Open Journal of Discrete Mathematics, 2011, 1, 153-159
doi:10.4236/ojdm.2011.13019 Published Online October 2011 (http://www.SciRP.org/journal/ojdm)
Reliability Analysis of Facility Systems Subject to
Edge Failures: Based on the Uncapacitated Fixed-Charge
Location Problem
Zongtian Wei1,2*, Huayong Xiao1
1Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, China
2Department of Mat hem at i cs, Xi’an University of Architecture and Technology, Xi’an, China
Received August 11, 2011; revised Sepetember 19, 2011; accepted September 30, 2011
Abstract
A facility system can be modeled by a connected graph in which the vertices represent entities such as sup-
pliers, distribution centers or customers and the edges represent facilities such as the paths of goods or in-
formation. The efficiency, and hence the reliability, of a facility system is to a large degree adversely af-
fected by the edge failures in the network. Such failures may be caused by various natural disasters or terro r-
ist attacks. In this paper, we consider facility systems’ reliability analysis based on the classical uncapaci-
tated fixed-charge location problem when subject to edge failures. For an existing facility system, we formu-
late two models based on deterministic case and stochastic case to measure the loss in efficiency due to edge
failures and give computational results and reliability envelopes for a specific example.
Keywords: Facility System, Reliability, Edge Failure, Uncapacitated Fixed-Charge Location Problem
1. Introduction
It is well known that facility location decisions are stra-
tegic in a supply chain, or more generally speaking, in a
facility system design, since facility location decisions
are costly and difficult to reverse, and their impact spans
a long time horizon.
We use the term “facility” here in its broadest sense.
That is, it is meant to include facilities such as factories,
warehouses, retail outlets, schools, hospitals, and satel-
lites, as well as transportation lines, cab les to name but a
few that have been analyzed in the research literature.
Every facility system in operation maybe faces various
disruptions. Such disruptions have begun to receive sig-
nificant attention from practitioners and researchers after
the terrorist attacks of September 11, 2001. Facility sys-
tem disruptions can have significant physical costs (e.g.,
damage to facilities, inventory, electronic networks, and
infrastructure) and subsequent losses due to downtime. A
recent study [1] estimates the cost of downtime (in terms
of lost revenue) for several online industries that cannot
function if their computers are down.
We view the structure of a facility system as a con-
nected graph in which the vertices represent facilities
such as subway stations, distribution centers, etc., and
the edges represent facilities such as the paths of goods
(e.g., transportation lines) or information (e.g., cables).
In this paper, we distinguish two kinds of facilities, e.g.,
“vertex facilities” and “edge facilities”. For simplicity,
we call them “vertices” and “edges”, respectively.
It is well known that, regardless of intentional strikes
or natural disasters, edges are easily to be damaged. In
most cases, facility system disruptions are caused by the
failures of edges, e.g., closure of highway because of the
inclement weather, traffic jam, road damage caused by
earthquak es or debris flo ws.
In this paper, adopting the facility location analysis
framework, we will mainly consider facility systems’
reliability analysis based on the classical uncapacitated
fixed-charge location problem (UFLP) when subject to
edge failures, whereby we consider an existing facility
system in which the facilities may or may not be located
optimally. The edges may be lost due to natural disasters
or terrorist attacks. We want to know the efficiency of
the remaining system under such circumstances. We will
formulate two models based on deterministic case and
stochastic case to measure the loss in efficiency due to
edge failures and give computation al results and reliabil-

Z. T. WEI ET AL.
154
ity envelopes for a specific example.
The remainder of the paper is organized as follows. In
Section 2 we review some related literature. In Section 3,
we formulate two reliability models based on the UFLP
and edge failures. We use a scenario-based algorithm to
compute a specific example and give the results and re-
liability envelopes in Section 4. Section 5 is a summary
of this paper.
In the following, by “loss” we refer to the edge disrup-
tions (failures) mentioned above or, sometimes the nec-
essary closure.
2. Literature Review
In this section, we briefly review the facility systems’
reliability under disruptions.
The concept of facility system reliability is related to
network reliability theory, which is concerned with cal-
culating or maximizing the probability that a graph re-
mains connected after random failures due to congestion ,
disruptions, or blockages. Typically, this literature con-
siders disruptions to the links of a network, but some
papers consider node failures [2], and in some cases the
two are equivalent. Given the d ifficulty in computing the
reliability of a given network, the g oal is of ten to find the
minimum-cost network with some desirable properties
like 2-connectivity [3,4], k-connectivity [5], or special
ring structures [6].
The reliability of a facility system is the probability
that all suppliers are operable [7]. Generally speaking,
the key differen ce between ne tworks reliability and facil-
ity systems reliability is that the former are primarily
concerned with connectivity; they consider the cost of
constructing the network but not the cost that results
from a disruption, whereas the latter consider both types
of costs and generally assume connectivity after a dis-
ruption [8].
The facility location problem is a classical, combina-
torial optimization problem to determine the number and
locations of a set of facilities and assign customers to
these in such a way that the total cost is minimized. Two
types of costs are considered in the problem. A setup cost
(facility cost) occurs while a facility is opened, and a
connection cost occurs while a customer is assigned to
the opened facility.
If an arbitrary number of customers can be connected
to a facility, the problem is called uncapacitated facility
location problem (UFLP) [9]. The UFLP is NP-hard [10]
and have been extensively studied. Lots of algorithms,
exact and heuristic, have been developed in the past
decades [1,11-13].
A number of papers in the location literature have ad-
dressed the problem of finding the optimal location of
protection devices to reduce the impact of possible dis-
ruptions to infrastructure systems.
For example, Carr et al. [14] presents a model for op-
timizing the placement of sensors in water supply net-
works to detect maliciously injected contaminants. James
and Salhi [1] investigate the problem of placing protec-
tion devices in electrical supply networks to reduce the
amount of out age time.
Church et al. [15] presented a model called the r-in-
terdiction median problem. This model can be used to
identify which r of the existing set of p-facilities, when
interdicted or lost impacts delivery efficiency the most.
Such a model can be used to identify the worst case of
loss, when losing a pre-specified number of facilities.
The model is restricted in two ways: it is based upon the
assumption that the terrorist or interdictor is successful in
each and every strike, and it is also based upon the as-
sumption that exactly r facilities will be struck and lost.
Such a model does address a worst case scenario, but it
does not exactly capture the issues that would be key to
understanding the range of failures and possible out-
comes.
In [16], the authors argued that first, it is important to
recognize that a strike or disaster may not impair a facil-
ity’s operation. That is, a terrorist strike may be success-
ful only a certain percentage of the time. The same is
true for a natural disaster. When it does occur, there is a
threat that operations at a facility may need to be sus-
pended, bu t it is not absolute e. Second, interdiction may
not be intelligent when the strikes impact a non critical
facility. Although it is important to model “worst-case”
scenarios, it is also important to model and understand
the range of possible failures and impacts. Therefore,
they proposed a family of models which can be used to
model the range of possible impacts associated with the
threat of losing one or more facilities to a natural disaster
or intentional strike. They show how to model determi-
nistic loss and probabilistic loss. In addition, they pre-
sented results associated with the application of the worst
case and the best case expected loss models to a data set.
There is a mature literature on reliable network (e.g.,
supply chain network (SCN)) design and analysis under
component failures. Unfortunately, so far we have not
found the explicit study of facility systems' reliability
subject to edge failures. In fact, the reliability, and hence
the efficiency, of a facility systems is to a large degree
adversely affected by failures of the edges. Such failures
may be caused by congestion, inclement weather, earth-
quakes, debris flows, sandstorms, strikes or terrorist at-
tacks. Thus the network based facility system reliability
models we will study are more practical and closer to the

155 Z. T. WEI ET AL.
reality of facility system management.
3. Facility System Reliability Analysis
Models
In this paper, we use the total operational cost as the ef-
ficiency measure of a facility system. The notion reli-
ability is defined to be the ratio of the system’s effi-
ciency and the efficiency after some edges have failed.
We distinguish two cases: deterministic and stochastic,
to formulate the UFLP-based reliability analysis models
that evaluate the efficiency of a facility system after
some edges have failed.
Suppose that we have a system of some operating fa-
cilities supplying a set of demand points. If each facility
can serve any assigned demand, then we can assign each
customer to their closest facility (as measured by cost or
distance). We can define weighted distance for a de-
mand-facility interaction as the distance from the de-
mand to their closest facility weighted by the number of
trips needed to su pply that demand from a facility utiliz-
ing some type of transport mode (e.g., truck). Thus, we
can measure the overall efficiency of the system as the
total truck-miles of travel needed to supply all of the
demand from the set of located facilities.
The exact opposite of the UFLP occurs when we con-
sider an existing system in which the facilities may or
may not be located optimaally.
When either closing or considering the loss of one or
more edges by a disaster, the basic question is what hap-
pens to the operating efficiency of the system. We can
measure this loss of efficiency by calculating the result-
ing increase in delivery cost (or loss of the system effi-
ciency) as a reliability envelope. The details will be dis-
cussed in Section 4.
The following are the notations for our formulations.
Sets
I: set of customers, indexed by i.
J: set of potential facility locations, indexed by j.
Parameters
hj: demand at customer . iI
fj: fixed cost of open a facility at . jJ
ce: “delivery ” cost per uni t per “l engt h” t hrough road e.

: weight of the fixed cost in the objective function.
We view a facility system as a weighted connected
simple graph , where ; E is the
edge set with the edges denoting goods or information
paths; H is the vertex weight set with the weight hij of
vertex i, denoting the demand of customer i, and D is the
edge weight set with the weight dij of edge , de-
noting the length (e.g., distance) between i and j under
the existing cond itions. By dij we also denote the shortest
path lengt h (or distance) between i and j if

,, ,GVEHD VIJ

,

,ij
Assume that X is an feasible solution of the UFLP, i. e . ,
1,
j
X

if a facility is established at location jJ

; 0,
otherwise. Let C be the opened facility (server) set cor-
responding to X, and
F
E be the potential failure
edge set, where the edge failure is defined as an edge
losses its designed function completely. Therefore, a
failed edge in G is equivalent to delete (or close) the
corresponding line from the facility system. We also as-
sume that edge failures are independent and multiple
edge failures may occur simultaneou sly.
3.1. The Deterministic Reliability Models
Let Sr be the set of scenarios corresponding to the closure
or deletion of r edges from G, i.e., every r
s
S explic-
itly specifies the failed r edges in F. Let dijs be the short-
est distance between customer i and facility j in scenario
s. Define


:
is ijs
NjCd

. Assume that in any
scenario r
s
S

, customer i is served by an opened fa cil-
ity jC

which is the nearest one from i if is N


.
Associated with each customer i is a per-unit penalty cost
i

that represents the cost of not serving the customer if
is
N


. i

may represent a lost-sales cost, or the cost
to pay a competitor to serve the customer temporarily.
We define the assignment variables as ijs Y1,

if cus-
tomer i is served by facility j in scenario s; 0, otherwise.
Assume that in any scenar io r
s
S
C, customer i is served by an opened facility j

which is the nearest one
from i.
We formulate the deterministic reliability model (DRM)
as the following integer-linear programming problem:

,,
1
min
is is
jjiijsijs ii
sS jCiINjCiIN
r
fXchdYh








 









..1, ,
ijs r
jC
s
tY iIsS

 

(1)


0,1 ,,,
ijs r
YiIjCsS

 (2)
For a given edge loss level r (the number of closed or
deleted edges), this model can be used to evaluate a fa-
cility system’s operational efficiency under the best case,
namely the minimal loss of the system’s efficiency.
Changing “min” to “max” in the objective function,
then we obtain the worst case model, that is the model to
measure the maximal increase in weighted distance un-
der the edge failure level r.
3.2. The Stochastic Reliability Models

ij
E

.
The reliability model formulated above is based upon a
deterministic analysis. Up to this point we have modeled
edge loss a certainty. We now consider the case where

Z. T. WEI ET AL.
156
loss is not a certainty upon an edge failure. Usually, the
chances of losing an edge are based upon some probabil-
ity. We wish to derive the maximal or minimal expected
efficiencies associated with an existing system. To do
this we need to identify both the worst case and the best
case expected outcomes.
Let
F
E be the target edge (the potential failure
edet of a
o set when
ge) sn attack. Assume that an attacker can hit
each edge in F at most once and that the edges in F will
be hit simultaneously.
Let Sr be the scenari

0rrF edges
in F have been attacked. Each r
s
Ss
fine
hich r
edges in F have been attacked. Depecifies w

isattackedinsce
s
EeFe

narios,
then any
s
s
F
E
nario s, can be used to represent a failed edge set in sceso we call
s
F
the sub-scenario of s.
Let j be a nearest opened flity to customer i and aci
s
ijF
de the shortest distance between them in scenario
efine

:
s
iF
NjCd . Assume that in
any scenario y an opened facil-
ity jC which is the nearest one from i if s
iF
N
b
DFs. s
ij
omer ied b
F
is servFs, cust


.
Let he failure probability of edge jF aft
attack. It is easy to see that scenario ccurs with
probability
pj be ter one
Fs o

\
1
s
sss
F
jj
jFjEF
pp



p
Denote the assignment variables as if cus-
to
e opened facility (server) set of an existing
fa
1,
s
ijF
Y
n scenarmer i is served by an open facility j iio Fs; 0,
otherwise.
Let C be th
cility system. We formulate the stochastic reliability
model (SRM) as following (The penalty cost i

are
defined in Section 3.1.):

,,
1
min
r
sss
s siFiF
s
jj
sS jC
FiijFijFi i
FE iIN jCiIN
fX
pchdYh


s



 
















 
..1, ,
s
ijFss
jC
s
tY iIFE

 
 (3)
E (4)
The objective function selects r edges fr
m
given failure level r, this model can be used to
ev
. The Reliability Envelopes
he models described above can be applied to a given
d the SRM to a
da

F0,1 ,,,
s
ijs s
YiIjCF
om F to mini-
ize the weighted distance expectation after r edges in F
have been attacked. Constraints (3) require that each
customer be served by at most one server in any scenario
Fs. Constraints (4) require the assignment variables to be
binary.
For a
aluate a facility system’s operational efficiency under
the best case. Changing “min” in the objective function
to “max”, then we obtain the worst case model, that is
the model to measure the maximal increase in expected
weighted distance under failure level r.
4
T
facility system over a range of edge loss level r. We use
the weighted distance to measure the efficiency of a fa-
cility system and efficiency is measured at 100% if all
edges are operating. If an edge is lost due to a natural
disaster, intentional strike or planned closure, then the
efficiency is lost and overall efficiency decreases. If
many edges exist, then there exist several possible out-
comes of losing just one edge. One can easily enumerate
each of the possible ways of losing one edge as well as
calculate the impact of each possible loss in terms of
changes in efficiency. The results of this series of calcu-
lations will define a range of losses from the best case
(i.e. the least decrease in efficiency) to the worst case (i.e.
the greatest decrease in efficiency). We then have a re-
gion defined by an upper curve and a lower curve, where
the upper and the lower curve represent the solutions of
the least or the greatest impact associated with a given
loss level, respectively. The region depicted between
these two curves can be defined as the operational enve-
lope or reliability envelope. For a given edge loss level,
this envelope specifies the range of possible system per-
formance from the best-case to the worst-case. Actual
performance will fall within this range.
In this section we apply the DRM an
ta set to generate reliability envelopes. Our data set is
derived from the 2008 China census data: a 49-node set
consisting of the capitals of all the provinces in China
plus the two special administrative regions Hong Kong
and Macau, as well as other 15 big cities. The demand of
city i, hi is settled to be the city’s administrative region
population divided by 10000. The transportation links
(edges) are set to the recent national highways and the
transportation costs per unit per length through different
roads are all set to 0.005c

. The fixed cost of facilities
setup are estimated by considering the factors such as
local labor price, facility size, and other natural condi-
tions.
By using the above data set we optimally solve an
UFLP with 0.7


in order to site an existing facility
system. Figuows the optimal solution, where the 8
distribution centers are city 4, city 7, city 11, city 20, city
24, city 26, city 28 and city 45, and the edges marked by
red color represent the delivery routes from each distri-
bution center to its customers.
re 1 sh

Z. T. WEI ET AL.
157
facilities and a poten-
tiaGiven this operating system of 8 then solve the SRM with edge failure probability p = 0.3 and p = 0.7, respectively. The solutions of the latter and
the corresponding reliability envelope are showed in Ta-
ble 2 and Figure 3, respectively.
l failure edges set which is consisted of 8 edges: F =
{1 (5,47), 2 (3,4), 3 (7,40), 4 (10,11), 5 (28,31), 6 (24,25),
7 (17,45), 8 (21,49)} (Notice that the sub-graph GF

is connected, i.e., both of is N and
s
iF
N are not

),
We solve the worst-case DRM and the t-case DRM.
The solutions are given in Table 1.
In Table 1, for each edge loss
bes
level, the objective
fu
volume 6 of Table 1
ar l efficien-
ci
atest difference
be
e DRM, we
nction values and efficiency for each case are also
given as a percentage, where 100% represents the oper-
ating level before edge failures.
Obviously, the edges shown in
e the most important objective of protection.
Figure 2 presents the values of operationa
es (in percent) as a graph, depicting the lower and the
upper boundaries of the reliability envelope. Notice that
the greatest marginal impact for the worst case occurs
when the edge loss level is small while for the best case
occurs when the edge loss level is great.
It is also important to note that the gre
tween the worst case and the best case of the envelope
occurs when the edge loss level is moderate.
By using the same data set as in the abovFigure 1. Optimal solution of a UFLP with α = 0.7.
Table 1. Results of the Worst-case DRM and the Best-Case DRM with α = 0.7.
Level Best-Case Worst-Case
r Objec. Value Failed Edges Efficiency Objec. Value Efficiency Failed Edges
0 31,890.22 - 100% 31,890.22 - 100%
1 31,995.78 3 99.67% 40,117.94 8 79.49%
2 32,105.80 1,3 99.33% 43,361.45 2,8 73.55%
3 32,241.20 1,3,6 98.91% 44,169.38 2,4,8 72.20%
4 32,484.10 1,3,5,6 98.17% 44,813.33 2,4,7,8 71.16%
5 33,079.71 1,3,4,5,6 96.40% 45,056.22 2,4,5,7,8 70.78%
6 33,723.66 1,3,4,5,6,7 94.56% 45,191.63 2,4,5,6,7,8 70.57%
7 37,179.48 1,2,3,4,5,6,7 85.77% 45,301.65 1,2,4,5,6,7,8 70.40%
8 45,407.21 1,2,3,4,5,6,7,8 70.23% 45,407.21 1,2,3,4,5,6,7,8 70.23%
0 1 2 3 4 5 6 7
8
0.75
0.8
0.85
0.9
0.95
1Probabilistic Reliability Envelope
Number of Potential Failure Edges
System Efficiency
01234567
8
0.7
0.75
0.8
0.85
0.9
0.95
1Deterministic Reliability Envelope
Number of Failed Edges
System Efficiency
Figure 3. Reliability envelope of solutions in Table 2.
Figure 2. Reliability envelope of solutions in Table 1.

Z. T. WEI ET AL.
158
Notice that the characteristics of this reliability enve-
lope are similar to that of the DRM except that, on the
same edge loss level, the efficiency losses of the SRM
are less than that of the DRM, since we assume that the
failure probability of an edge under a strike is 1 in the
DRM. We can also observe from Table 2 that, for a
given edge attacked level, the most important edges of
protection are shown in volume 6.
5. Summary and Conclusions
In this paper, we propose two types of scenario based
facility location models in order to analyze the reliability
of an existing
ur
done by placing extra
s or tunnels which
adding a surveil-
dge failures to the efficiency of a facility sys-
te
the overall supplement of the facility system will
de
en solving the example since
w
be.
facility system when subject to edge fail-
es. We distinguish deterministic and stochastic cases to
formulate and compute a specific example. Reliability
envelopes in these two different cases are also given. The
information in the reliability envelopes can be very use-
ful in looking at ways to protect a facility system. Whe-
ther the protection is against a natural disaster or inten-
tional strike, reducing the probability of success even by
modest amounts could have an impact on system effi-
iency. For example, this could be c
strength in key sections such as bridge
paced in disaster-prone areas, or bys
lance system with guards to help protect against an in-
truder. Either techniques may not completely eliminate a
loss, by reduce the edge failure probability to zero, but
such strategies may generate more benefits in terms of
improved expected system operating efficiencies than
what it might cost. Therefore, the value of our analysis
could lead to higher levels of safety as well as efficient
levels of resource allocation for security measures
(whether that involves a possible natural disaster or an
attacker).
While a large body of literature focus on the reliable
and robust facility system design and analysis under
component failures, the existing works are mainly con-
centrated on the “node” (e.g., suppliers, distribution cen-
ters) losses. To the best of our knowledge, researchers
and practitioners have not paid enough attention to the
impact of e
Table 2. Results of the worst case SRM and the
level Best-Case
m by so far. Comparing to the related works done in
this field, our work have at least the following innova-
tions.
Firstly, combining edge failures into facility systems
reliability analysis is more realistic than only considering
vertex failures. In fact, natural disasters or intentional
attacks damage the edges of a facility system more easily.
Secondly, in the PMP and other uncapacitated facility
location problems, when one or more “nodes” have
failed,
crease dramatically but the total demand does not
change. If in this situation all demands must to be met,
then every node must has no any capacity limit. However,
the capacity of nodes are designed a priori, when a vertex
failure happen, how can it’ s adjacen t nodes guarantee the
increased demands, let alone more than one vertex fail-
ure occur simultaneously?
The recovery time for a failed edge maybe shorter than
that for a failed vertex, but this is not always the case. So
we do not explicitly point out the time horizon in our
models. In addition, the evaluation of edge failure prob-
ability is important and difficult, and we will discuss this
question in another work. We set the failure probability
of all edges as the same wh
e aimed to demonstrate the impact of edge failures to a
facility system efficiency. Naturally, we need to consider
the further research directions as follows.
We assume that the edge failures are independent each
other, but in practice, once an edge failed, the function of
its adjacent edges will be impacted. Modeling the reli-
able facility systems and related problems under this
situation are worthy of study.
st case SRM with edge failure probability 0.7
Worst-Case
r Objec. Value Attacked Edges Efficiency Objec. Value Attacked Edges Efficiency
0 31,890.22 - 100% 31,890.22 - 100%
1 31,912.22 1 99.93 % 36,826.85 8 86.60 %
2 31,952.84 1,6 99.80 % 39,097.31 2,8 81.57 %
3
32,092.22 99.37 % 39,791.40 2,4,80.14 %
1,3,,6,7 2,3,47,8
1,2,3,6,7 2,3,4,7,8
8 39,993.41 1,2,3,4,5,6,7,8 79.74 % 39,993.41 1,2,3,4,5,6,7,8 79.74 %
31,995.07 1,3,6 99.67 % 39,469.42 2,4,8 80.80 %
4 1,3,5,6 7,8
5 32,390.03 1,3,4,5,6 98.46 % 39,888.56 2,4,5,7,8 79.95 %
6 32,712.01 4,59 7.49 %39,930.78 ,5,79.86 %
7 35,056.77 4,5,90.97 % 39,971.40 5,6,79.78 %

Z. T. WEI ET AL. 159
Wely coe analysis of an existiny
systemn thislthough tpact of l-
ures tofacili is less than that of vertes,
especially when the stem is large, the
design of a reliable/robust facility system cog
edge failures iimportant problem. Weo
study the facil reliams co
both ee failuertex futu
54
lhi, “Tabu Search Heuristic
Protection Devices on Elec-
trical Supply Tree Networks,” Journal of Combinatorial
onnsider thg facilit
i paper. Ahe imedge fai
a ty systemx failure
scale of a facility synsiderin
s also an will als
ity systembility problensidering
dgres and vfailures in the re.
We do not consider the edge capacity constraint in our
models, so it is obvious that not every edge failure will
change the objective function value. For example, the
potential failure edges are all in the delivering paths of
the optimal solution shown in Figure 1. Despite these
potential failure edges are selected according to reality,
thre exist other potential failure edges. Due to the failure
probability of these edges are relatively small and the
consideration of computational time, we omit them.
When consider the capacity of edges, the case will be
different. Modeling and algorithm of problems in this
case may be interacted.
Finally, our models are based on a classical facility
location problem, the UFLP. Next, we will consider edge
failures in more extensive reliable facility location prob-
lems, e.g., CFLP (see [11]) etc.
6. Acknowledgements
This paper was supported by the Open Fund of Xi’an
Jiaotong University (No.2010-4), the SXESF (No. 09JK
5) and the BSF (No.JC0924).
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