now represent BiPSNIP as:

minx,θ,θ2,v

∑ω∈�

pωθω,

subject to x ∈ X,

θω ≥ γ ωk pk(1 − xk)vω

k , k ∈ AD2ω, ω ∈ �,

(16a)θω ≥ γ ω

k qkxkvωk , k ∈ AD2ω, ω ∈ �,

(16b)Constraints (15a), (15b), (15c), ω ∈ �.

The BiPSNIP model (16) differs from the BiSNIPmodel (11) in that for each ω, constraints (16a) and (16b)are now enforced only at the checkpoint k that evaderω selects via vω

k . (In model (11) this is not necessarysince the identical-perceptions assumption leads to con-straints (11a) and (11b) being tight at the maximum of their

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Models for nuclear smuggling interdiction 11

right-hand-side values for each ω.) Optimality of vωk with

respect to the evader’s perception is ensured by Con-straints (15a)–(15c). As stated, model (16) is a nonlinearMIP due to the product of the binary variables xk and vω

k .This is easily linearized as captured in the following linearMIP:

minx,θ,θ2,v,y,z

∑ω∈�

pωθω,

subject to x ∈ X,

θω ≥ γ ωk pkyω

k , k ∈ AD2ω, ω ∈ �,

(17a)θω ≥ γ ω

k qkzωk , k ∈ AD2ω, ω ∈ �,

(17b)θ2ω ≥ r2ω

k (1 − xk), k ∈ AD2ω, ω ∈ �,

(17c)∑k∈AD2ω

vωk = 1, vω

k ≥ 0, k ∈ AD2ω, ω ∈ �,

(17d)

θ2ω =∑

k∈AD2ω

r2ωk yω

k , ω ∈ �,

(17e)yω

k ≤ vωk , k ∈ AD2ω, ω ∈ �,

(17f)0 ≤ yω

k ≤ 1 − xk, k ∈ AD2ω, ω ∈ �,

(17g)zω

k ≥ vωk + xk − 1, k ∈ AD2ω, ω ∈ �.

(17h)

Constraints (17a) and (17b) are equivalent to con-straints (16a) and (16b) under the linearization of (1 −xk)vω

k by yωk via constraints (17f) and (17g) and the lin-

earization of xkvωk by zω

k in constraints (17h). The optimalityconditions from Constraints (15a)–(15c) for the smuggler’sbehavior are captured in Constraints (17c)–(17e) and againinclude the linearization of (1 − xk)vω

k by yωk .

We have assumed the evader’s response in BiPSNIP isunique for each x ∈ X , and in this case there is no ambigu-ity in how the evader’s behavior is being modeled. However,consider model (17) when this is not the case. Due to thesimultaneous minimization over all the decision variableswith respect to the interdictor’s objective function, amongthe optimal responses of the evader (separately, for eachω ∈ �), the decision most beneficial to the interdictor is se-lected. So, when indifferent with respect to two paths (underthe evader’s perception) the evader effectively cooperateswith the interdictor. Now, this might happen assuming thesmuggler is unaware of the interdictor’s perception, but evenin this case it is clearly an optimistic assumption from the in-terdictor’s point of view. An alternative formulation whichinvolves introducing a nested “minx,θ maxθ2,v,y,z” could beformulated to yield the corresponding uncooperative orpessimistic solution, but we do not pursue this here. Thisissue is not specific to the interdiction setting, and suchmethods for dealing with nonunique responses have been

investigated in the bilevel programming literature (Bard,1991).

5. Step inequalities for BiSNIP

Our initial attempts to solve the BiSNIP model (12) usingbranch-and-bound (B&B) codes indicated that BiSNIP’sLP relaxation can be loose. Hence, we seek to tighten theformulation with a class of valid inequalities that we termstep inequalities. In this section, we develop the inequalitiesand explain them from an intuitive perspective. Then, wedescribe a separation procedure that, given a solution toan LP relaxation of BiSNIP, can efficiently identify a mostviolated step inequality or prove that there are no violatedinequalities. We report our computational experience oniteratively generating step inequalities using this separationprocedure to help solve BiSNIP.

Consider the BiSNIP model (12), and let T(ω) ={k1, k2, . . . , k�} ⊆ ADω satisfy:

rωk1

≥ rωk2

≥ · · · ≥ rωk�

> 0. (18)

We define a step inequality on T(ω) as:

θ̄ω ≥ rωk1

− (rω

k1− rω

k2

)xk1 − · · · − (

rωk�

− 0)xk�

. (19)

When � = 1, the one-step inequality given by Inequal-ity (19) is simply an existing constraint in model (12). Ingeneral, when � ≥ 2 the step inequalities are not redundant,at least when x takes on fractional values in the convex hullof X . Consider a two-step inequality, i.e., Equation (19)with � = 2: θ̄ω ≥ rω

k1− (rω

k1− rω

k2)xk1 − (rω

k2− 0)xk2 . If xk1 =

xk2 = 0 then the smuggler will select k1 and θ̄ω = rωk1

by thetwo-step inequality. If xk1 = 1 then the two rω

k1terms in the

step inequality cancel and θ̄ω “steps down” to the resid-ual term rω

k2(1 − xk2 ) so that the two-step inequality col-

lapses to a one-step inequality, i.e., an existing constraintin model (12). More generally, the step inequality exploitsthe ordering among different evasion paths and effectivelyreduces the maximal evasion probability one step a timeas further sensors are installed. Of course, the value of thestep inequality is that it removes fractional solutions thatare otherwise feasible to the LP relaxation of BiSNIP. Thiscan be viewed by continuously increasing xk1 from zero toone and then continuously increasing xk2 from zero to oneand so on.

There are an exponential number of step inequalities,and adding all possible step inequalities to BiSNIP is outof the question. So, we instead iteratively solve the linearprogramming relaxation of BiSNIP and add step inequali-ties on an as-needed basis. This procedure is repeated untilno violated step inequalities remain. The separation problemfor step inequalities requires that given (xlp, θ̄ lp), a feasiblesolution to the LP relaxation of model (12), we either iden-tify a most violated step inequality for each ω or determinethat none are violated. We restrict the set T(ω) defining thestep inequality to include k1, where rω

k1= maxk∈ADω rω

k , and

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12 Morton et al.

to satisfy the ordering condition given by Condition (18).Then, to maximize the right-hand side of Inequality (19)over such T(ω) ⊂ ADω we solve:

νω = minT(ω)⊂ADω

∑ki∈T(ω)

(rω

ki− rω

ki+1

)xlp

ki, (20)

for each ω ∈ �, where rωk�+1

≡ 0. We solve model (20) effi-ciently by solving a shortest-path problem on an acyclicnetwork G(V, E) in which V = ADω ∪ {k|ADω|+1} withrω

k|ADω |+1≡ 0. The edge set E contains a directed arc from

node ki to kj only if rωki

≤ rωkj

; the associated arc length is(rω

kj− rω

ki)xlp

kj. We solve the shortest-path problem from node

k|ADω|+1 to node k1 over G(V, E) and obtain optimal valueνω. The nodes from V on a shortest path define T∗(ω) solv-ing Equation (20). If θ̄ω,lp ≥ rω

k1− νω then there are no vi-

olated step inequalities for ω at (xlp, θ̄ lp). Otherwise, thecheckpoints T∗(ω) define a most violated step inequalityfor ω.

Table 1 shows our computational results for a test prob-lem with 85 origins, 263 customs checkpoints, nine des-tinations, and |�| = 306 scenarios. We assume cij = 1 forall (i, j) ∈ AD, so that the budget constraint is simply acardinality constraint, and we solve our test problem forvarious values of the budget b on a 1.7 GHz, Dell Xeondual-processor machine with 2 GB of memory. In our testproblem, the average cardinality of ADω is about 20. Theseparation procedure for the step inequalities was coded inC++ and the MIP problems were solved with the CPLEX

Concert Technology libraries (version 9.0).All MIPs were solved with a relative tolerance of 0.1%,

i.e., the B&B algorithm was terminated when 100 × (z̄ −z)/z ≤ 0.1. Here, z̄ is the objective function value of the

Table 1. Computational results for: (i) solving model (12) directlyusing CPLEX: and (ii) iteratively adding violated step inequali-ties to the initial LP relaxation and then proceeding with B&B.“Rel. gap (%)” is 100 × (zIP − zLP)/zLP, “Comp. time” reports to-tal computation time in seconds, “no. of ≥” reports the number ofstep inequalities generated and “Iters.” reports the total numberof major iterations

CPLEX B&B With step inequalities

Rel. gap. Rel. gap. No. ofb (%) Comp. time (%) Comp. time ≥ Iters.

30 20.2 489 0.02 8 465 540 20.9 559 0.00 6 405 450 22.8 1913 0.00 11 657 860 25.7 7779 0.33 17 651 1770 27.6 35428 1.02 98 639 780 27.7 13251 0.00 10 489 690 28.7 10130 0.28 15 530 5

100 29.0 10923 0.34 16 522 6110 28.0 3215 0.19 13 594 8120 26.4 256 0.45 14 419 8

B&B algorithm’s current incumbent solution, and z is theB&B’s current lower bound on the MIP’s optimal value.We use zIP to denote z̄ when B&B terminates, and we usezLP to denote the optimal value of the initial LP relaxation,i.e., z at the beginning of B&B. In Table 1, the zLP valuesused to compute the “Rel. gap” are those of model (12)’sLP relaxation under the “CPLEX B&B ” heading and thoseof the same LP relaxation after the addition of the step in-equalities under the “With step inequalities” heading. Weiteratively added step inequalities in a sequence of “ma-jor iterations” until the maximum violation was less than10−6. The computation times reported under “With stepinequalities” include the time to generate the step inequal-ities and the time to solve the resulting MIPs. As Table 1shows, the use of step inequalities can considerably tightenthe optimal value of the initial LP relaxation, and our resultssuggest that in turn, this can significantly reduce requiredcomputational effort, particularly on the most challenginginstances.

We close this section by noting that in Pan (2005), thestep inequalities described here have been extended, withcomputational success, to the SNIP model on a generalnetwork. However, to do so first requires detailed devel-opment of an L-Shaped decomposition method for SNIP,which is beyond the scope of this paper.

6. Conclusions

We have described two types of stochastic network inter-diction models whose solution can be used to select sitesto install sensors for detecting smuggled nuclear material.This work is motivated by the US Department of Energy’sSecond Line of Defense Program. In both models, our goalis to minimize the probability that a smuggler can success-fully travel through an underlying transportation networkundetected. Our two models differ with respect to whetherthe interdictor and smuggler have the same (SNIP) or dif-fering (PSNIP) perceptions of the detection probabilitieson the network’s arcs. In SNIP, the smuggler is aware of thelocations of all sensors that have been installed. PSNIP cancapture the case in which the smuggler is aware of only asubset of the sensor locations. For both SNIP and PSNIPwe developed the special case of the model in which sen-sors can only be installed at border crossings of a singlecountry, namely Russia. Our experience with the formerspecial case, BiSNIP, indicates significant computationalbenefit from using a new class of valid inequalities calledstep inequalities.

Acknowledgements

This research was partially supported by the National Sci-ence Foundation under grants DMI-0217927 and DMI-0228419. The authors thank Bill Charlton of Texas A&M

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Models for nuclear smuggling interdiction 13

University, Steven M. Dinehart of the Los Alamos NationalLaboratory SLD team, and Bruce Pentola of the Depart-ment of Energy’s National Nuclear Security Administra-tion (NA-265). We are also grateful to three anonymous ref-erees and an Associate Editor whose suggestions improvedthe paper.

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Biographies

David Morton is an Associate Professor in the Graduate Program inOperations Research and Industrial Engineering at The University ofTexas at Austin. He graduated from Stetson University with a B.S.in Mathematics and Physics and from Stanford University with anM.S. and Ph.D. in Operations Research. His research interests includestochastic optimization of complex systems that contain significantuncertainties.

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Feng Pan is a technical staff member in the Risk Analysis and Deci-sion Support Systems Group in Los Alamos National Laboratory. Hegraduated from the University of Wisconsin at Madison with a B.S. inMathematics and from the University of Texas at Austin with a Ph.D.in Operations Research and Industrial Engineering. His research inter-ests are in stochastic programming and include developing applications,algorithms and theory.

Kevin Saeger is the group leader of the Risk Analysis and Decision Sup-port Systems Group in Los Alamos National Laboratory. He graduatedfrom Tri-State University with a B.S. in Aerospace Engineering and fromThe Massachusetts Institute of Technology with a Ph.D. in Aeronauticsand Astronautics. His research interests are in stochastic programmingand robust optimization techniques for risk reduction in socio-technicalsystems.

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