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Evasive Flow Capture: Optimal Location of

Weigh-in-Motion Systems, Tollbooths, and Security

Checkpoints

Nikola Markovi´

c

Department of Civil and Environmental Engineering, University of Maryland, College Park, Maryland, USA

Ilya O. Ryzhov

Department of Decision, Operations, and Information Technologies, Robert H. Smith School of Business,

University of Maryland, College Park, Maryland, USA

Paul Schonfeld

Department of Civil and Environmental Engineering, University of Maryland, College Park, Maryland, USA

The ﬂow-capturing problem (FCP) consists of locating

facilities to maximize the number of ﬂow-based cus-

tomers that encounter at least one of these facilities

along their predetermined travel paths. The FCP liter-

ature assumes that if a facility is located along (or

“close enough” to) a predetermined path of a ﬂow of

customers, that ﬂow is considered captured. However,

existing models for the FCP do not consider targeted

users who behave noncooperatively by changing their

travel paths to avoid ﬁxed facilities. Examples of facil-

ities that targeted subjects may have an incentive to

avoid include weigh-in-motion stations used to detect

and ﬁne overweight trucks, tollbooths, and security

and safety checkpoints. This article introduces a new

type of ﬂow-capturing model, called the “evasive ﬂow-

capturing problem” (EFCP), which generalizes the FCP

and has relevant applications in transportation, revenue

management, and security and safety management. We

formulate deterministic and stochastic versions of the

EFCP,analyzetheir structural properties, studyexactand

approximate solution techniques, and show an applica-

tion to a real-world transportation network. © 2014 Wiley

Periodicals, Inc. NETWORKS, Vol. 000(00), 000–000 2014

Keywords: ﬂow capture; facility location; noncooperative users;

checkpoints; tollbooths; inspection; network ﬂows; stochastic ﬂows

1. INTRODUCTION

Theﬂow-capturingproblem(FCP) is an importantclass of

network facility location models, in which demand is deﬁned

in terms of ﬂows of customers traveling between their ori-

gin and destination nodes. The objective of the FCP is to

locate a given number of facilities to maximize the number

Received February 2014; accepted October 2014

Correspondence to: N. Markovi´

c; e-mail: nikola@umd.edu

Contract grant sponsor: NSF; Contract grant number: 1335416

DOI 10.1002/net.21581

Published online in Wiley Online Library (wileyonlinelibrary.com).

©2014 Wiley Periodicals, Inc.

of ﬂow-based customers who encounter at least one facility

on their preplanned travel paths. The FCP was independently

introduced by Hodgson [25] and Berman et al. [11], and has

been extensively studied within operations research, various

areasof engineering, economics, and geography.Someofthe

applications of the original FCP and its variants included the

optimal location of bank ATMs [11], vehicle inspection sta-

tions [23, 27], trafﬁc counting points [53], rail park-and-ride

facilities [28], and alternative-fuel stations [33, 34].

Existing FCP models assume that, if a facility is located

along (or “close enough” to) a predetermined path of a ﬂow

of customers, then that ﬂow is considered captured. The liter-

ature on ﬂow capture does acknowledge that implementation

ofcertainﬁxedfacilitiescould encourage the targeted users to

avoid them by changing their travel paths. For example, Mir-

chandanietal.[38]arguethattruckerstransportinghazardous

materials may ﬁnd out or guess the locations of inspection

stations and try to avoid them by changing their routes. How-

ever, existing models for the FCP are not able to handle such

noncooperative behavior.

We address the problem of locating facilities that tar-

geted ﬂows may have an incentive to evade by changing their

travel paths. Examples of such facilities include the weigh-

in-motion (WIM) stations that are used to detect and ﬁne

overweight trucks, tollbooths, and security and safety check-

points. In this article, we introduce a new model, called the

“evasive ﬂow-capturing problem” (EFCP), which assumes

that a ﬂow can travel along multiple paths as long as the

detour is not too large, and that a targeted ﬂow chooses to

travel along the shortest path not covered by a facility.

We make the following contributions:

1. We introduce and mathematically formulate the EFCP,

which has broad applications in transportation, revenue

management, and security and safety management. One

consequence of noncooperative behavior is that any

NETWORKS—2014—DOI 10.1002/net

solution always incurs greater (or equal) costs under the

EFCP objective than the FCP objective.

2. We prove several properties which show that EFCP is

structurally different from FCP, and generally indicate

that there is signiﬁcant value in solving the mixed-

integer formulation where possible, rather than relying

onheuristics.Forexample,weshowthatagreedyheuris-

tic for EFCP can perform arbitrarily poorly, in marked

contrast to FCP, where it is widely used and enjoys the-

oretical performance guarantees. At the same time, we

prove that a partial linear relaxation of EFCP will always

yieldanoptimalsolution,thus,considerablyreducing the

computational effort needed to solve the mixed-integer

program.

3. We extend the deterministic EFCP to account for ﬂows

whose intensities and degrees of evasiveness are uncer-

tain. We show, under certain independence assumptions,

that evasiveness drives the value of the stochastic solu-

tion, and we identify structural properties that make the

stochastic EFCP more computationally tractable.

4. We show an application of EFCP to a relevant real-world

problem of allocating WIM stations. Numerical results

are conducted on an actual transportation network and

include realistically estimated inputs. In addition, we

contrast EFCP and FCP through numerical experiments,

anddemonstratethatsolutionsoptimalforFCPdo poorly

whentargetedsubjectstrytoavoidthefacilities,showing

that EFCP adds considerable value.

We proceed as follows. In section 2, we review the lit-

erature on FCP. Section 3 describes our main motivating

application of EFCP and states the assumptions. In sections

4 and 5, we formulate and analyze deterministic and stochas-

tic EFCP. Section 6 provides a realistic case study where

we illustrate an application of EFCP on a road network of

Nevada designated for large commercial vehicles. In section

7, we show additional numerical experiments on simulated

problems. Section 8 draws conclusions and discusses other

straightforward applications of the proposed models.

2. LITERATURE REVIEW

Many FCPs were proposed since the FCP was ﬁrst intro-

duced. Below, we summarize the characteristics of various

FCPs found in the literature. Different aspects of these

problems include:

1. Deviations from preplanned trips where a ﬂow is consid-

ered captured not only if a facility (e.g., gas station and

restaurant) is located along the predetermined path of a

ﬂow, but also in its relative proximity [7, 31].

2. Limited capacity of the facilities [5, 50], as well as

decisions about the size of facilities [49].

3. Temporal aspects such as time spent in a facility [5],

determining service start times [48], and multiperiod

planning where decisions about the facility locations are

made over several years [18].

4. Multiple counting of consumers in which the level of

consumption depends on the number of facilities (e.g.,

billboards)thatcustomersencounter[3],andconsumers’

preference for obtaining a service at the beginning,

middle, or end of their trips [56].

5. Probabilisticinformationaboutthetravelorigins,turning

movements to visit facilities, and customer arrival and

service rates [5, 9, 10, 42].

6. Competition between facilities that have the same or

different owners [8, 52].

7. Synthesis with demand coverage, where ﬂow cap-

ture (e.g., intercepting customers along their trips) is

addressed jointly with covering ﬁxed customers residing

at nodes [6, 26].

The introduction of the FCP and its variants also ini-

tiated work seeking more efﬁcient problem formulations

[16, 31, 39, 51, 55], as well as developing exact and approx-

imate solution techniques [23, 24, 36] for efﬁciently solving

realisticproblem instances. Further information about30 dif-

ferentFCPs is providedby[55]. Despite thislargebody of lit-

erature, previous models on ﬂow capture have not accounted

for the noncooperative behavior of ﬂows, which naturally

arises in applications where targeted ﬂows have an incentive

to avoid the facilities. Note that this represents the ﬂip side of

the common deviation FCP in which facilities are located to

enabletheﬂowstodeviatefromtheirshortestpathstotrytobe

serviced by the facilities. With the noncooperative ﬂows, the

opposite is the case, and facilities should be located to make

it hard to deviate in such a way as to avoid being captured.

In this article, we focus on the problem of optimally allo-

cating WIM facilities on road networks, an important appli-

cation for road infrastructure preservation and maintenance.

However, much like the FCPs described above, the EFCP

concept encompasses many variants that could include dif-

ferent objectives (e.g., cost minimization in WIM allocation,

proﬁt maximization in tollbooth allocation, or risk minimiza-

tion in locating security and safety checkpoints), constraints,

temporal aspects, and treatment of information. We believe

that the EFCP, like the FCP, possesses broad applicability.

3. BACKGROUND ON WIM ALLOCATION

Truckers overload their vehicles to increase their produc-

tivity and thus their proﬁts. However, these extra proﬁts for

the truckers come at the expense of severe pavement and

environmental damages, whose costs are passed to soci-

ety as a whole. The total damage due to overweight trucks

costs taxpayers millions of dollars every year in maintenance

and rehabilitation. For example, only the pavement damage

attributed to overweight trucks in California was roughly

estimated at $20–$30 million per year [44]. An effective

way of reducing this damage is to implement WIM systems,

designed to detect overweight trucks (Fig. 1). As a truck

drives over a WIM scale, the category of truck, axle weights,

velocity, and other data are recorded and stored by the WIM

system. The information gathered by a WIM system can be

associated with the truck license plate and registration num-

ber through the use of high-speed cameras. These data can

then be transmitted to the weight-enforcing authorities and

trucks violating weight restrictions can be cited [44]. The

2 NETWORKS—2014—DOI 10.1002/net

FIG. 1. Real-time image data are monitored on a computer in a ﬁxed facility or a vehicle. When a suspect truck

is identiﬁed, an enforcement unit can intercept and weigh the truck to conﬁrm the violation. [Color ﬁgure can be

viewed in the online issue, which is available at wileyonlinelibrary.com.]

WIM stations are uncapacitated and collect data 24/7, mak-

ing them much more efﬁcient than static weigh stations that

may have limited hours of operation and where considerable

queuing delays may occur.

WIM technology is expensive and cannot be implemented

on every road link. Recent implementations of WIM check-

points reveal that their location in a road network is deter-

mined by prioritizing the most damaged road links. Such an

approach was taken in Montana, where ofﬁcials reported an

estimatedreduction of annual pavementdamageby$700,000

[47]. This intuitive approach toward allocating WIM systems

canbeimprovedbydevelopingmodelsthatoptimizetheloca-

tion of WIM checkpoints. Several such models are found

in the literature [1, 46], but they are built on the assump-

tion that trucks travel along the shortest paths from their

origins to their destinations and that locating WIM check-

points along trucks’ shortest paths sufﬁces to enforce weight

control. However, this simplifying assumption misrepresents

the real world, where truck drivers quickly learn the location

of checkpoints, communicate with other truckers,and start

avoiding the checkpoints by taking detours (see [20] for a

discussion of the empirical evidence). If this fact is ignored

in allocating WIM checkpoints, then the implementation of

WIM technology can potentially result in greater damage

due to additional vehicle-miles traveled. We call this phe-

nomenon, the WIM paradox [12] and show an example in

Figure 2.

In formulating the EFCP for WIM allocation, we make

four assumptions that are outlined below to clarify relations

incorporated in the mathematical formulations:

1. The damage produced by a truck ﬂow (i.e., a group of

trucks with the same origin and destination) increases

linearly with the distance traveled across the network.

This is clearly the case for pavement and environmental

damage. (This should not be confused with the nonlinear

relation between the weight of a vehicle and the per mile

damage it produces.)

2. A truck ﬂow fcan travel along kf-shortest paths from its

origin to destination. The number kfcan be determined

so that the (kf+1)th-shortest path would represent

an excessive detour for truckers (i.e., that the cost of

taking such a long detour would exceed the beneﬁt from

FIG. 2. WIM paradox: if there is a relatively small detour, trucks traveling from A to B will bypass the WIM

checkpoint and produce greater damage due to the longer distance traveled. [Color ﬁgure can be viewed in the

online issue, which is available at wileyonlinelibrary.com.]

NETWORKS—2014—DOI 10.1002/net 3

FIG. 3. Example of an excessive detour and WIM allocations that do (do not) capture ﬂow from A to B. [Color

ﬁgure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

overloading the truck). For example, kfcan be deter-

mined so that the (kf+1)th-shortest path is 30% longer

than the shortest path (Fig. 3).

3. A truck ﬂow is considered captured if at least one WIM

checkpoint is located along each of the kfpaths. There

is no excessive damage associated with captured ﬂows.

4. An uncaptured ﬂow travels along the shortest of its kf

paths that have not been covered by checkpoints because

that minimizes the truckers’ cost (Fig. 3).

Finally, it should be noted that WIMs are located on links,

whereas almost all models on the FCP locate facilities on

nodes. In practice, covering a node with WIM technology

would require deployment of WIMs along all inbound or all

outbound road links, which would be suboptimal in many

cases. Therefore, we propose a link-based formulation for

our EFCP.

4. DETERMINISTIC EFCP

Weintroduce the EFCPwhile assuming that all the param-

eters are known with certainty. First, we provide nonlinear

and linear formulations of the problem and verify that the

two formulations are equivalent. Second, we study relations

between EFCP and FCP. Third, we consider the structural

properties of the EFCP and contrast them with those of the

FCP. Fourth, we propose exact and approximate solution

methods to tackle this problem.

4.1. Problem Formulation

LetG(N,A)be a bidirectional road transportationnetwork,

whereNis a setof nodes andAis a setof arcs (i,j). Wedenote

byF, a setof truck ﬂowsanddeﬁne Pfas aset of paths,which

contains kf-shortest paths for the ﬂow f∈F. Let Ap

fbe the

set of arcs along path p∈Pfof ﬂow f∈F. Additionally, let

wij denote the cost of implementing and maintaining a WIM

checkpoint at arc (i, j) and let cp

fbe the excessive damage

cost incurred if ﬂow f∈Fpasses unintercepted along path

p∈Pf. Let xij be a binary variable equal to 1 if a WIM

checkpoint is located at arc (i, j) and 0 otherwise. Moreover,

we deﬁne x=xij|(i,j)∈Aand w=wij |(i,j)∈Aas

vectors of |A|elements.

We can now deﬁne the EFCP for WIM allocation as the

minimization problem

P1 : min

x∈{0,1}|A|wTx+Q(x)

where Q(x)is an oracle that, given an allocation of check-

points x, computes the cost of excessive damage associated

with uncaptured ﬂows. Since assumption 4 speciﬁes that

these ﬂows seek to minimize their travel distance, this cost

is computed by adding up the damage that uncaptured ﬂows

would produce by traveling along their shortest paths not

covered by WIMs. This can be done with a simple algorithm

which ﬁnds the shortest paths of uncaptured ﬂows such that

(i,j)∈Ap

fxij =0, and adds up the corresponding cp

fvalues.

Note that, given assumptions 1 and 4, trucks’ travel

distance minimization coincides with the minimization of

damage. This allows us to write problem P1 as a single-

level binary linear program. To do so, we introduce three sets

of auxiliary binary variables, which are used to (1) check

whethera ﬂow is captured and (2)direct the uncapturedﬂows

alongtheshortestunmonitoredpathswhile accounting for the

corresponding damage.

yp

f=⎧

⎪

⎨

⎪

⎩

1 if at least one WIM station is located

along path p∈Pfof ﬂow f∈F

0 otherwise

yf=⎧

⎪

⎨

⎪

⎩

1 if at least one WIM station is located

along all paths p∈Pfof ﬂow f∈F

0 otherwise

zp

f=⎧

⎪

⎨

⎪

⎩

1ifﬂowf∈Ftravels unintercepted

along path p∈Pf

0 otherwise

4 NETWORKS—2014—DOI 10.1002/net

Now we can formulate the WIM allocation problem as a

linear binary integer program:

P2 : min

xij,yp

f,yf,zp

f∈{0,1}

(i,j)∈A

xijwij +

f∈F

p∈Pf

zp

fcp

f(1)

s.t.

(i,j)∈Ap

f

xij ≥yp

f∀f∈F,p∈Pf(2)

zp

f≤1−yp

f∀f∈F,p∈Pf(3)

(i,j)∈Ap

f

xij ≤|Ap

f|·yp

f∀f∈F,p∈Pf(4)

yf≤yp

f∀f∈F,p∈Pf(5)

p∈Pf

zp

f≥1−yf∀f∈F(6)

Theobjective(1)minimizestheinvestmentcostandexces-

sive damage due to overweight trucks whose paths are not

all covered by at least one WIM station. Constraints (2)–(4)

ensure that, if at least one WIM is allocated along a path of a

ﬂow (yp

f=1), the ﬂow cannot pass unintercepted along that

path (zp

f=0). Constraint (5) guarantees that yfcan take a

value of 1 only if all the corresponding paths are covered by

atleast one WIM. Constraint (6)requires unintercepted ﬂows

to contribute excessive damage costs to the objective value.

The above linearization includes three sets of auxiliary

binary variables and ﬁve additional sets of constraints. The

following result demonstrates that the two formulations are

indeed equivalent. We present a full proof below; the tech-

nique, which is based on the separability of the second-stage

objective, will also be used in later proofs.

Proposition1. Problems P1 and P2 are equivalent.

Proof. Toprovethis,weneedtoshowthatQ(x)=¯

Q(x),

where

¯

Q(x)=min

yp

f,yf,zp

f∈{0,1}⎧

⎨

⎩

f∈F

p∈Pf

zp

fcp

fs.t. constraints (2)to (6)⎫

⎬

⎭

Note that the above minimization is separable in f, whence

¯

Q(x)=

f∈F

¯

Q(x,f)

where

¯

Q(x,f)=min

yp

f,yf,zp

f∈{0,1}

p∈Pf

zp

fcp

f

s.t. constraints (2)to (6)forﬁxed f

Next, we partition each set Pfinto sets P1

fsuch that

(i,j)∈Ap

fxij ≥1forp∈P1

f,andP2

fsuchthat(i,j)∈Ap

fxij =0

for p∈P2

f. It follows that

1. For p∈P1

f, constraints (4) and (3) imply yp

f=1 and

zp

f=0, respectively;

2. For p∈P2

f, constraints (2) and (3) imply yp

f=0 and

zp

f≤1, respectively.

Now, note that (5) is deﬁned over p∈Pf, and so is the

summation in (6). We can determine ¯

Q(x,f)depending on

whether the set P2

fis empty:

1. If P2

f=∅, then constraint (5) implies yf= 0 and con-

straint (6) is equivalent to p∈P2

fzp

f≥1. In this case, we

have

¯

Q(x,f)=min

zp

f∈{0,1}⎧

⎪

⎨

⎪

⎩

p∈P2

f

zp

fcp

fs.t.

p∈P2

f

zp

f≥1⎫

⎪

⎬

⎪

⎭

=min

p∈P2

fcp

f.

2. If P2

f=∅, then constraint (5) implies yf≤1 and con-

straint (6) is equivalent to p∈P1

fzp

f≥1−yf. Since

zp

f=0, for all p∈P1

f, (6) implies yf≥1. Hence, yf=1

and ¯

Q(x,f)=0.

The two cases can be summarized as

¯

Q(x,f)=⎧

⎨

⎩

min

p∈P2

fcp

f,P2

f=∅;

0, P2

f=∅.

Finally, recall that ¯

Q(x)=f∈F¯

Q(x,f)and note that

minp∈P2

fcp

ffor P2

f=∅corresponds to the shortest of the

paths not covered by WIMs. This is precisely the deﬁnition

of Q(x).■

4.2. Relation to FCP

Recallthat the FCP locates facilitiestomaximize the num-

ber of ﬂow-based customers that encounter these facilities

along their predetermined travel paths. Here we consider a

case with a variable number of facilities and note that max-

imizing a weighted sum of captured ﬂows is equivalent to

minimizing the weighted sum of uncaptured ﬂows (see [19]

for a similar argument regarding the maximal covering loca-

tion problem). Using our notation, we can formulate this

variant of FCP as

FCP: min

xij,yf∈{0,1}

(i,j)∈A

xijwij +

f∈F

(1−yf)cf

s.t.

(i,j)∈Af

xij ≥yf∀f∈F

whereAfisthe set of arcsalongthe single predetermined path

ofaﬂow,andxij and yfare as deﬁned earlier. (Referring to

the deﬁnition of yf, note that Pfis here a singleton containing

the predetermined path of a ﬂow.)

In the following part, we provide two propositions that

(1) demonstrate that P2 encompasses FCP, and (2) establish

NETWORKS—2014—DOI 10.1002/net 5

relationsbetweensolutionstoEFCP and FCP. The ﬁrst result

will be further used to analyze the computational complex-

ity of EFCP. The second result will be additionally illustrated

laterin numerical examples, whichshowthatallocations sug-

gestedby FCPdo poorly in asetting where ﬂowstry to evade

facilities. Due to space considerations, the proofs here and

throughout are moved to the Appendix.

Proposition2. For kf= 1, problem P2 reduces to FCP.

Proposition3. Considerthe FCPwhere ﬂows travelalong

their shortest paths and let FCP(x)denote the value of a

facility allocation xin this problem. Similarly, let EFCP(x)

denote the value of a facility allocation xin EFCP (prob-

lem P2). Then, EFCP(x)≥FCP(x)for any feasible x∈

{0,1}|A|.

4.3. Structural Properties

Since problem P1 represents minimization of a set func-

tion, it is relevant to know whether this set function is

submodular or supermodular. These properties are widely

studied in the FCP literature due to their implications for

computational tractability. On the one hand, submodular set

functions can be minimized in strongly polynomial time

[29, 45]. On the other hand, a simple greedy heuristic is

guaranteed to perform well when applied to minimization of

supermodular functions. This guarantee is extensively used

in the FCP literature and is stated below for completeness.

Before we proceed, recall that a set function his nondecreas-

ing, submodular, and supermodular if for all S⊂T⊂Nand

k/∈Twe have

1. nondecreasing: h(S)≤h(T)

2. submodular: h(T∪{k})−h(T)≤h(S∪{k})−h(S)

3. supermodular: h(T∪{k})−h(T)≥h(S∪{k})−h(S)

(i.e., – his submodular)

Theorem1 (Nemhauser et al. [40]).Consider the opti-

mization problem

Z∗=max

S⊂N,|S|≤mh(S).

Let ZGbe a value returned by the greedy heuristic that

sequentiallyselectselementsinNthatmyopicallyimprovethe

objective function. If h(S) is submodular and nondecreasing,

then

ZG

Z∗≥1−1−1

mm

≥1−1

e≈0.63.

Numerous papers on FCP show that the problem of locat-

ing mfacilities to maximize the weighted sum of captured

ﬂows can be expressed using the framework of Theorem 1

[3, 7, 8, 10, 23]. This result guarantees that a greedy heuris-

tic will quickly provide solutions for FCP that are within

37% of the optimum. Numerical comparison with exact

solution techniques, for example, branch and bound, shows

that a greedy algorithm performs exceptionally well yielding

optimal or near optimal solutions [7, Table 2].

However, although EFCP is closely related to FCP, it is a

substantially more complex problem. Our next result shows

that EFCP is neither submodular (thus, existing results on

polynomialcomplexitydonotapply)norsupermodular(thus,

a greedy heuristic is not guaranteed to perform well). In fact,

we show later that a greedy heuristic can perform arbitrarily

poorly for EFCP.

Proposition4. The objective function of problem P1 is

nonsubmodular, nonsupermodular, and nonmonotonic.

Proposition 4 indicates that standard solution approaches

for FCP are not guaranteed to work well in EFCP. We now

address the computational complexity of EFCP.

Proposition5. Problem P2 is NP-hard.

Note that problem P1 minimizes the total investment in

WIMs and excessive damage associated with unintercepted

overweighttrucks.Whilethisisareasonableeconomicobjec-

tive,mostworkon FCP considersa ﬁxed number of facilities,

and focuses on placing them to maximize the number of

capturedcustomers(whichisequivalenttominimizingexces-

sive damage associated with uncaptured ﬂows). Thus, we

also consider a variant of P1 whose objective function only

includes excessive damage, not the cost of implementing the

facilities. We denote this by

P1: min

x∈{0,1}|A|

(i,j)∈Axij≤m

Q(x).

It is straightforward to show that the structural properties

of P1 obtained in proposition 4 also hold for P1. Addition-

ally, for kf= 1, problem P1transforms into a classic FCP,

which is known to be NP-hard [11].

4.4. Solution Techniques

Formulation P2 represents a binary integer program,

which can be tackled by any mathematical programming

softwareusingbranch-and-bound-based algorithms. Here we

showthatthebinaryvariablesyfandzp

fcanbelinearlyrelaxed

withoutalteringthe optimal solution orthe valueof the objec-

tivefunction.Inour empirical study,we found that thispartial

linear relaxation typically reduces solution time for P2 by

10–15%. In addition, we propose a tighter formulation of P2,

which enables linear relaxation of all the variables except xij

at the cost of additional constraints. These results are sum-

marized in the following two theorems. Before we proceed,

it should be noted that relaxation of binary variables was also

explored within FCP [32, 55] as well as the closely related

maximal covering location problem [19].

Theorem2. Let EFCP1

LR denote a partial linear relaxation

of EFCP (problem P2), such that yf,zp

f≥0and xij,yp

f∈

6 NETWORKS—2014—DOI 10.1002/net

{0,1}. Moreover, let x∗

EFCP1

LR denote its optimal solution with

objective value EFCP1

LR(x∗

EFCP1

LR ). Then, x∗

EFCP1

LR =x∗

EFCP

and EFCP1

LR(x∗

EFCP1

LR )=EFCP(x∗

EFCP).

Remark1. The partial linear relaxation stated in Theorem

2 reduces the number of binary integer variables from |A|

xij

+

|F|

yf

+2·

f∈F

|Pf|

yp

f&zp

f

to |A|+f∈F|Pf|.

In the following result, we show that we can addition-

ally relax yp

f, provided that we tighten formulation P2.In

this case, however, the relaxation comes at the cost of addi-

tional constraints. Whether it will run faster than EFCP1

LR is

problem-dependent.

Theorem3. Let EFCP2

LR denote a partial linear relaxation

of EFCP (problem P2), such that

1. Constraints (4) are replaced with constraints

xij ≤yp

f∀f∈F,p∈Pf,(i,j)∈Ap

f(7)

2. Allauxiliaryvariablesarelinearlyrelaxedyp

f,yf,zp

f≥0,

whereas the facility location variables are kept binary

xij ∈{0,1}.

Moreover, let x∗

EFCP2

LR denote its optimal solution with

objective value EFCP2

LR(x∗

EFCP2

LR ). Then, x∗

EFCP2

LR =x∗

EFCP

and EFCP2

LR(x∗

EFCP2

LR )=EFCP(x∗

EFCP).

Remark2. The partial linear relaxation stated in Theorem

3 reduces the number of binary integer variables from |A|+

|F|+2·f∈F|Pf|to |A|. However, the total number of

constraints is increased from |F|

(6)

+4·

f∈F

|Pf|

(2)−(5)

to |F|+3·

f∈F|Pf|+f∈Fp∈Pf|Ap

f|.

Wealsoconsidertheperformanceofagreedyheuristicthat

introduces checkpoints at the best current locations as long

as the WIM implementation improves the objective function.

Recall that such heuristics are often used in the FCP, where

they can be guaranteed to perform within 37% of optimality.

However, in the EFCP, the greedy heuristic cannot be guar-

anteed to perform within any fraction of the optimal value.

Our numerical experiments in section 7 include cases where

the heuristic performs very poorly.

Proposition6. For any 0<ε<1, there exists an

instance of EFCP (problem P1) for which EFCP(x∗

EFCP)≤

ε.EFCP(xG), where xGrepresents the allocation of check-

points found by the greedy heuristic.

Moreover,we considerproblem P1andagreedyheuristic

that places a given number of facilities (e.g., mfacilities) in

the best current position, as in [7]. We show that a bound

cannot be determined for this greedy algorithm either. Our

numerical experiments in section 7 also include instances

where it performs poorly.

Proposition7. For any ε>0, there exists an instance

of EFCP (problem P1) for which EFCP(x∗

EFCP)≤

ε.EFCP(xG), where xGrepresents the allocation of check-

points found by the greedy heuristic.

5. STOCHASTIC EFCP

The EFCP proposed in section 4 represents an optimiza-

tion problem in which all the parameters are assumed to be

known with certainty. For example, the damage that a ﬂow

produces (i.e., the parameter cp

f) and its willingness to avoid

facilities (i.e., the size of the set Pfcontaining shortest paths)

are assumed to be known. However, in real-world appli-

cations, this information could be obtained through expert

opinion or data collection, which result in different estimates

or realizations of these parameters. To address the case when

cp

fandPfare not knownwith certainty,wepropose a stochas-

tic extension of EFCP and develop an efﬁcient formulation

of this problem.

It should be noted that per mile pavement and environ-

mental damages that a ﬂow produces vary with the number

and types of vehicles within a ﬂow, excessive loads, climate,

and weather. Conversely, the willingness to avoid facilities

may depend on both psychological and economic factors

(e.g., price of gasoline, driver’s hourly pay, trucker’s abil-

ity to overload the truck which depends on demand, the age

of the truck, tires, types of loads, and road conditions). In

our analysis, we will assume that these two parameters are

independent.

5.1. Problem Formulation

Suppose that we wish to allocate facilities, given ﬂows

whose intensities and willingness to evade facilities are ran-

dom. This problem ﬁts into the basic idea of two-stage

stochastic programming, where decisions are based on prob-

abilistic data available at the time the decisions are made

[14]. In our problem, we distinguish between two inde-

pendent sources of uncertainty. Let ξ=ξf|f∈Fbe a

vector of discrete random variables denoting unit intensities

of ﬂows f∈F(i.e., damage or risk produced per unit of

distance traveled). Similarly, let η=ηf|f∈Fbe a vec-

tor of discrete random variables denoting the willingness of

ﬂows to evade facilities. This quantity could be deﬁned as a

percentage by which drivers are willing to increase the dis-

tance traveled (e.g., 20% of the shortest path). A particular

realization of these random parameters will be denoted by

ω∈ω1,...,ωR. As a result, in the stochastic extension,

NETWORKS—2014—DOI 10.1002/net 7

we will have Pf(ω),yp

f(ω),yf(ω),zp

f(ω), and cp

f(ω), asso-

ciated with each realization. Recall that, by assumption 1, we

can write

cp

f(ω) =lp

f(ω)ξf(ω),

where lp

f(ω) is the predetermined length of path p∈Pf(ω).

The set Pf(ω) itself is determined by the realization ηf(ω),

independently of ξf(ω).

To streamline the presentation, we focus on a two-stage

stochastic EFCP with a ﬁxed number of facilities, a gener-

alization of problem P1presented earlier. This problem is

deﬁned as

SP1: min

x∈{0,1}|A|

EQ(x,ξ,η)

s.t.

(i,j)∈A

xij ≤m

where Q(x,ξ(ω),η(ω)) is again the oracle that, given an

allocation xof checkpoints, computes the excessive damage

associated with a particular realization of ξand η. Problem

SP1can be linearized similarly to P1; however, we ﬁrst

show that some of the randomness inherent to SP1can be

removed without altering the problem. In particular, we show

that, under independence assumptions, stochastic ﬂow inten-

sities(i.e., per mile damage orrisk) can be replaced with their

means while preserving the randomness associated with the

willingness of targeted subjects to evade the facilities. This

result reduces the noise and enables us to consider fewer

scenarios and thus solve the problem much more efﬁciently.

Theorem4. Suppose that ξand ηare independent, and let

ξ=E[ξ]denote the expected intensity of ﬂows (i.e., damage

or risk per distance traveled). Then the following holds:

min

x∈{0,1}|A|

(i,j)∈Axij≤m

EQ(x,ξ,η)=min

x∈{0,1}|A|

(i,j)∈Axij≤m

EQ(x,ξ,η).

Also, the two problems have the same optimal solution x∗.

Now, we linearize SP1while considering that some

of the paths remain the same for different realizations of

ηf. For each ﬂow f, let ˜ω1

f,...,˜ωR

fbe an ordering of the

realizations ω1,...,ωRsuch that ηf(˜ωr

f)≥ηf(˜ωr−1

f), and

thus Pf(˜ωr−1

f)⊆Pf(˜ωr

f). Essentially, for every ﬂow, we

are sorting the set of possible values for ηfin increasing

order. Next, we formulate the scenario-based constraints

recursively, while assuming for notational convenience that

Pf(˜ω0

f)=∅and yf(˜ω0

f)=1. We refer to this problem as

SP2and formulate it as follows:

First Stage:

min

xij∈{0,1}

E

f∈F

˜

Q(x,ξf,ηf,f)(8)

s.t.

(i,j)∈A

xij ≤m(9)

Second Stage:

˜

Q(x,ξf,ηf(˜ωr

f),f)=min

yp

f(˜ωr

f),yf(˜ωr

f)∈{0,1}

zp

f(˜ωr

f)∈{0,1}

p∈Pf(˜ωr

f)

zp

f(˜ωr

f)cp

f(˜ωr

f)

(10)

s.t.

(i,j)∈Ap

f(˜ωr

f)

xij ≥yp

f(˜ωr

f)∀p∈Pf(˜ωr

f)/Pf(˜ωr−1

f)(11)

zp

f(˜ωr

f)≤1−yp

f(˜ωr

f)∀p∈Pf(˜ωr

f)/Pf(˜ωr−1

f)(12)

(i,j)∈Ap

f(˜ωr

f)

xij ≤|Ap

f(˜ωr

f)|·yp

f(˜ωr

f)∀p∈Pf(˜ωr

f)/Pf(˜ωr−1

f)

(13)

yf(˜ωr

f)≤yp

f(˜ωr

f)∀p∈Pf(˜ωr

f)/Pf(˜ωr−1

f)(14)

yf(˜ωr

f)≤yf(˜ωr−1

f)(15)

p∈∪r

r=1Pf(˜ωr

f)

zp

f(˜ωr

f)≥1−yf(˜ωr

f)(16)

Program (8)–(16) describes the same relations as (1)–(6),

but includes recursively deﬁned path-based constraints. In

this regard, the newly introduced constraint (15) ensures that

each ﬂow fcan be captured in the rth realization only if it is

alsocapturedin realization r−1, which includesfewerpaths.

5.2. Stochastic versus Deterministic EFCP

The following two remarks imply that, after we apply

Theorem 4 and formulate recursively the second stage, the

two-stage stochastic EFCP becomes only slightly more difﬁ-

cult than the deterministic EFCP with the largest realizations

of ηf.

Remark3. Let ηf(˜ωR

f)denote the largest realization of the

willingness of a ﬂow to avoid facilities. Then the two-stage

stochastic EFCP deﬁned with (8)–(16) includes

•|A|

xij

+|F|·R

yf(ω)

+2·

f∈F

|Pf(˜ωR

f)|

yp

f(ω) &zp

f(ω)

binary variables,

•1

(9)

+4·

f∈F

|Pf(˜ωR

f)|

(11)−(14)

+2·|F|·R

(15)−(16)

constraints.

Remark4. Consider (8)–(16) given a single realization of

a ﬂow’s willingness to avoid facilities, ηf(˜ωR

f). This case rep-

resents a deterministic EFCP. In this setting, constraint (15)

becomes redundant, so the deterministic EFCP includes

•|A|

xij

+|F|

yf(ω)

+2·

f∈F

|Pf(˜ωR

f)|

yp

f(ω) &zp

f(ω)

binary variables,

•1

(9)

+4·

f∈F

|Pf(˜ωR

f)|

(11)−(14)

+|F|

(16)

constraints.

8 NETWORKS—2014—DOI 10.1002/net

The above remarks imply that the two-stage stochastic

EFCP includes more ﬂow-based variables and constraints

[i.e., yf(ω) and (15)–(16)]. However, the number of vari-

ables of type zf

p(ω) and yf

p(ω), as well as constraints of type

(11)–(14), is the same in the stochastic EFCP as in the deter-

ministic problem that uses ˜ωR

fas the sole realization for

ﬂow f. Furthermore, because the EFCP formulation is ﬂow-

separable, we can generate variables and constraints for each

ﬂow individually. Thus, for a ﬁxed f, we only need Mf≤R

variables of type yf(ω), where Mfis the number of possible

values that ηfcan take. Overall, this reduces the size of the

problem, since, for each ﬁxed ﬂow f, we only need to add

one yf(ω) variable for multiple scenarios in which ηfhas

the same value. This makes the two-stage stochastic EFCP

only slightly more difﬁcult than the deterministic problem,

provided that Theorem 4 and recursive formulation of the

second stage are applied.

In our experiments (further described in section 7), we

found that the recursive formulation makes the problem con-

siderablyeasiertosolveinextensiveformusingmathematical

programming software. We brieﬂy note that it can also be

solved using, for example, the integer L-shaped method [35],

as it can be shown that the optimality conditions for this

method are satisﬁed by the stochastic EFCP. However, in our

experience, it was much more efﬁcient to simply solve the

recursive formulation in extensive form, using the partial lin-

ear relaxations stated in Theorems 2 and 3. These relaxations

apply to the stochastic EFCP, where they relax even more

variables than in the deterministic model.

5.3. The Value of the Stochastic Solution

Let VSS denote the value of the stochastic solution, which

represents the beneﬁt from solving the two-stage stochastic

EFCP over solving its deterministic counterpart in which all

random parameters are replaced with their expected values

[13]. Recalling formulation SP1and Theorem 4, we can

formally deﬁne

VSS =EQ(x,ξ,η)−EQ(x∗,ξ,η),

where

x∗=argmin

x∈{0,1}|A|

(i,j)∈Axij≤m

EQ(x,ξ,η);

x=argmin

x∈{0,1}|A|

(i,j)∈Axij≤m

Q(x,ξ,η).

In the following proposition, we show that one can design

an instance of EFCP with an arbitrarily large VSS.

Proposition8. Foranyﬁnite ε>0, thereexists an instance

of the two-stage stochastic EFCP for which VSS >ε.

Remark5. Given the network topology and willingness of

ﬂows to evade facilities, VSS =0 if realizations of ηare

such that |Pf(ω)|=1 for all ω∈. This result follows from

the deﬁnition of VSS, as the observation that EQ(x,ξ,η)=

Q(x,ξ,η)when |Pf(ω)|=1 for all ω∈. It follows that

VSS is always zero in FCPs with random intensities.

6. CASE STUDY FOR DETERMINISTIC EFCP

Inthissection,weprovideacasestudyontheroadnetwork

of Nevada, including roads designated for large commercial

vehicles, truck ﬂows simulated based on data available in

the literature, and realistically estimated damage produced

by overweight trucks. We use this illustrative case study to

demonstrate the potential value of EFCP, compared to both

FCP and a real-world facility implementation, in a realis-

tic problem. We begin by explaining how we estimate the

excessive damage and then discuss other inputs.

6.1. Excessive Damage Estimation

Inthissection,weestimatetheparametercp

fwhichdenotes

the excessive damage cost if ﬂow f∈Fpasses unintercepted

along path p∈Pf. We have already argued that overweight

trucks damage the pavement and environment. Thus, we esti-

mate cp

fby roughly computing the aforementioned damage

costs associated with loads that exceed legal limits.

Pavementdamagedependsonmanyfactors including axle

weights, axle conﬁguration, pavement structure, and climate.

Since detailed information about the pavement structure and

climate may not be available for the entire transportation

network, the pavement damage can be estimated based on

equivalent single axle loads (ESAL). This method allows dif-

ferent axle types (single, tandem, and tridem) to be summed

together and is widely used in pavement design since it pro-

videsareasonablyaccurateindicatorofthepavementdamage

[44]. ESAL may be estimated with the formula

ESAL =α(W/α)

80 4.2 (17)

where αis the number of individual axles in an axle group

(for steering and single α= 1; for tandem α= 2; for tridem α

=3)andWis the weight of an axle [kN]. In computing the

excessivepavementdamage,the followingaxle loads [44] are

usedas legal limits forfour different axle groups:steering (55

kN, 0.21 ESAL), single (88 kN, 1.49 ESAL), tandem (151

kN, 1.57 ESAL), and tridem (233 kN, 2.65 ESAL).

Table 1 provides an example of how we compute the

excessive pavement damage associated with a 17 t truck that

has front steering and rear single axle. The assumed gross

truck weight distribution is 38:62 between the front steer-

ing and rear single axle, the same as the maximum axle load

ratio in kN (e.g., 55:88). In particular, Table 1 provides the

axle weights in kN and corresponding ESAL computed with

Equation (17). The obtained ESAL are compared with the

limits to get the excessive ESAL. Finally, the excessive pave-

ment damage of 6.12 cents per mile for this particular truck is

computed assuming the fee of 4 cents per ESAL-mile, which

is adjusted for inﬂation from [2].

NETWORKS—2014—DOI 10.1002/net 9

TABLE 1. Computing excessive pavement damage: An example.

Weight (kN) ESAL ESAL Limits Excess ESAL Total Excess ESAL Excessive Pavement Damage (cent/mi)

Steering 64.12 0.39 0.21 0.18 0.18 + 1.35 = 1.53 1.53 ×4=6.12

Single 102.59 2.84 1.49 1.35

Environmental damage includes accidents (fatalities,

injuries, and property damage), emissions (air pollution and

greenhouse gases), noise, and unrecovered costs associated

with the provision, operation, and maintenance of public

facilities[22]. Weassume the averageenvironmentaldamage

cost of 1.53 cents per ton-mile, which is adjusted for inﬂation

from[22]. Thus, assuming that thetruck from Table 1 is over-

loaded by 2.7 t, the corresponding excessive environmental

damage is 4.13 cents per mile.

6.2. Road Network, Flows, and Other Inputs

The proposed model is tested on the road network of

Nevada. We consider road links that are state designated for

surface transportation assistance act (STAA) vehicles. Since

manyof the observed roadlinks are nonseparated,we assume

that xij =xji as in an undirected graph. Hence, the road net-

work we observe includes 205 nodes and 221 edges, most

of which have 2 or 4 lanes. The relevant data are extracted

from Matlog [30], which contains the Oak Ridge National

Highway Network [43].

The truck ﬂows along three major transit routes are speci-

ﬁed based on data from the Federal Highway Administration

[21] and from [4]. They include 5,000 trucks/day on I-

15 (southwest of Reno—Salt Lake City) and I-80 (passing

through Las Vegas), as well as 2,000 trucks/day along the

route stretching from northwest of Reno to south of Las

Vegas. We also randomly generate 59 local truck ﬂows with

their origins and destinations at least 50 miles apart. More-

over, the number of trucks within the ﬂow is sampled from

a Poisson distribution with mean 50 trucks/day. We consider

10 types of trucks with different numbers and combinations

of axles. Table 2 provides truck weights, weight limits, and

the assumed percentages within the total ﬂow for each truck

type. Since truck weights are typically bimodally distributed

[47] due to imbalanced ﬂows, we simulate trucks assuming

that60% are traveling with heavy loads and40%are traveling

with light loads (e.g., empty or nearly empty trucks return-

ing to their origins). Discrete distributions of load weights in

tons are provided in Table 2. The expected number of over-

weight trucks generated based on the assumed inputs from

Table 2 is 4.5% of the total number of trucks. The percentage

is within the range reported in the literature, such as 2.6% for

California [44] and 8.8% for Montana [47].

Yen’s [54] k-shortest path algorithm is used to ﬁnd kf-

shortest loopless paths, such that the (kf+1)th-shortest path

is at least 20% longer than the shortest path. Thus, kfvaries

considerably with ﬂows. For example, kf= 5 for transit ﬂow

passingthrough Las Vegas,whereaskf=910for ﬂow travers-

ing Nevada east–west. The 59 local truck ﬂows are randomly

generated so that their origins and destinations are 50 miles

apart and the maximum number of paths that needs to be con-

sidered equals 30 (i.e., the kf= 30 shortest path is more than

20% longer than the shortest path).

A single set of ﬂows, trucks, and truck loads is gen-

erated using Monte Carlo simulation. The corresponding

excessive damage cp

fis computed for all the ﬂows and their

paths, as described in section 6.1. Finally, the WIM cost

includes the cost of hardware and software, implementation,

maintenance, recalibration, ofﬁce, and personnel. Available

references indicate that total cost can vary considerably

depending on the technology (e.g., sensors, cameras) and

location (e.g., state within the same country). We provide

numerical results for WIM cost ranging between $10,000

and $360,000 per lane per year. In our judgment, $60,000 per

lane-year is currently the most realistic cost, since the cost

TABLE 2. Parameters for simulating truck ﬂows.

Loads

Type Class (FHWA) Number of Axles Empty Truck Weight (t) Light (t) Heavy (t) Weight Limit (t) Percent of Total Flow

S 5 2 6 B(3, 0.45) B(15, 0.40) 14.3 9

S 6 3 8 B(4, 0.50) B(22, 0.45) 20.6 17

T 8 3 10 B(5, 0.45) B(25, 0.40) 23.1 3

T 8 4 13 B(6, 0.45) B(31, 0.40) 29.4 4

T 9 5 15 B(7, 0.45) B(39, 0.40) 35.7 46

T 10 6 16 B(9, 0.50) B(50, 0.45) 43.9 3

T 11 5 15 B(8, 0.50) B(46, 0.45) 40.7 7

MT 12 6 18 B(9, 0.50) B(53, 0.45) 47.0 3

MT 13 8 21 B(12, 0.50) B(68, 0.45) 59.0 4

MT 13 7 20 B(11, 0.50) B(59, 0.45) 52.7 4

S: single unit truck, T/MT: single/multitrailer truck, B(n,k): binomial distribution.

10 NETWORKS—2014—DOI 10.1002/net

TABLE 3. Optimal results for different WIM costs.

WIM Costs ($/lane-year) x∗

EFCP (links covered) WIM Systems ($/year) Excessive Damage ($/year) Total Cost ($/year) CPU Time (s)

10,000 30, 32, 93, 130, 216 140,000 26,947 166,947 4.23

60,000 32, 62, 93, 130 720,000 56,370 776,370 4.46

110,000 32, 93, 130 880,000 403,640 1,283,640 4.01

160,000 105, 164 960,000 681,633 1,641,633 4.15

210,000 105, 164 1,260,000 681,633 1,941,633 3.92

260,000 105, 164 1,560,000 681,633 2,241,633 4.08

310,000 105 620,000 1,723,607 2,343,607 4.03

360,000 no WIMs 0 2,349,907 2,349,907 4.00

of only WIM inroad equipment ranges between $7,000 and

$12,000per lane-year depending on thetechnology (adjusted

for inﬂation from [15]).

6.3. Results and Numerical Comparison of EFCP and

FCP

We implemented the binary program (1)–(6) in GAMS

23.5 and solved it using the GAMS/CPLEX solver for mixed

integer programs on a PC with an AMD Athlon 3300 GHz

processor with 4 GB of RAM. The optimal results for dif-

ferent WIM costs are provided in Table 3 and some of the

corresponding allocations x∗

EFCP are shown in Figure 4. To

simplify the comparison, the links in Table 3 and throughout

this section are denoted with tags (e.g., 1–221 for 221 road

links), rather than with their origin and destination nodes. All

the results are obtained within a few seconds of computation

time, as indicated in the last column of Table 3.

Now let us observe what would happen if we applied FCP

to determine the optimal allocation of WIM checkpoints.

First, we apply FCPwhere ﬂows travel along their short-

est paths and provide the optimal WIM allocation x∗

FCPwith

the corresponding objective function FCPx∗

FCP. Second,

we evaluate this solution for the EFCP where kfis deter-

mined so that the (kf+1)th-shortest path is at least 20%

longer than the shortest path. We denote this value by EFCP

x∗

FCPand contrast it with the optimal value EFCP x∗

EFCP.In

line with proposition 3, we see that the FCPobjective sig-

niﬁcantly underestimates the cost incurred by x

FCP due to

noncooperative behavior.

The last column in Table 4 indicates that the solution

obtained from FCPperforms poorly in the evasive setting.

The graphical comparison and dispersion of the uncaptured

ﬂows for the two solutions is provided in Figure 4. This com-

parison indicates that truck ﬂows simply bypass the facilities

allocated with FCP. For example, Figure 4e shows that ﬂow

traversingNevadaeast–westbypassesthe implemented facil-

ity at a small increase in travel distance. A similar situation

occurs in Figure 4c, but at a higher increase in driving dis-

tance that also includes greater excessive damage associated

with the same transit ﬂow.

Table 4 also clearly illustrates the WIM paradox, accord-

ing to which the inefﬁcient use of WIM technology actually

causes excessive damage (and total system cost) to increase.

In particular, the allocation x∗

FCPbased on the FCPincurs a

cost of approximately $2.9–$3.9 M/year for the WIM tech-

nology cost of $110–$360 k/lane-year. Conversely, Table 3

indicates a total cost of roughly $2.4 M/year when no WIM

technology is implemented. Hence, the FCPallocation is

counterproductive, and actually incurs greater total cost than

a solution that includes no WIMs at all. This clearly demon-

strates the potential pitfalls of using FCPin settings where

users behave noncooperatively.

6.4. Comparison of EFCP to the Real-World Solution

Here we look at the real-world implementation of static

weigh systems in Nevada and contrast the current locations

withthosesuggestedbyEFCPforWIMallocation.Weshould

note several grounds for caution in interpreting this compari-

son. First, locations of static weigh scales are more restricted

thanthoseofWIMsbecause static scales require considerable

land for their ramps and truck queues. Thus, the authori-

ties may have considered only a subset of the links that we

consider (e.g., only links that are further away from towns).

The reasons for this could be the land ownership and price,

or space availability. Second, we focus on road links that

are designated for STAA vehicles. Conversely, in allocating

static scales the authorities may have considered additional

roads (i.e., not only roads designated for STAA vehicles) as

potential bypasses. Third, the assumed intensities of truck

ﬂows are based on recent references, but the ﬂows may have

been different when the static weigh stations were originally

implemented. Furthermore, our experiment includes some

randomly simulated local truck ﬂows.

Since Nevada currently has three static stations [41], we

apply problem P1for m= 3 to minimize the excessive dam-

age. The real-world implementation and optimal solution for

EFCPareshowngraphicallyinFigure5 together with the cor-

responding excessive damage. The main difference between

the two solutions is due to (not) capturing the transit ﬂow

between northwest of Reno and south of Las Vegas (note that

the two checkpoints in Figure 5b are grouped together to cap-

ture this ﬂow). Thus, under the assumptions of the model, the

optimal solution for EFCP outperforms the real-world imple-

mentation by about $670,000/year. While the exact dollar

amount reﬂects the assumptions made in our experiments,

it suggests that there is signiﬁcant economic potential in

modeling evasive transportation ﬂows.

NETWORKS—2014—DOI 10.1002/net 11

FIG. 4. Comparison of uncaptured ﬂows given FCPand EFCP allocations. Solid lines denote uncaptured ﬂows

and their widths are proportional to damage. Dotted lines denote road links without overloaded trucks. [Color

ﬁgure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

12 NETWORKS—2014—DOI 10.1002/net

TABLE 4. Comparison of EFCP and FCPfor different WIM costs.

WIM Costs ($/lane-year) x∗

FCP(links covered) FCP(x∗

FCP)($/year) EFCP(x∗

FCP)($/year) EFCP(x∗

EFCP)($/year) EFCP(x∗

EFCP)

EFCP(x∗

FCP)

10,000 30, 37, 62, 130, 138 152,791 667,245 166,947 0.250

60,000 62, 107, 130 699,362 1,911,587 776,370 0.406

110,000 62, 154 1,102,999 2,874,007 1,283,640 0.447

160,000 154 1,367,532 3,094,203 1,641,633 0.530

210,000 154 1,567,532 3,294,203 1,941,633 0.589

260,000 154 1,767,532 3,494,203 2,241,633 0.641

310,000 154 1,967,532 3,694,203 2,343,607 0.634

360,000 154 2,167,532 3,894,203 2,349,907 0.603

The real-world solution suggests that practitioners, unlike

theFCPs,haveconsideredthatoverloadedtruckswould try to

evade the checkpoints, as they have placed them at locations

that cannot be avoided at a small increase in driving distance.

These locations include links close to border crossings and

other areas where the road network is not well connected. As

it happens, however, the optimal allocation for three stations

issomewhatcounter-intuitive,asitispreferabletoimplement

2 of 3 checkpoints very close together, instead of spreading

them out across the network. This case suggests that EFCP

can be a useful decision support tool with the potential to

yield better solutions than those based on human judgment

and intuition.

7. NUMERICAL EXPERIMENTS

We present additional numerical experiments for both the

deterministic and stochastic EFCP. These problems use sim-

ulated data and are intended to provide additional insight into

the performance of different solution techniques.

7.1. Deterministic EFCP

We consider a set of simulated problems, where the

random instances are based on the entire road networks of

Nevada and Vermont. We use 400 and 200 randomly simu-

lated ﬂows (respectively) as well as the following differently

speciﬁed values:

1. Willingness of ﬂows to avoid facilities (i.e., kfis deﬁned

so that the (kf+1)th-shortest path is 1.1 or 1.2 times

longer than the shortest path);

2. Cost of facilities for problem P1, or number mof

facilities for problem P1.

The numerical results are summarized in Table 5, which

indicates that the partial linear relaxation proposed in Theo-

rem 2 reduced the computation time in 47/64 cases. In 4/64

cases, it made no difference, and in 13/64 cases, it increased

the computation time. The average reduction, calculated over

all64 instances, was13%, while the median reduction among

64 instances was 11%. Table 5 also illustrates the results

of propositions 6 and 7: although the greedy heuristic often

performs well, and typically runs in less than 1 s, there are

probleminstanceswhere it performs extremely poorly.More-

over,theperformanceofthegreedyheuristicismuchworsein

problem P1. As in section 6, the FCPsolution also signiﬁ-

cantly underperforms in EFCP. We omit the running times

of the tighter formulation proposed in Theorem 3, as the

FIG. 5. Comparison of real-world locations of weigh stations with those suggested by EFCP. Solid lines denote

uncaptured ﬂows and their widths are proportional to damage. Dotted lines denote road links without overloaded

trucks. [Color ﬁgure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

NETWORKS—2014—DOI 10.1002/net 13

TABLE 5. Deterministic EFCP: Summary of results for Nevada and Vermont.

Problem P2 Problem P2

State Threshold WIM(k$) EFCP (s) EFCP1

LR (s) EFCP(x∗

EFCP)

EFCP(xgreedy)

EFCP(x∗

EFCP)

EFCP(x∗

FCP)mEFCP (s) EFCP1

LR (s) EFCP(x∗

EFCP)

EFCP(xgreedy)

EFCP(x∗

EFCP)

EFCP(x∗

FCP)

Nevada 1.1 10 34 24 0.86 0.75 2 65 34 0.95 0.79

60 9 8 0.89 0.94 4 101 28 0.65 0.49

110 15 9 0.87 0.99 5 21 16 0.59 0.35

160 13 13 0.86 0.99 7 52 35 0.71 0.61

210 17 14 0.84 0.99 8 45 43 0.64 0.51

260 20 23 0.91 0.84 10 20 20 0.59 0.35

310 32 34 0.97 0.88 11 28 28 0.72 0.74

360 7 7 1.00 0.93 13 22 19 0.58 0.19

1.2 10 507 410 0.77 0.60 2 481 388 0.95 0.49

60 779 517 0.68 0.46 4 762 421 0.49 0.17

110 353 297 0.73 0.57 5 505 514 0.44 0.12

160 772 475 0.76 0.66 7 553 670 0.39 0.52

210 537 405 0.83 0.72 8 719 371 0.36 0.33

260 336 298 0.89 0.61 10 542 577 0.37 0.24

310 340 301 0.95 0.61 11 445 437 0.43 0.64

360 273 259 1.00 0.83 13 462 390 0.52 0.20

Vermont 1.1 5 57 67 0.83 0.28 2 229 80 0.92 0.77

7.5 82 88 0.88 0.38 5 13 11 0.49 0.27

10 77 82 0.88 0.42 6 15 13 0.54 0.26

20 22 21 0.72 0.48 8 37 42 0.58 0.22

30 23 22 0.72 0.54 11 37 41 0.49 0.16

40 25 23 0.81 0.67 14 37 31 0.35 0.06

50 24 22 0.92 0.81 17 58 57 0.22 0.01

60 42 28 1.00 0.85 20 57 12 0.00 0.00

1.2 5 3,117 2,071 0.69 0.24 2 3,457 3,312 0.99 0.81

7.5 3,542 3,710 0.92 0.28 5 586 490 0.38 0.23

10 1,439 1,279 0.78 0.31 6 994 818 0.40 0.22

20 907 804 0.57 0.41 8 1,079 772 0.43 0.18

30 1,089 984 0.68 0.50 11 987 848 0.28 0.11

40 1,574 1,472 0.80 0.62 14 814 950 0.11 0.04

50 1,414 1,272 0.92 0.71 17 1,140 962 0.05 0.01

60 1,011 1,393 1.00 0.76 20 593 492 0.00 0.00

EFCP refers to solving problem P2 (i.e., a pure binary integer program) EFCP1

LR refers to solving the partial linear relaxation from Theorem 2 Threshold 1.1

or 1.2 implies that the (kf+1)th-shortest path is at least 1.1 or 1.2 times longer than the shortest path.

increased number of constraints led to slower computation

times for these problem instances.

7.2. Stochastic EFCP

We randomly simulate 200 ﬂows, all with the same

expected intensity of ξf=200 units/mile. Moreover, we

assume that ηfcan take values {1,1.1,1.2}with equal prob-

ability. The event that ηf(ω) =1 corresponds to the case

where the ﬂow only travels along its shortest path. The other

two cases represent a willingness to exceed the shortest path

by 10% and 20%, respectively.

We solve (1) the deterministic counterpart of the problem

(i.e.,ηf=1.1)and (2) the two-stagestochasticEFCP.Table6

contrasts xwith x∗for different values of m. Moreover, we

evaluate xover the three scenarios, EQ(x,ξ,η), and compute

VSS. We see that VSS>0in62%oftheproblem instances.

The last column of Table 6 indicates that the cost reduction

achievedbysolvingthestochasticEFCPis15.5% on average,

but it can be as high as 100%.

Table 7 evaluates several solution techniques. The partial

linear relaxation proposed in Theorem 2 reduced the

computation time in 34/37 instances, with the average

improvementoverall37instancesbeing15%,andthemedian

improvement being 19%. We have also argued previously

that the stochastic EFCP is only slightly more difﬁcult than

the deterministic problem for ηf(˜ωR

f), provided that the sec-

ondstage is formulatedrecursivelyand Theorem 4 is applied.

Table 7 indicates that computation times for the two cases are

fairly similar (SP1

LR vs. EFCP1

LR for ηf(ωR)). In 16 instances,

the stochastic EFCP took more time, in 13 instances the

stochastic EFCP took less time, and in 8 instances, the dif-

ference in solution times was within 1 s. On average over 37

instances,thedeterministicEFCPtook3%longer;themedian

ofthese values was0%. The Wilcoxonsigned-ranktest failed

to reject the null hypothesis of zero median in the difference

between the stochastic and deterministic EFCP at the default

5% signiﬁcance level (the computed P-value was 0.46).

We also implemented the integer L-shaped method [35]

and applied it to the same problem instances. However, this

method did not ﬁnd optimal solutions in the vast majority

of instances within the 4-h time limit, suggesting that it is

more efﬁcient to solve the extensive form of the stochastic

14 NETWORKS—2014—DOI 10.1002/net

TABLE 6. Stochastic EFCP: Computing VSS for the road networks of Nevada and Vermont.

State mxEQ(x,ξ,η) x∗EQ(x∗,ξ,η) 100·VSS

EQ(x,ξ,η)

Nevada 1 79 8,453,873 79 8,453,873 0.00

2 21, 33 4,755,020 21, 33 4,755,020 0.00

3 21, 33, 66 3,617,553 21, 33, 88 3,439,407 4.92

4 33, 62, 67, 79 2,118,247 33, 69, 77, 79 2,090,827 1.29

5 33, 62, 67, 79, 88 1,445,200 33, 68, 69, 79, 83 1,297,647 10.21

6 28, 33, 62, 67, 79, 88 1,224,200 28, 33, 68, 69, 79, 83 899,860 26.49

7 21, 28, 33, 62, 67, 84, 88 689,373 21, 28, 33, 66, 67, 84, 88 671,767 2.55

8 21, 28, 33, 62, 67, 84, 88, 103 509,300 21, 28, 33, 62, 67, 84, 88, 103 509,300 0.00

9 21, 29, 33, 42, 63, 67, 84, 88, 103 381,720 24, 28, 33, 63, 68, 79, 88, 93, 97 355,680 6.82

10 24, 29, 33, 38, 63, 68, 79, 88, 93, 97 256,540 24, 28, 33, 38, 63, 68, 79, 88, 93, 97 256,540 0.00

11 16, 21, 29, 33, 63, 64, 70, 81, 84, 88, 103 210,193 16, 21, 29, 33, 63, 64, 70, 81, 84, 88, 103 210,193 0.00

12 16, 20, 21, 29, 33, 63, 64, 77, 81, 84, 88, 103 175,127 16, 20, 22, 29, 33, 63, 64, 77, 81, 84, 88, 103 153,273 12.48

13 16, 20, 22, 29, 33, 63, 64, 70. 81, 84, 88, 103, 112 102,633 16, 20, 22, 29, 33, 63, 64, 70, 81, 84, 88, 103, 112 102,633 0.00

14 16, 20, 22, 29, 33, 38, 63, 64, 77, 81, 84, 88, 103, 112 74,233 16, 20, 22, 29, 33, 63, 64, 70, 81, 84, 88, 101, 104, 112 71,280 3.98

15 16, 20, 21, 29, 33, 38, 63, 64, 77, 81, 84, 88, 95, 103, 112 60,153 16, 20, 22, 29, 33, 38, 63, 64, 70, 81, 84, 88, 101, 104, 112 42,880 28.72

16 14, 20, 22, 29, 33, 38, 52, 63, 70, 79, 88, 93, 97, 101, 102, 112 83,767 16, 21, 27, 29, 31, 33, 53, 63, 64, 70, 81, 84, 88, 101, 104, 112 25,500 69.56

17 10, 16, 20, 29, 33, 35, 38, 62, 70, 71, 79, 88, 93, 97, 101, 102, 112 164,827 16, 21, 27, 29, 31, 33, 35, 63, 64, 70, 81, 84, 88, 95, 101, 104, 112 11,720 92.89

18 14, 16, 21, 27, 29, 31, 33, 35, 37, 62, 70, 79, 88, 93, 97, 101, 102, 112 140,993 16, 21, 29, 30, 33, 37, 38, 53, 63, 64, 70, 81, 84, 88, 95, 101, 104, 112 0 100.0

Vermont 1 107 3,553,473 45 3,434,947 3.34

2 26, 88 3,069,940 8, 45 3,051,600 0.60

3 36, 82, 115 2,394,960 36, 82, 115 2,394,960 0.00

4 36, 82, 88, 115 1,935,313 36, 82, 88, 115 1,935,313 0.00

5 26, 37, 82, 88, 115 1,071,427 26, 37, 82, 88, 115 1,071,427 0.00

6 26, 37, 82, 88, 107, 115 938,660 26, 37, 45, 82, 88, 115 931,473 0.77

7 26, 37, 82, 88, 95, 115, 146 780,660 26, 37, 82, 88, 95, 115, 146 780,660 0.00

8 34, 45, 58, 87, 88, 95, 97, 146 657,773 34, 43, 45, 58, 87, 88, 95, 97 633,800 3.64

9 34, 45, 58, 77, 87, 88, 95, 97, 146 532,087 34, 43, 45, 58, 77, 87, 88, 95, 97 508,113 4.51

10 34, 42, 45, 58, 63, 77, 87, 88, 97, 111 371,613 34, 42, 45, 58, 63, 77, 87, 88, 97, 111 371,613 0.00

11 34, 42, 45, 58, 80, 88, 95, 97, 107, 114, 146 277,067 34, 42, 45, 58, 80, 87, 88, 95, 97, 114, 146 271,200 2.12

12 34, 42, 45, 58, 80, 88, 95, 97, 106, 107, 114, 146 197,247 34, 42, 45, 58, 80, 88, 95, 97, 107, 114, 131, 146 194,427 1.43

13 27, 38, 42, 45, 58, 80, 88, 95, 97, 107, 114, 131, 146 123,320 27, 38, 42, 45, 58, 80, 88, 95, 97, 107, 114, 131, 146 123,320 0.00

14 27, 38, 42, 45, 58, 80, 87, 88, 95, 97, 108, 114, 127, 146 85,440 27, 38, 42, 45, 58, 80, 87, 88, 95, 97, 108, 114, 127, 146 85,440 0.00

15 27, 38, 42, 45, 58, 80, 87, 88, 95, 97, 108, 114, 127, 146, 164 52,000 8, 27, 38, 42, 45, 58, 80, 87, 88, 95, 97, 108, 114, 127, 146 52,000 0.00

16 27, 38, 42, 44, 45, 58, 80, 87, 88, 95, 97, 108, 114, 127, 146, 164 41,080 8, 26, 37, 38, 45, 55, 59, 68, 82, 87, 88, 95, 108, 115, 127, 147 30,847 24.91

17 8, 27, 38, 41, 44, 45, 58, 80, 87, 88, 95, 97, 108, 114, 129, 130, 146 25,220 8, 26, 37, 38, 45, 55, 59, 68, 82, 87, 88, 95, 108, 115, 129, 130, 147 20,107 20.27

18 27, 38, 42, 44, 45, 58, 80, 87, 88, 95, 97, 108, 114, 129, 138, 146, 164, 175 19,820 8, 26, 37, 38, 45, 55, 59, 68, 82, 87, 88, 95, 108, 115, 129, 138, 147, 170 9,587 51.63

19 26, 35, 36, 38, 41, 44, 45, 82, 85, 88, 93, 95, 96 105, 114, 115, 129, 130, 164 32,127 26, 37, 38, 41, 45, 55, 59, 68, 82, 87, 88, 104, 108, 112, 115, 129, 130, 138, 147 0 100.0

NETWORKS—2014—DOI 10.1002/net 15

TABLE 7. Computation times for stochastic EFCP and comparison with the deterministic EFCP.

Nevada (18 instances) Vermont

mSP(s) SP1

LR (s) EFCP1

LR for ηf(ωR)(s) SP (s) SP1

LR (s) EFCP1

LR for ηf(ωR)(s)

1 84 59 60 760 231 121

2 74 63 62 4,983 13,134 3,738

3 96 144 92 7,073 3,949 2,643

4 80 61 247 18,945 7,926 2,454

5 74 60 56 835 709 662

6 72 59 54 1,125 899 1,079

7 65 62 56 1,197 827 744

8 114 88 53 1,513 940 619

9 63 51 54 3,273 1,160 915

10 60 50 48 1,716 1,352 1,239

11 90 77 77 981 868 593

12 66 52 86 1,551 1,038 1,306

13 60 49 59 614 564 609

14 59 48 49 628 558 605

15 60 49 49 706 521 651

16 58 48 48 713 615 961

17 58 47 47 799 1,206 1,226

18 59 48 47 1,175 1,034 1,096

19 605 472 545

EFCP with the recursive formulation. We have consequently

omitted the integer L-shaped method from Table 7. We

also omitted the second relaxation from Theorem 3, as the

increased number of constraints led to longer computation

times.

8. CONCLUSIONS

Wecontribute to the literature on facility location by intro-

ducing a new type of ﬂow-capturing model in which targeted

subjectshaveanincentivetoavoidthefacilities.Theproposed

EFCP generalizes the previously studied FCP, but includes

structurallydifferent properties that,for example, can cause a

greedy heuristic to perform arbitrarily poorly. The two prob-

lems are also contrasted in a realistic case study and the

numerical comparison indicates that results optimal for the

FCP perform poorly in the setting where targeted ﬂows try to

avoid the facilities. These results, as well as the wide appli-

cability of the EFCP in transportation, revenue management,

and security and safety management, show the relevance of

the proposed ﬂow-capturing model.

We also propose a stochastic extension of EFCP where

intensities of ﬂows and their willingness to avoid facilities

are characterized with scenarios that could be obtained either

through data collection or expert opinion. We exploit the

structural properties of the problem to reduce it to an instance

which is only slightly more difﬁcult than the deterministic

EFCP. This is crucial for efﬁciently solving the stochastic

EFCPbecause the classic solutionmethod (integer L-shaped)

fails to ﬁnd the optimal solutions to real-world problems

in a reasonable amount of time. Moreover, the stochas-

tic EFCP is contrasted with its deterministic counterpart

through numerical experiments. This comparison shows that

stochastic solutions add considerable value, which motivates

the application of the stochastic EFCP.

The line of research proposed in this article could be use

to

1. Improve the current practice of transportation agencies

in locating WIM systems that consists of simply pri-

oritizing the most damaged road links. The proposed

EFCP for WIM allocation could both speed up the deci-

sion making process of highway agencies and provide

more cost-effective solutions that (1) reduce govern-

mentexpendituresforroadmaintenanceand(2)decrease

environmental damage due to overweight commercial

vehicles.

2. Improve toll collection for transportation agencies

through optimal allocation of tollbooths. The EFCP for

WIM allocation can be readily applied to allocation of

tollbooths in a road transportation network. This appli-

cation would only require different estimation of the

parameter cp

f, which would represent the lost revenue

and road deterioration associated with those ﬂows that

bypass the tollbooths.

3. Improve safety management through optimal allocation

ofsecuritycheckpoints(e.g.,inspectionstationsforvehi-

cles transporting hazardous material). The EFCP for

WIM allocation can be directly applied in allocating

ﬁxed security and safety checkpoints to manage risk.

This application may include allocation of a ﬁxed num-

ber of facilities (i.e., problem P1) to minimize the risk

associated with unintercepted ﬂows. In such a setting, cp

f

would represent an estimated risk.

One limitation of this work is the preprocessing required

to ﬁnd kf-shortest paths for each ﬂow. This approach is rea-

sonable for highway road networks, as we have shown in our

case study. However, it would be more difﬁcult to apply this

approach to well-connected road networks (e.g., urban areas

16 NETWORKS—2014—DOI 10.1002/net

like Manhattan) due to a very large number of possible paths.

In such cases, an alternative cut-based formulation could per-

haps circumvent the issue of the large number of path-based

variables that would currently arise in instances involving

well-connectednetworks.Another way to copewiththe well-

connected networks would be to apply network aggregation

techniques to reduce the size of the network and hence the

number of shortest paths to be considered within the EFCP.

APPENDIX

Proof of proposition 2

For kf=1,wehave|Pf|=1 and thus

1. We can omit condition ∀p∈Pffrom constraints (2)–(5).

We can also drop the superscript pfrom the formulation,

as well as the summation in (6).

2. Variables yfand yp

fare equivalent by construction and

thus constraint (5) can be omitted.

3. Constraints (3) and (6), zf≤1−yfand zf≥1−yf,

imply zf=1−yf.

Now we can replace zffrom (1) with 1 −yfand omit (3)

and (6). This reduces (1)–(6) to the following mathematical

program:

min

xij,yf∈{0,1}

(i,j)∈A

xijwij +

f∈F

(1−yf)cf

s.t.

(i,j)∈Af

xij ≥yf∀f∈F

(i,j)∈Af

xij ≤|Af|·yf∀f∈F

Note the following relations deﬁned with the two above

inequalities:

1. If (i,j)∈Afxij =0 then the ﬁrst inequality implies yf=

0;

2. If (i,j)∈Afxij ≥1 then the second inequality implies yf

=1.

Considering that the objective function would force yfto

take the value of 1 whenever (i,j)∈Afxij = 0, we can omit

the second inequality.

Proof of proposition 3

Note that constraint (5) hold for all p∈Pf, and thus,

yf≤yp∗(f)

f, where p∗(f)denotes the shortest path of a ﬂow.

This relation and constraint (6) imply p∈Pfzp

f≥1−yp∗(f)

f.

First, we include cp

fin the summation on the left-hand side

and multiply the right-hand side with cp∗(f)

f. Note that we are

allowedtodo this becausecp

f≥cp∗(f)

fforall p∈Pf(based on

assumption 1 as well as the deﬁnition of p∗(f)as the shortest

path of a ﬂow). Second, we sum the obtained inequality over

all the ﬂows, yielding

f∈F

p∈Pf

zp

fcp

f≥

f∈F

(1−yp∗(f)

f)cp∗(f)

f.

Finally, we add the facility cost to both sides of the last

inequality and note that

(i,j)∈A

xijwij +

f∈F

p∈Pf

zp

fcp

f≥

(i,j)∈A

xijwij

+

f∈F

(1−yp∗(f)

f)cp∗(f)

f,

whence EFCP(x)≥FCP(x)as required.

Proof of proposition 4

We prove this by a counterexample. Let wbe a vector of

zeros. Consider a case of a single ﬂow fwith its kf-shortest

paths indexed p=1,...,kf. Let Sdenote an allocation of

facilitiescoveringonly the ﬁrst r-shortest paths.Let Tdenote

an allocation of facilities covering the ﬁrst r-shortest paths

like in S, as well as shortest paths indexed p=r+2,...,kf−

1. Moreover, let h(S) denotes the objective value of P1 given

allocationS.Clearly,wehaveS⊂T⊂Aandh(S)=h(T)=

cr+1

fbecause the ﬂow travels along the shortest unmonitored

path, which is path r+ 1 in both cases. Now, let k/∈Tbe the

location of a checkpoint such that only the (r+1)-shortest

path is intercepted and observe the following:

h(T∪{k})−h(T)=ckf

f−cr+1

f

h(S∪{k})−h(S)=cr+2

f−cr+1

f

The above equalities imply h(T∪{k})−h(T)≥h(S∪

{k})−h(S)because ckf

f≥cr+2

f. Thus, submodularity does

not hold for all S⊂T⊂Aand k/∈T.

To show that the function is neither supermodular nor

monotonic, let Tdenote an allocation of facilities covering

the ﬁrst r-shortest paths like in S, as well as shortest paths

indexed p=r+2,...,kf. Then,

h(T∪{k})−h(T)=0−cr+1

f≤0

h(S∪{k})−h(S)=cr+2

f−cr+1

f≥0

The above expressions imply that the objective function

of P1 is not monotonic. Moreover, since this time we have

h(T∪{k})−h(T)≤h(S∪{k})−h(S), supermodularity does

not hold for all S⊂T⊂Aand k/∈T.

Proof of proposition 5

To prove that problem P2 is NP-hard, we reduce a known

NP-hard problem, namely the problem of “locating unca-

pacitated inspection stations” (LUIS), studied in [38], to an

instance of problem P2. The goal of this problem is to place

NETWORKS—2014—DOI 10.1002/net 17

the smallest possible number of inspection stations needed

to cover all truck ﬂows (thus ensuring that all trucks are

inspected). Using our notation, it can be written as

LUIS: min

xij∈{0,1}

(i,j)∈A

xij

s.t.

(i,j)∈Af

xij ≥1

where(i, j)arearcsin a graph,fdenotes a truck ﬂow,Afisthe

set of arcs along the single predetermined path of a ﬂow, and

Aand xij are as deﬁned earlier. Given an arbitrary instance of

LUIS,we constructaninstance of P2whoseoptimal solution

yields an optimal solution to LUIS.

First, we let wij = 1 and let cf=(i,j)∈Awij for all f. Then

the problem

LUIS: min

xij,yf∈{0,1}

(i,j)∈A

xijwij +

f∈F

(1−yf)·cf

s.t.

(i,j)∈Af

xij ≥yf

is an instance of FCP. In this formulation, the variable yf

equals 1 if ﬂow fis captured and 0 otherwise. However, if

we do not capture ﬂow f, we incur a penalty cfthat exceeds

the cost of implementing a station on each arc. Therefore, the

optimal solution to LUISnever leaves any ﬂows uncaptured,

andwill remain unchanged if we require yf= 1, in which case

LUIS and LUISare identical. Since FCPis an instance of

P2 with kf= 1, we conclude that P2 is NP-hard.

Proof of Theorem 2

To prove this, it sufﬁces to show that for any ﬁxed binary x

andy=yp

f|p∈Pf,f∈Fwhichsatisfy(2)and (4), the two

problems have the same optimal second-stage value. First,

observe that for ﬁxed binary xand ywhich satisfy (2) and

(4), the objective function of problem EFCP1

LR corresponds

to

(i,j)∈A

xijwij +

f∈F

B(y,f),

where

B(y,f)=min

yf,zp

f≥0

p∈Pf

zp

fcp

f

s.t. zp

f≤1−yp

f∀p∈Pf

yf≤yp

f∀p∈Pf

p∈Pf

zp

f≥1−yf

We proceed by partitioning each set Pfinto P1

fand P2

f,

such that yp

f=1 for p∈P1

fand yp

f=0 for p∈P2

f.Now,

note that B(y,f)can be determined based on whether the set

P2

fis empty. Using arguments similar to those in the proof

of proposition 1, we conclude that, for ﬁxed xand y, the

objective function of EFCP1

LR is given by

(i,j)∈A

xijwij +

f∈F

B(y,f),

where

B(y,f)=⎧

⎨

⎩

min

p∈P2

fcp

f,P2

f=∅;

0, P2

f=∅.

Finally,weobservethatf∈FB(y,f)correspondstoQ(x)

from problem P1. This implies that for ﬁxed xand y, prob-

lems EFCP1

LR and EFCP have the same optimal second-stage

values. Thus, the same xand yoptimize both problems.

Proof of Theorem 3

We ﬁrst prove that for any allocation x,wehave

EFCP2

LR(x)=EFCP(x). We show this by working through

the constraints of EFCP2

LR similar to proposition 1. Again,

the objective is separable in f, whence

EFCP2

LR(x)=wTx+

f∈F

¯

Q(x,f)

where

¯

Q(x,f)=min

yp

f,yf,zp

f≥0

p∈Pf

zp

fcp

f

s.t. (2)–(3),(5)–(7)for ﬁxed f.

We partition each set Pfinto sets P1

fsuch that

(i,j)∈Ap

fxij ≥1forp∈P1

f,andP2

fsuchthat(i,j)∈Ap

fxij =0

for p∈P2

f, and observe that

1. For p∈P1

f, constraints (7) and (3) imply yp

f≥1 and

zp

f=0;

2. For p∈P2

f, constraints (2) and (3) imply yp

f=0 and

zp

f≤1, respectively.

Now we can compute ¯

Q(x,f)similar to proposition 1.

Thus, we omit the corresponding steps and conclude that the

objective function of EFCP2

LR(x)can be given as

EFCP2

LR(x)=wTx+

f∈F

¯

Q(x,f)

where

¯

Q(x,f)=⎧

⎨

⎩

min

p∈P2

fcp

f,P2

f=∅;

0, P2

f=∅.

The above expression for EFCP2

LR(x)matches the objec-

tive function of P1 and is thus equivalent to the objective of

P2. The desired result follows.

18 NETWORKS—2014—DOI 10.1002/net

Proof of Theorem 4

Recall from the Proof of proposition 1, that for a ﬁxed

feasible xand realization ω, the oracle provides the damage

associated with a particular ﬂow f∈F, given by

Q(x,ξ(ω),η(ω),f)=⎧

⎨

⎩

min

p∈P2

f(ω) cp

f(ω),P2

f(ω) =∅;

0, P2

f(ω) =∅.

whereP2

f(ω) isasetofpathssuchthat(i,j)∈Ap

f(ω) xij =0.For

any fwith P2

f(ω) =∅, let sf(ω) =argminp∈P2

f(ω) cp

f(ω)

be the shortest unmonitored path of ﬂow f. Note that both P2

f

andsfdepend on xaswell as ω,becausedifferentrealizations

η(ω) can change the set of paths that need to be covered,

which also affects the performance of a ﬁxed allocation x.

Let us now deﬁne a random variable df, such that

df(ω) =lsf(ω)

f,P2

f(ω) =∅;

0, P2

f(ω) =∅.

The total damage produced by all the ﬂows is computed

as

Q(x,ξ(ω),η(ω)) =

f∈F

df(ω)ξf(ω),

wheredfis a functionof the shortest unmonitoredpath which

depends on the ﬁxed allocation xas well as the realization of

ηf. Conversely, ξf(ω) is the intensity of ﬂow f(i.e., per mile

damage or risk). Based on the assumed independence of ξ

and η,wehave

EQ(x,ξ,η)=E

f∈F

dfξf

=

f∈F

E[df]E[ξf]

=E

f∈F

dfξf

and the result follows.

Proof of proposition 6

Let 0 <ε<1, and suppose that there is a single ﬂow

fthat can travel along at least two arc-disjoint paths. In this

case, the optimal value can be expressed as EFCP (x∗

EFCP)=

min(c1

f,(i,j)∈Swij), whereSistheleastexpensiveallocation

of facilities that covers all the paths of ﬂow f. Moreover, the

greedy heuristic is initialized with a solution that includes

no facility implementation and the corresponding damage

c1

f. Since facility implementation only worsens the objective

functionin the ﬁrstiteration (i.e., a ﬂowdivertsand/orfacility

cost is incurred), the greedy heuristic stops after the ﬁrst pass

and returns the solution EFCP(xG)=c1

f.

Recall that c1

frepresents the excessive damage produced

if ﬂow ftravels along the shortest path and note that c1

fcan

be arbitrarily high depending on the intensity of the ﬂow

and length of the path. Suppose that c1

f=2

ε(i,j)∈Swij and

observe that

EFCP(x∗

EFCP)

EFCP(xG)=min(c1

f,(i,j)∈Swij)

c1

f

=(i,j)∈Swij

2

ε(i,j)∈Swij =ε

2

The above equality shows that for any 0 <ε<1,

there exists an instance of P1 for which EFCP(x∗

EFCP)≤

ε·EFCP(xG).

Proof of proposition 7

Let ε>0, and consider a completely connected network

with m+ 1 nodes. Suppose that there are mﬂows with the

sameorigin node Oand mdistinctdestination nodes. Further-

more, suppose ﬂows can travel from Oto their destinations

through all mremaining nodes (i.e., these nodes are not “too

far” apart and thus all possible paths are acceptable). Clearly,

the optimal solution consists of locating mfacilities along m

links adjacent to node Oand thus EFCP(x∗

EFCP)=0.

Conversely, the greedy heuristic is initialized with a solu-

tion that includes no facility implementation and all ﬂows

traveling freely from Oto their mdestination nodes. In the

ﬁrst step, the greedy heuristic tries implementing a facility

on all the links. However, placing a facility on any of the

links adjacent to node Oyields an increased excessive dam-

agesincethecorrespondingﬂowdiverts.Thegreedyheuristic

proceedsby implementing allmfacilities on links connecting

mdestinationnodes,withoutinterceptinganyﬂows.Thus,we

have EFCP(xG)=m

f=1c1

f.

Inthedescribedcase,wehaveEFCP(x∗

EFCP)/EFCP(xG)=

0<ε. Thus, for any ε>0, there exists an instance of P1

for which EFCP(x∗

EFCP)≤ε·EFCP(xG).

Proof of proposition 8

Let ε>0, and assume that m= 1. Now suppose there is a

single ﬂow that can travel along two arc-disjoint paths. Let l

denotethelength of the shorter, and let γ·lbe thelengthof the

longer path (γ>1). Furthermore, let ηdenote the maximum

distance that a ﬂow is willing to travel to avoid facilities.

Assume that ηhas two possible realizations, P(η =γ·l)=δ

and P(η =l)=1−δ, where δ<1.

Since η<γ×l, in the deterministic counterpart of

EFCP, the ﬂow fcan travel only along the shorter path.

Thus, ¯ximplies implementation of a facility anywhere along

this path. The corresponding expected cost is computed as

EQ(x,ξ,η) =(1−δ)×0+δ×(γ ×l×ξ) =δ×γ×l×ξ,

where ξis the expected unit damage (i.e., per mile dam-

age). Conversely, x∗implies implementation of the facility

along the shorter path if δ·γ≤1, or along the longer path if

δ·γ>1. Assume that δ·γ>1andnotethatthe expectedcost

forthe corresponding optimal solution isEQ(x∗,ξ,η) =l·ξ.

NETWORKS—2014—DOI 10.1002/net 19

In the aforementioned case, the value of the stochastic

solution is given as

VSS =EQ(x,ξ,η) −EQ(x∗,ξ,η)

=l·ξ·(δ ·γ−1).

Finally, note that we can deﬁne parameters l,ξ,δand γ(such

that δ<1 and δ·γ>1) to make VSS in the above example

arbitrarily large. Thus, for any ﬁnite ε>0, we can design an

instanceof the two-stagestochastic EFCP suchthat VSS >ε.

ACKNOWLEDGMENTS

The authors thank the Editors and three anonymous

reviewers for their very valuable suggestions. A discus-

sion with Richard Church on linear relaxations is also

acknowledged.

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