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Evasive Flow Capture: Optimal Location of Weigh-in-Motion Systems, Tollbooths, and Security Checkpoints

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The flow-capturing problem (FCP) consists of locating facilities to maximize the number of flow-based customers that encounter at least one of these facilities along their predetermined travel paths. The FCP literature 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 facilities 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, analyze their structural properties, study exact and approximate solution techniques, and show an application to a real-world transportation network. © 2014 Wiley Periodicals, Inc. NETWORKS, 2014
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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 fF. Let Ap
fbe the
set of arcs along path pPfof flow fF. 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 fFpasses unintercepted along path
pPf. 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 pPfof flow fF
0 otherwise
yf=
1 if at least one WIM station is located
along all paths pPfof flow fF
0 otherwise
zp
f=
1ifowfFtravels unintercepted
along path pPf
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 +
fF
pPf
zp
fcp
f(1)
s.t.
(i,j)Ap
f
xij yp
ffF,pPf(2)
zp
f1yp
ffF,pPf(3)
(i,j)Ap
f
xij ≤|Ap
fyp
ffF,pPf(4)
yfyp
ffF,pPf(5)
pPf
zp
f1yffF(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}
fF
pPf
zp
fcp
fs.t. constraints (2)to (6)
Note that the above minimization is separable in f, whence
¯
Q(x)=
fF
¯
Q(x,f)
where
¯
Q(x,f)=min
yp
f,yf,zp
f{0,1}
pPf
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 1forpP1
f,andP2
fsuchthat(i,j)Ap
fxij =0
for pP2
f. It follows that
1. For pP1
f, constraints (4) and (3) imply yp
f=1 and
zp
f=0, respectively;
2. For pP2
f, constraints (2) and (3) imply yp
f=0 and
zp
f1, respectively.
Now, note that (5) is defined over pPf, 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 pP2
fzp
f1. In this case, we
have
¯
Q(x,f)=min
zp
f{0,1}
pP2
f
zp
fcp
fs.t.
pP2
f
zp
f1
=min
pP2
fcp
f.
2. If P2
f=∅, then constraint (5) implies yf1 and con-
straint (6) is equivalent to pP1
fzp
f1yf. Since
zp
f=0, for all pP1
f, (6) implies yf1. Hence, yf=1
and ¯
Q(x,f)=0.
The two cases can be summarized as
¯
Q(x,f)=
min
pP2
fcp
f,P2
f=∅;
0, P2
f=∅.
Finally, recall that ¯
Q(x)=fF¯
Q(x,f)and note that
minpP2
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 +
fF
(1yf)cf
s.t.
(i,j)Af
xij yffF
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 STNand
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
SN,|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
Z111
mm
11
e0.63.
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)Axijm
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
f0and 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·
fF
|Pf|
  
yp
f&zp
f
to |A|+fF|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
ffF,pPf,(i,j)Ap
f(7)
2. Allauxiliaryvariablesarelinearlyrelaxedyp
f,yf,zp
f0,
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·fF|Pf|to |A|. However, the total number of
constraints is increased from |F|

(6)
+4·
fF
|Pf|
  
(2)(5)
to |F|+3·
fF|Pf|+fFpPf|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|fFbe a
vector of discrete random variables denoting unit intensities
of flows fF(i.e., damage or risk produced per unit of
distance traveled). Similarly, let η=ηf|fFbe 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
ωω1,...,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 pPf(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)Axijm
EQ(x,ξ,η)=min
x{0,1}|A|
(i,j)Axijm
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(˜ωr1
f), and
thus Pf(˜ωr1
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
fF
˜
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}
pPf(˜ωr
f)
zp
f(˜ωr
f)cp
f(˜ωr
f)
(10)
s.t.
(i,j)Ap
f(˜ωr
f)
xij yp
f(˜ωr
f)pPf(˜ωr
f)/Pf(˜ωr1
f)(11)
zp
f(˜ωr
f)1yp
f(˜ωr
f)pPf(˜ωr
f)/Pf(˜ωr1
f)(12)
(i,j)Ap
f(˜ωr
f)
xij ≤|Ap
f(˜ωr
f)yp
f(˜ωr
f)pPf(˜ωr
f)/Pf(˜ωr1
f)
(13)
yf(˜ωr
f)yp
f(˜ωr
f)pPf(˜ωr
f)/Pf(˜ωr1
f)(14)
yf(˜ωr
f)yf(˜ωr1
f)(15)
p∈∪r
r=1Pf(˜ωr
f)
zp
f(˜ωr
f)1yf(˜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 r1, 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
+|FR

yf(ω)
+2·
fF
|Pf(˜ωR
f)|
  
yp
f(ω) &zp
f(ω)
binary variables,
1

(9)
+4·
fF
|Pf(˜ωR
f)|
 
(11)(14)
+2·|FR
 
(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·
fF
|Pf(˜ωR
f)|
 
yp
f(ω) &zp
f(ω)
binary variables,
1

(9)
+4·
fF
|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 MfR
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)Axijm
EQ(x,ξ,η);
x=argmin
x{0,1}|A|
(i,j)Axijm
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 .
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 fFpasses unintercepted
along path pPf. 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 xfor 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 ηfR)). 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) xEQ(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 ηfR)(s) SP (s) SP1
LR (s) EFCP1
LR for ηfR)(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 pPffrom 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), zf1yfand zf1yf,
imply zf=1yf.
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 +
fF
(1yf)cf
s.t.
(i,j)Af
xij yffF
(i,j)Af
xij ≤|AfyffF
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 pPf, and thus,
yfyp(f)
f, where p(f)denotes the shortest path of a flow.
This relation and constraint (6) imply pPfzp
f1yp(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
fcp(f)
fforall pPf(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
fF
pPf
zp
fcp
f
fF
(1yp(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 +
fF
pPf
zp
fcp
f
(i,j)A
xijwij
+
fF
(1yp(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,wehaveSTAandh(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
fcr+1
f
h(S{k})h(S)=cr+2
fcr+1
f
The above equalities imply h(T{k})h(T)h(S
{k})h(S)because ckf
fcr+2
f. Thus, submodularity does
not hold for all STAand 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)=0cr+1
f0
h(S{k})h(S)=cr+2
fcr+1
f0
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 STAand 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 +
fF
(1yf)·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|pPf,fFwhichsatisfy(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 +
fF
B(y,f),
where
B(y,f)=min
yf,zp
f0
pPf
zp
fcp
f
s.t. zp
f1yp
fpPf
yfyp
fpPf
pPf
zp
f1yf
We proceed by partitioning each set Pfinto P1
fand P2
f,
such that yp
f=1 for pP1
fand yp
f=0 for pP2
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 +
fF
B(y,f),
where
B(y,f)=
min
pP2
fcp
f,P2
f=∅;
0, P2
f=∅.
Finally,weobservethatfFB(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+
fF
¯
Q(x,f)
where
¯
Q(x,f)=min
yp
f,yf,zp
f0
pPf
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 1forpP1
f,andP2
fsuchthat(i,j)Ap
fxij =0
for pP2
f, and observe that
1. For pP1
f, constraints (7) and (3) imply yp
f1 and
zp
f=0;
2. For pP2
f, constraints (2) and (3) imply yp
f=0 and
zp
f1, 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+
fF
¯
Q(x,f)
where
¯
Q(x,f)=
min
pP2
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 fF, given by
Q(x,ξ(ω),η(ω),f)=
min
pP2
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) =argminpP2
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)) =
fF
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
fF
dfξf
=
fF
E[df]E[ξf]
=E
fF
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. 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=γ·l)=δ
and P=l)=1δ, 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) =(1δ)×0+δ××l×ξ) =δ×γ×l×ξ,
where ξis the expected unit damage (i.e., per mile dam-
age). Conversely, ximplies implementation of the facility
along the shorter path if δ·γ1, 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··γ1).
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 .
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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