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Locating capacitated facilities to maximize captured demand

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We consider the problem of locating a set of facilities on a network to maximize the expected number of captured demand when customer demands are stochastic and congestion exists at facilities. Customers travel to their closest facility to obtain service. If the facility is full (no more space in the waiting room), they attempt to obtain service from the next-closest facility not yet visited from its current position on the network. A customer is lost either when the closest facility is located too far away or all facilities have been visited. After formulating the model, we propose two heuristic procedures. We combine the heuristics with an iterative calibration scheme to estimate the expected demand rate faced by the facilities: this is required for evaluating objective function values. Extensive computational results are presented.
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IIE Transactions (2007) 39, 1015–1029
Copyright
C
“IIE”
ISSN: 0740-817X print / 1545-8830 online
DOI: 10.1080/07408170601142650
Locating capacitated facilities to maximize captured demand
ODED BERMAN
1,
,RONGBING HUANG
2
, SEOKJIN KIM
3
and MOZART B. C. MENEZES
4
1
Joseph L. Rotman School of Management, University of Toronto, 105 St. George Street, Toronto, ON, Canada M5S 3E6
E-mail: berman@rotman.utoronto.ca
2
School of Administrative Studies, York University, Atkinson Building, 4700 Keele Street, Toronto, ON, Canada M3J 1P3
3
Department of Business Administration, Millersville University, P.O. Box 1002, Millersville, PA 17551-0302, USA
4
Department of Logistics and Operations Management, HEC-School of Management, Paris, 1 Rue de la Liberation, F-78351 Jouy
en Josas Cedex, France
Received February 2006 and accepted October 2006
We consider the problem of locating a set of facilities on a network to maximize the expected number of captured demand when
customer demands are stochastic and congestion exists at facilities. Customers travel to their closest facility to obtain service. If the
facility is full (no more space in the waiting room), they attempt to obtain service from the next-closest facility not yet visited from its
current position on the network. A customer is lost either when the closest facility is located too far away or all facilities have been
visited. After formulating the model, we propose two heuristic procedures. We combine the heuristics with an iterative calibration
scheme to estimate the expected demand rate faced by the facilities: this is required for evaluating objective function values. Extensive
computational results are presented.
Keywords: Location, congestion, optimization, queueing
1. Introduction
One of the most important objectives for profit and non-
profit organizations when locating service facilities in a ge-
ographical region is to capture as much potential customer
demand as possible. Strategically, most companies locate
their facilities to maximize their market share and to create
an entry barrier to potential competitors. Without effec-
tive location of facilities, it would be difficult to maintain a
high-market-share position. Most non-profit organizations
attempt to make their facilities more accessible to potential
customers. Regardless of how attractive or useful an organi-
zations products and services may be, if customers cannot
reach them with a desired level of service, the chance of
success is greatly diminished. In this paper, we specifically
attempt to capture maximal customer demand from the
viewpoint of a decision maker at a strategic level.
We assume that customers are concentrated at discrete
points (or nodes) on a network. The weight of a node rep-
resents the demand intensity at that node. It reflects the to-
tal number of potential customers and should be estimated
through an appropriate statistical analysis. The inter-arrival
times of customers at a node are uncertain and drawn from
some distribution specific to that node. A customer who
wishes to obtain service travels to the next “eligible” facil-
Corresponding author
ity (the next-closest facility which has not yet been visited)
if the facility is sufficiently close to her current position on
the network. Otherwise, she gives up service and leaves the
system. If the facility is full (no more space in the waiting
room) when visited, she again considers traveling to the next
eligible facility and so on. Service times at a facility are as-
sumed to be drawn from some distribution identical across
all servers in the network. For simplicity, it is assumed that
each facility hosts a single server.
The problem belongs to a class of optimization prob-
lems, called Location Problems with Stochastic Demands
and Congestion (LPSDC), which attempt to find optimal
locations for a set of facilities in the presence of stochas-
tic demands and potential congestion at the facilities on
a discrete undirected network. A comprehensive and yet
detailed overview of LPSDC is provided by Berman and
Krass (2002). LPSDC models are primarily involved with
two types of uncertainty: (i) the actual amount and timing
of the demand generated by customers; and (ii) a possible
loss of demand or delay of service due to congestion at
service facilities.
It is well known that two important classes of determin-
istic location models in the literature are coverage type and
median type. The two classes have their direct parallels in
the LPSDC context. For median type models the readers
can refer to Berman et al. (1990). Our paper belongs to the
class of coverage-type models.
0740-817X
C
2007 “IIE”
1016 Berman et al.
There are two subclasses of coverage problems: set cov-
ering and maximal covering problems. Most coverage-type
models addressing congestion trace their origins to the
model developed by Daskin (1983). Since the busy fraction
of a server is exogenously given, the stochastic behavior of
an underlying system is not explicitly captured in Daskins
model. However, the model has led to several subsequent
models that have attempted to integrate congestion explic-
itly. Following the initial work of Larson (1975), Batta et al.
(1989) attempt to relax the assumption that busy fractions
of servers are statistically independent.
Pursuing an alternative approach, ReVelle and Hogan
(1989) focus on a local region of a network and attempt to
capture the stochastic behavior of the region more explicitly.
Specifically, region i is the set of nodes within a prespecified
distance of node i.AsinDaskin (1983), the number of busy
servers in a region is assumed to be binomially distributed.
However, the busy fraction of a server is not server specific
butregion specific, i.e., there are different busy fractions
in different regions. Batta et al. (1989) examine both set
covering and maximal covering problems in the framework
of covering-location models for emergency situations that
require multiple response units. Given an upper bound on
exponential service times, Ball and Lin (1993) derive a site-
specific upper bound for busy fractions to provide solutions
satisfying a required level of availability at each node. Fol-
lowing ReVelle and Hogans assumption that the busy frac-
tions are region specific, Marianov and ReVelle (1994) treat
aregion having k servers within as an M/M/k/k system.
Borras and Pastor (2002) provide an ex-post evaluation of
the availability level on the three models in the literature by
simulation and also suggest a new model formulated simi-
lar to the Ball–Lin model incorporating the estimate for the
busy fraction of the ReVelle–Hogan model.
In a closely related paper (Berman, Krass and Wang,
2006) where customers travel only to the closest facility, two
types of demand loss addressed are: “lack of coverage”—
which occurs when none of the facilities are close enough
to customers’ location to provide a sufficient level of con-
venience and “lack of service”—which occurs when a cus-
tomer finds the visiting facility full and thus is lost to the
system.
In this paper, we explicitly allow a dissatisfied customer
(or a “lost” customer due to “lack of service” in Berman,
Krass and Wang, 2006) to visit the next-closest facility from
her current position, so that she may possibly visit several
facilities on the network. Therefore, this paper may portray
better a common customer behavior. The problem of lo-
cating facilities taking advantage of this customer behavior
can be useful in various situations:
1. When the company providing service or product is a
monopoly. For example in Ontario, Canada, liquor is
monopolized by the government which operates LCBO
(Liquor Control Board of Ontario) stores. These stores
are often congested and it is quite common to see cus-
tomers traveling to another LCBO when the current one
is full.
2. Services with loyal customers. For example, customers
having an account with a particular bank tend to stay
loyal since requiring services from other banks incurs
extra cost or inconvenience. Another example in Canada
is gas stations. One of the largest companies is Petro
Canada which gives customers reward points. Therefore,
customers who decide not to obtain gas from a full Petro
Canada station will tend to go to another Petro Canada
station even if there is a competitor gas station around
the corner.
3. In a case of customers that are not fully loyal and exis-
tence of fierce competition, we can use a high sensitivity-
to-distance parameter in the function describing the
fraction of customers willing to visit a facility (defined in
the next section) to take into account the competition.
This paper also extends recent work by Berman, Krass
and Menezes (2006a, 2006b), which allow multiple visits to
facilities assuming that facilities can be disrupted and thus
unavailable to provide service. The objective is to minimize
the total expected cost of traveling. In contrast to our pa-
per, the facilities considered in Berman, Krass and Menezes
(2006a, 2006b) are not congested.
Our main contribution is two-fold: first we introduce the
problem of redirecting demand, when facilities are con-
gested, in a network setting; second, we develop an approx-
imation algorithm to compute the objective function value
that is very difficult to compute analytically and thus avoid
the expensive time cost of simulation.
The rest of this paper is organized as follows: in Section
2weformulate the main problem. Next, in Section 3, we
present the approximation algorithm to evaluate the ob-
jective function. In Section 4 two heuristics are introduced
and in the subsequent section we show some computational
results. Section 6 discusses two natural extensions to the
model and a comparison of several models. We conclude
with final remarks in Section 7.
2. Problem formulation
Let G = (N, E)beanetwork, where N ={1, 2, ···, n} is
the set of nodes and E is the set of edges. The fraction of
the total population associated with node i N is denoted
by w
i
. The demand process at node i is Poisson distributed
with rate λw
i
.For simplicity of notation and without loss
of generality, we assume that λ = 1.
There are p facilities and in each facility there is a single
server who performs service according to an exponential
distribution with rate µ.Facilities are located only at nodes.
A facility is defined by a pair (j, i)where j is a facility and i is
its location. The facility is full when there are c customers
in the system and thus no new customers are allowed to
enter. We denote by d(x, y) the shortest distance between
Locating capacitated facilities 1017
x and y, x, y G and by d(i, L) the shortest distance from
node i to the closest facility in a set of nodes L, i.e., d(i, L) =
min
jL
d(i, j).
Thus, facility j operates as an M/M/1/c queueing sys-
tem. If the number of customers q
j
at facility j is c,avisiting
customer does not enter facility j for service.
The fraction of demands from any node willing to visit
facility j from customers’ current location i is denoted by a
non-increasing function f (d
ij
).
In this paper we consider the piecewise linear convex
function given by
f (d
ij
) = max {1 d
ij
/(αd
max
), 0},
where α>0isthe distance sensitivity of customers and
αd
max
> 0isathreshold distance. We assume that cus-
tomers never visit any facility located beyond the threshold
distance. A typical choice of d
max
is the diameter of the net-
work. Thus, a fraction 1 f (d
ij
)ofcustomers are lost. By
using α
i
(instead of α)inf (d
ij
)wecan implicitly take compe-
tition into account when customers are not fully loyal. We
note that other functional forms of f are available in the
literature, e.g., an exponential decay function f (d) = e
αd
,
a linear decay function f (d) = (u
i
d)/(u
i
l
i
)where l
i
and
u
i
are lower and upper bounds of d such that f (l
i
) = 1 and
f (u
i
) = 0, see Berman, Krass and Wang (2006).
A solution is defined by a pair (S, L), where S =
{1, 2,...,p} is the set of facilities and L ={L(1),...,L(p)}
={L
1
,...,L
p
} is a multi-set of locations, where L(j) (or
L
j
)isthe location of facility j S.Wedenote by ρ
(j)
the
blocking fraction of facility j, i.e., the fraction of time
that q
j
= c,where q
j
is the number of customers at fa-
cility j. Let S
(i)
={S
(i)
1
, S
(i)
2
,...,S
(i)
p
} be the sequence of
facilities to be visited by a customer from node i and
L
(i)
={L
(i)
1
, L
(i)
2
,...,L
(i)
p
} be the set of locations correspond-
ing to S
(i)
, i.e., L
(i)
j
= L(S
(i)
j
). By convention, we let S
(i)
0
= 0,
ρ(S
(i)
0
) = 1, and L
(i)
0
= i.Acustomer from node i first at-
tempts to enter service at facility S
(i)
1
which is located at
L
(i)
1
. Since multiple facilities are allowed at a node, L is a
multi-set. For example, if L ={1, 2, 2, 4}, the second and
third facilities are both located at node 2.
The long-run fraction of demand from node i visiting
facility S
(i)
1
located at L
(i)
1
is w
i
f (d(L
(i)
0
, L
(i)
1
)). If the number
of customers q(S
(i)
1
)atfacility S
(i)
1
is less than c, the customer
from node i enters the facility for service. Otherwise, the
customer visits the next facility S
(i)
2
or leaves the system with
probability 1 f (d(L
(i)
1
, L
(i)
2
)). Define B
i
(S
(i)
j
)tobethe long-
run fraction of time that facility S
(i)
j
is available to provide
service to demand originating from node i,while facilities
S
(i)
0
, S
(i)
1
, ···, S
(i)
j1
are not available (having c customers in
each). The expected demand V
i
(L) originating from node i
and captured by all p facilities is
V
i
(L) = w
i
p
j=1
B
i
(S
(i)
j
)
j
k=1
f
d
L
(i)
k1
, L
(i)
k

. (1)
Since it is impossible to obtain the B
i
(S
i
j
)values analytically,
Equation (1) can not be used to calculate V
i
(L)practically.
We approximate B
i
(S
i
j
)by
1 ρ
S
(i)
j

j
k=1
ρ
S
(i)
k1
,
where (1 ρ(S
(i)
j
)) is the long-run fraction of time that
the jth facility is available and
j
k=1
ρ(S
(i)
k1
)isthe long-
run fraction of time that the first j 1 facilities are
unavailable. This is clearly an approximation since it
assumes that ρ
1
, ···
p
are independent. Therefore, con-
sidering the approximation proposed, the expected de-
mand V
i
(L) originating from node i and captured by all p
facilities is
V
i
(L) = w
i
p
j=1
1 ρ
S
(i)
j

j
k=1
ρ
S
(i)
k1
j
k=1
f
d
L
(i)
k1
, L
(i)
k

= w
i
p
j=1
1 ρ
S
(i)
j

j
k=1
ρ
S
(i)
k1
f
d
L
(i)
k1
, L
(i)
k

,
(2)
and the total expected demand captured by p facilities is
V(L) =
iN
V
i
(L). (3)
The problem is then to maximize the total expected demand
captured by p facilities:
(P1) V(L
) = max
LN
{V(L):|L|=p}.
Note again that (P1) is just an approximation to the orig-
inal problem. This approximation is shown to be very good
(in Section 5, we compare the results obtained by the heuris-
tic algorithms developed for Equation (2) with the results
obtained by simulation using Equation (1)).
Given a fixed set of facility locations, let Q =
(q
1
, q
2
,...,q
p
)bethe state of the system where q
j
is the
number of customers at facility j, j = 1, ···, p. There are
(c + 1)
p
different states of the system for queues of max-
imum size of c which is a huge number. For example, for
queues of maximum length of five customers and ten facil-
ities the total number of states is over 60 000 000.
3. Calibrating blocking fractions
The huge number of system states makes it very difficult
to solve (P1) analytically. Furthermore, the blocking frac-
tion of a facility is dependent on the other (p 1) facilities.
So, by changing one single element in the facility location
1018 Berman et al.
set the steady-state distributions must be recalculated. Our
problem combines the complications of classical location
problems (most of which are NP-complete) and the dy-
namics of queueing systems. A special case of our problem,
when the maximum queue size (or equivalently the service
rate) is infinite, is equivalent to the p-median problem which
is an NP-hard problem.
The incorporation of stochastic aspects into location de-
cisions inevitably leads to intractable formulations which
often requires simplifying assumptions on the problem and
further reasonable approximations for the most important
quantities of interest. Considering the complicated nature
of the problem, we propose two heuristic procedures to im-
prove the objective function value and to keep the feasibility
of solutions at each iteration with computational efficiency.
To do so, we need to approximate the blocking fraction ρ
(j)
of facility j.Foran M/M/1/c queueing system (c.f., e.g.,
Gross and Harris (1985)), we have:
ρ
(j)
=
(λ
(j)
)
c
(1 λ
(j)
)
1 (λ
(j)
)
c+1
if λ
(j)
= µ,
(c + 1)
1
if λ
(j)
= µ,
(4)
where λ
(j)
, the mean demand rate for facility j,isthe sum-
mation of node-to-facility demand assignments to facility
j, i.e., how many customers from all nodes visit facility j S
per unit time. Unfortunately, the exact quantity λ
(j)
is not
known. We propose an iterative approximation procedure
to calibrate the demand rates, λ
(j)
, j,which we will use in
expression (4). We refer to it as Demand Assignment and
Calibration (DAC).
Foragiven set of facility locations and a specific demand
node i,wedesignate the closest facility to i as its first pre-
ferred. If two or more facilities are equally close to demand
node i,wewill choose the facility with the smallest index as
its first preferred. Denote by FP ={f
1
, ···, f
v
} the set of all
facilities that can be chosen as the first preferred by some
demand nodes. Notice that v p since as we mentioned in
the last section, multiple facilities are allowed in a node.
Therefore, it is possible that some facilities will never be
chosen as the first preferred.
A customer visiting her first preferred, say facility f
k
= j,
will follow the same route as all other customers whose
first preferred is facility f
k
. Let SP
k
={SP
1
k
, ···, SP
p
k
} be
the sequence of preferred facilities in which SP
1
k
= f
k
FP
(k = 1, ···,v). Note that once f
k
is chosen, SP
k
is fully de-
termined (i.e., among facilities SP
t
k
, ···, SP
q
k
, SP
t
k
is the
closest facility to facility SP
t1
k
for t = 2, ···,v). Then for
a customer whose first preferred is f
k
, her possible patroniz-
ing sequence is facility f
k
facility SP
2
k
··· facility
SP
p
k
.Ofcourse, she may be served at some facility which
is not at full capacity or just leave the system before visit-
ing all other facilities in SP
k
. Let LP
k
={LP
1
k
, ···, LP
p
k
} be
the set of locations corresponding to SP
k
.Wecan use SP
k
and LP
k
to keep track of customers’ patronizing informa-
Fig. 1. A 10-node example for using DAC.
tion. As an example consider the network depicted in Fig. 1.
Shortest distances and weights are given in Table 1. Suppose
p = 3, S ={1, 2, 3} and L ={7, 3, 9}.Itiseasy to verify that
FP ={1, 2, 3}.WehaveSP
1
={1, 3, 2}, SP
2
={2, 3, 1},
and SP
3
={3, 1, 2}; LP
1
={7, 9, 3}, LP
2
={3, 9, 7} and
LP
3
={9, 7, 3}.Acustomer from either nodes 1, 2, 3, 4, 6
will have SP
2
={2, 3, 1} and will choose facility 2 (at node
3) as her first preferred. Ignoring distance sensitivity, she
will obtain service if facility 2 is not at full capacity; other-
wise, she will visit facility 3 (at node 9). If facility 3 is also
at full capacity, she will attend facility 1 (at node 7).
We refer to a tour traveled from a demand node or a
facility to its next-preferred facility as a “leg”. There are no
first leg demands for those facilities not in FP. Define λ
(j)
m
as the demand rate originated from the first m legs faced by
facility j.Wewould like to estimate λ
(j)
p
.Toobtain λ
(j)
p
we
will have to estimate λ
(j)
m
, m = 1, ···, p, sequentially starting
with λ
(j)
1
. Therefore, we have:
λ
(j)
1
=
{iN: S
(i)
1
=j}
ω
i
f
d
i, L
(i)
1

if j FP,
0ifj ∈ FP.
(5)
Given λ
(j)
1
,wecompute an estimate for ρ
(j)
1
using Equa-
tion (4). In order to estimate λ
(j)
2
,weneed to consider all
the demands visiting facility j as their first preferred one
and also consider all the demands visiting facility j as their
Locating capacitated facilities 1019
Table 1. Shortest distance matrix and fraction of total population
Node
Node 1 2 3 4 5 6 7 8 9 10
10 25.57 34.00 12.53 83.28 12.07 83.45 102.74 74.28 77.28
225.57 0 36.10 37.47 85.38 13.49 85.55 77.17 48.72 79.38
334.00 36.10 0 43.02 49.28 32.37 49.45 73.08 44.25 43.28
412.53 37.47 43.02 0 92.29 24.19 92.46 114.64 86.19 81.51
583.28 85.38 49.28 92.29 0 81.64 19.49 59.76 57.40 16.14
612.07 13.49 32.37 24.19 81.64 0 81.81 90.66 62.21 75.65
783.45 85.55 49.45 92.46 19.49 81.81 0 40.27 37.91 35.63
8 102.74 77.17 73.08 114.64 59.76 90.66 40.27 0 28.83 75.90
974.28 48.72 44.25 86.19 57.40 62.21 37.91 28.83 0 73.54
10 77.28 79.38 43.28 81.51 16.14 75.65 35.63 75.90 73.54 0
Weight 0.0767 0.048 0.0709 0.0578 0.07 0.0373 0.1909 0.1759 0.1881 0.0844
second one, i.e.,
λ
(j)
2
= λ
(j)
1
+
{k:SP
2
k
=j}
λ
(SP
1
k
)
1
ρ
(SP
1
k
)
1
f
d
LP
1
k
, LP
2
k

,
j = 1, ···, p. (6)
As can be seen in Equation (6), the demands that will visit
facility j as the second preferred one must have visited their
most-preferred one first, find it full and have the willingness
to travel the distance to the second-most-preferred facilities.
Once we have λ
(j)
2
,itiseasy to obtain ρ
(j)
2
using Equa-
tion (4). Repeating the process, we have the estimate of the
λ
(j)
m
(m = 2, ···, p)asfollows:
λ
(j)
m
= λ
(j)
m1
+
{k:SP
m
k
=j}
λ
(SP
1
k
)
1
m1
n=1
ρ
(SP
n
k
)
n
f
d
LP
n
k
, LP
n+1
k

,
j, m = 2, ···, p. (7)
The procedure above give us approximations to the true
demand rates λ
(j)
p
and blocking fractions ρ
(j)
p
for all j. This
is a first-round approximation which we will refine later.
Therefore, if ρ
(j)
1
is the blocking fraction for facility j, then
λ
(j)
2
from Equation (6) would be the true demand rate ob-
tained from the first two legs faced by facility j. Based on
this observation, since we would like to obtain steady-state
measures, our second round starts with the results returned
by the first-round approximation, i.e., ρ
(j)
1
of the second
round is set to be equal to ρ
(j)
p
, j obtained at the end
of the first round. Again Equation (7) is used to find λ
(j)
2
and then Equation (4) is used to find ρ
(j)
2
and proceeding in
this way using Equations (7) and (4) to find the values of λ
(j)
m
and ρ
(j)
m
for all j and 2 m p,tofinalize the second-round
approximation.
Similarly, if ρ
(j)
1
and ρ
(j)
2
are the real blocking fractions
for facility j, λ
(j)
3
from Equation (7) will be the true demand
rate obtained from the first three legs. Therefore, at the third
round, we set ρ
(j)
1
and ρ
(j)
2
to be equal to ρ
(j)
p
of the second
round and apply Equations (7) and (4) to obtain λ
(j)
3
and
ρ
(j)
3
sequentially. We perform the same process until our
algorithm is terminated at the pth round.
Given facility location pair (S, L), the DAC algorithm
can be stated as follows:
Procedure DAC
Step 1. Derive FP. SP
k
, LP
k
,where k = 1, ···, |FP| (car-
dinality of FP).
Step 2. Set i := 1, m := 2. Calculate λ
(j)
1
using Equation (5)
and ρ
(j)
1
using Equation (4), j.
Step 3. Calculate λ
(j)
m
using Equation (7) and ρ
(j)
m
using
Equation (4), j.
Step 4. If m < p, set m := m + 1 and do Step 3 until m = p.
Step 5. Set ρ
(j)
1
:= ρ
(j)
p
, ···
(j)
i
:= ρ
(j)
p
, j.
Step 6. If i < p, set i := i + 1, m := i and go to Step 3 oth-
erwise ρ(j):= ρ
(j)
p
, j.
Now let us reconsider the example depicted in Fig. 1. We
assume c = 3, µ = 0.3, d
max
= 114.64 (maximum among
all shortest paths) and the parameter of the decay function
α = 1.
Consider the facility location vector (7, 3, 9). Again,
we have FP ={1, 2, 3}, SP
1
={1, 3, 2}, SP
2
={2, 3, 1} and
SP
3
={3, 1, 2}. The corresponding locations are LP
1
=
{7, 9, 3}, LP
2
={3, 9, 7} and LP
3
={9, 7, 3}. Set i = 1.
Step 2 of Procedure DAC calculates λ
(j)
1
for j ∈{1, 2, 3}.
Recall that nodes {1, 2, 3, 4, 6} are closer to facility 2 than
to any other facility. If we compute for each one of these
nodes the weight times the fraction of demand that will
travel to facility 2 and add the results we obtain λ
(2)
1
= 0.220
613. Using Equation (5) we find λ
1
={0.307 166, 0.220 613,
0.319 758} and, using Equation (4), ρ
1
={0.258 92, 0.148
729, 0.274 402}.
Step 3, m = 2, using Equation (7) and (4) we find λ
2
=
{0.365 896, 0.220 613, 0.393 141} and ρ
2
={0.328 588, 0.148
729, 0.358 458}.
1020 Berman et al.
In Step 4, m = 3, back to Step 3 we find λ
3
={0.370 731,
0.243 306, 0.393 141} and ρ
3
={0.334 009, 0.177 685, 0.358
458}.
Step 5 fixes the first leg blocking fraction to ρ
1
={0.334
009, 0.177 685, 0.358 458},which was the last blocking frac-
tion found in iteration 1 (i = 1). After Step 3 is performed
two times for m = 2 and 3, we have new values for vectors
λ
3
={0.389 988, 0.251 786, 0.412 502} and ρ
3
={0.355 087,
0.188 587, 0.378 665}.
Avoiding unnecessary details, the final result is ρ = ρ
3
=
{0.360 103, 0.191 375, 0.384 562}. The captured demand is
then calculated using the objective function, which returns
avalue of 71.8%. A natural question is: how close Procedure
DAC output is to the result obtained by simulation? The
simulation output is 70.5%, a relative error of less than 2%.
Procedure DAC has two major virtues. First, it yields
very close approximations to the true blocking fractions
(numerical computations show us, as reported later in this
paper, that absolute values of the differences between es-
timates for blocking fractions obtained by the DAC pro-
cedure and those obtained by simulation are very small).
Second, it achieves computational efficiency. Given the
shortest-distance matrix and facility locations, for each
node, to find the closest facility, it requires p 1 compar-
isons. After finding the closest facility, it takes at most
p 1 comparisons to check whether or not this facility
is in FP. Therefore, FP can be obtained in O(np) time.
Since SP
1
k
= f
k
for any k ∈{1, ···,v},ittakes p 1 com-
parisons to obtain SP
2
k
. Therefore, obtaining SP
k
for any k
needs (p 1) + (p 2) + ···+2 = (p 2)(p + 1)/2 com-
parisons. In other words, we can get SP
k
for all k in O(p
3
) ef-
fort. Actually, if we sort the distances among p facilities first,
we can obtain SP
k
even faster. However, as can be seen in the
following, this will not affect the overall complexity. In Step
2, λ
(j)
1
for all j FP can be calculated in O(np ) time and ρ
(j)
1
for all j can be calculated in O(p) time for a given c.Forgiven
m and j, Equation (7) can be obtained in O(p
2
). Thus, Step 3
takes O(p
3
) time (complexity of calculating Equation (4) is
dominated by Equation (7)). Both Steps 4 and 6 call Step 3
again, so the total effort on Equation (7) is O(p
5
). The over-
all complexity of Procedure DAC is max{O(p
5
), O(np )}.
Unless p is large, the algorithm is quite fast.
4. Heuristics
As mentioned earlier, (P1) is very difficult to solve. In this
section we take advantage of the precision of Procedure
DACintwo main heuristic procedures: a Greedy Heuristic
(GH) and a Parametric Heuristic (PH). We also present
aRandomized Heuristic (RH) for benchmark purposes.
RH generates randomly a given number of location of p
facilities, evaluates each one of them using Procedure DAC
and chooses the one that maximizes the captured demand.
Next we describe GH and PH.
4.1. The GH
This heuristic is simple and efficient. We first consider a sin-
gle facility location. Once we find the best single location,
say node i N, using complete enumeration, we locate all
remaining facilities at node i and calculate the objective
function value using Procedure DAC. Now we remove one
facility from node i and insert it at a node that improves the
objective function value the most. If there is no improve-
ment, we leave the facility at node i and stop the algorithm;
otherwise, we remove another single facility from node i re-
peating the procedure until either there is no improvement
or there are no facilities left at node i.
Procedure GH
Step 1. Using complete enumeration obtain the best loca-
tion for a single facility. Denote this node by i.
Step 2. Set S :={1, 2, ···, p}, L :={i, i, ···, i}. Call Proce-
dure DAC to approximate λ and ρ, and calculate
the objective function value denoted by V
using
Equations (2) and (3) where (S, L)isthe input pa-
rameter. Set j := 1, k := 1, L
0
:= L.
Step 3. Call Procedure DAC and obtain the objective
function value using Equations (2) and (3) where
(S, L
k1
\{i}∪{j})isthe input parameter. Call it
V(S, L
k1
, j).
Step 4. If j < n, then set j := j + 1 and go back to Step 3.
Step 5. Denote by V
k
= max
jN
{V(S, L
k1
, j)}, i.e., the
best objective function value returned by Step
3 and Step 4 and denote by j
= arg max
jN
{V(S, L
k1
, j)}.
Step 6. If V
k
= V
, i.e., j
= i, Stop and L
k1
is the “opti-
mal” solution. Otherwise, set L
k
:= L
k1
\{i}∪{j
},
and V
:= V
k
.
Step 7. If k < p, then set j := 1, k := k + 1 and go back to
Step 3. Otherwise, L
p
is the “optimal” solution.
Now let us consider again the example depicted in Fig. 1,
with no changes in the problem parameters.
The GH starts by choosing the best single location in
Step 1 by complete enumeration. The best location turns
out to be node 7. Step 2 places all three facilities at that
node (L = (7, 7, 7)) and V
= 0.6383.
In a series of passages through Steps 3 and 4 together
the location of the third facility is evaluated for all possible
locations in N. The best objective function value is returned,
the location vector (7, 7, 3), in Step 5. Step 6 records this
solution as the best tentative solution and facility 3 is fixed
at node 3 and V
= 0.7089.
The algorithm returns to Steps 3 and 4, the location of
the second facility is evaluated for all possible locations in
N. Step 5 records the best location for the second facility, in
this example (7, 9, 3), and Step 6 includes it as best current
solution and V
= 0.7180.
In the final evaluation, when we test the location of facil-
ity 1, it is found that we can not do better by moving that
Locating capacitated facilities 1021
facility from node 7. Thus, the GH returns (7, 9, 3) with
acaptured demand of 71.8%. The actual optimal objec-
tive function value for this example obtained by complete
enumeration is 72.2% achieved from the location vector
{7, 9, 6}.
4.2. The PH
Recall that:
f (d(i, L)) = max
0, 1
d(i, L)
αd
max
,
where L is the location set of facilities and d(i, L) =
min
jL
{d(i, j)}.
Suppose that the first-leg demand rate, λ
(j)
1
,issignificantly
smaller than the service rate for each j. Then only a small
fraction of customers would travel to the second facility
and thus traveling to more than one facility can be ignored.
In this case, the solution of the k-median problem should
provide a good solution to our problem.
However, for sufficiently large values of d
max
,ifλ
(j)
1
is
significantly larger than the service rate for each j, most
customers would be willing to travel long distances and
thus intuitively it would make sense to locate all facilities at
the 1-median solution which minimizes the total distance
traveled.
Based on this intuitive reasoning we developed a heuristic
that works as follows: first, we solve the p-median problem
with weights w
i
(1 f (d(i, L))). Let L be the obtained loca-
tion set. Then we locate the servers accordingly to L and
calculate V
p
= V (L). Next we solve a (p 1)-median prob-
lem, locate the servers as the solution indicates and locate
the extra server at the location with the largest blocking
fraction among the p 1 ones. If the cost of this new so-
lution, V
(p1)
,isless than V
p
,wecontinue by solving a
(p 2)-median problem, etc. The process is repeated until
either no improvement is made or there are no more prob-
lems to solve. The best solution is recorded and accepted as
the solution to the PH.
The intuition behind this procedure is that, instead of
considering the blocking fractions as functions of location,
we are implicitly estimating the “blocking fraction as a pa-
rameter. That is, when p distinct locations are considered
(the solution of the p-median) we are implicitly assuming
that there is a low blocking fraction so no more than one
facility location is visited. When we consider a solution of
(p j) distinct facility locations for some 0 < j < p,weim-
plicitly assume that more than one facility will be visited
before obtaining service, thus we want to increase the “reli-
ability” of the solution by adding the extra j facilities to the
location with the largest blocking fraction. As we shall see
in the following, this procedure works quite well with some
values of d
max
because the procedure ignores customers’
sensitivity to distance.
We now state the integer programming formulation used
in the PH. The decision variables in the model are
x
ij
=
1ifnode i is served by the facility at node j,
0 otherwise.
y
j
=
1ifafacility is located at node j,
0 otherwise.
We formulate an integer program as follows:
(MP) max
n
i=1
n
j=1
ω
i
(1 f (d(i, j)))x
ij
,
subject to
n
j=1
x
ij
= 1, i = 1, ···, n, (8)
x
ij
y
j
, i, j = 1, ···, n, (9)
n
j=1
y
j
= k, (10)
x
ij
, y
j
∈{0, 1}, i, j = 1, ···, n, (11)
where k (equal to p j + 1inthe jth iteration) is the
number of facilities.
Formulation (MP) corresponds to the classical k-median
problem where we use minimization with “distance”
f (d(i, j)) instead of d(i, j). Therefore, we can use any heuris-
tic for the k-median problem to solve (MP). In this paper
we use a Lagrangian Relaxation Heuristic (LRH) to solve
(MP) (see Narula et al. (1977) and Daskin (1995) for more
details). We note that Lagrangian relaxation is one of the
most computationally attractive heuristics for the k-median
problem. Lagrangian relaxation using a subgradient opti-
mization method gives a very tight lower bound most of the
time (less than 0.15%, see Daskin (1995)).
Before presenting the PH algorithm we want to show that
using only (MP) can deliver a very bad solution to (P1).
Consider V
H
as the best objective function value among
the GH, PH and the RH. Define RE = (V
H
V
M
)/V
H
,
where V
M
is the objective function value returned when the
solution to (MP) is applied.
Proposition 1. Let IN be the set of all possible instances of
(P1), then:
RE
SUP
= sup{RE[I]|I IN}=1. (12)
Proof. Obviously, RE cannot be greater than unity. Con-
sider a network of size n with c = 1 and α 0, implying
that customers are very sensitive to distance and are not
willing to travel farther than their own home nodes. Also
let p = n, w
i
= δ for i ∈{2, ..., n} and w
1
= 1 (n 1)δ.It
is easy to see that the solution to (MP) will be {1, ..., n} for
any choice of µ. Thus, the captured demand in this system
1022 Berman et al.
is given by (recall that λ = 1):
δ
µ
δ + µ
(n 1) + (1 (n 1)δ)
µ
(1 (n 1)δ) + µ
µ
1 + µ
as δ 0.
That solution is independent of the number of nodes in
the network. However, when δ 0, the greedy solution is
to locate all servers on node 1. Suppose further that n
, implying that the M/M// queueing system will
capture every demand even for a small positive value of µ.
Thus, the relative error for this specific instance is given by
RE = 1
µ
1 + µ
as δ 0 and n →∞.
It follows that as µ 0, RE 1asclaimed in Equation
(12) above.
The result above shows that for an unbalanced network
(nodes with high variability of weights) and large cus-
tomers’ sensitivity to travel, the use of the classical approach
to location problems can deliver a poor solution when the
objective is to maximize captured demand under a queue-
ing setting. However, if each facility has a large capacity
in comparison to the total demand then the two problems
become equivalent.
Now we are ready to state the PH.
Procedure PH
Step 1. Set k := p + 1, V
= 0, L
=∅.
Step 2. Set k := k 1. If k = 0 Stop.
Step 3. Call Procedure LRH to solve (MP) and obtain the
solution
ˆ
X = x
1
, ···, ˆx
k
) (may not be optimal).
Step 4. Set m := k. Set S :={1, 2, ···, m} and L =
ˆ
X.If
m = p,gotoStep 7, otherwise repeat Step 5 and
Step 6 until m = p.
Step 5. Call Procedure DAC to approximate λ
(j)
and ρ
(j)
(j = 1, ···, m) using (S, L)asthe input parameters.
Step 6. Suppose that ρ
(j
)
is the largest among ρ
(j)
(j =
1, ···, m), i.e., facility j
is the one with the largest
blocking fraction (ties are broken arbitrarily). Set
m := m + 1, S = S ∪{m}, L = L ∪{ˆx
j
}.
Step 7. Using the location pair (S,L), call Procedure DAC,
and compute the objective function value V
k
using
Equations (2) and (3). If V
k
> V
, set V
= V
k
and
L
= L.GotoStep 2. Otherwise, Stop.
4.3. An upper bound
The LRH works by adjusting the Lagrangian multipliers to
narrow the gap between the lower and upper bounds. Define
V
to be the objective function value of the Lagrangian
dual problem when k = p in Procedure PH. Define ρ
(a, p)
to be the blocking fraction or the proportion of time an
M/M/p/cp queue has cp customers, when customers arrive
at rate a. Then we have the following proposition.
Proposition 2. The function (1 ρ
(V
, p)) is an upper
bound for the maximum captured demand.
Proof.
The upper bound used is calculated using the fol-
lowing two observations:
1. All customers in order to obtain service have to go at
least to the closest facility. Therefore, V
is an upper
bound on the amount of demand that will arrive to the
first preferred facility, because we do not consider any
leg traveled other than the first.
2. If there is no cost to travel from the first facility to the
second facility and onwards, no additional demand is
lost because of distance sensitivity, and we know that
the actual queueing system cannot be more efficient in
capturing demand than an M/M/p/cp system. These
two observations together complete the proof.
As will be shown in the following section, both heuris-
tics perform well when evaluated using the upper bound
developed above. We will also show that the PH is not as
good as the GH in terms of time efficiency for large-scale
problems. However, in terms of solution quality, although
in most cases the greedy procedure provides a superior so-
lution, there is no dominance.
5. Computational experiments
In order to test the two heuristic algorithms, an extensive set
of computational experiments was conducted. All runs were
performed on a PC equipped with 677 MHZ processor and
128M RAM. The procedures were coded in ANSI C. The
problem data used in the experiments were generated ran-
domly as follows. The Cartesian coordinates of the nodes
were generated over the interval (0, 100) uniformly. Then
nodes were connected randomly until a tree was formed.
Finally, a random number of links (this random integer
number is between zero and n(n 1)/2 (n 1), where n
is the cardinality of the network) were added to the tree
generated to create a network. All demand weights were
generated over the interval (0. 1) randomly. The length of
each link was calculated using the Euclidean distance for-
mula. For all problem instances, we ensured that no two
instances shared a common random seed.
First, we investigate how good the Procedure DAC is,
compared to simulation. As described above, we randomly
generated 180 instances with 30 and 50 nodes each. For each
instance we generated a random solution and computed
the objective function value obtained using DAC and the
analogous value obtained using simulation. The results are
shown in Tables 2 and 3. The column “Relative error”gives
intervals of the difference between the objective function
value obtained using DAC and simulation divided by the
latter. The column “Frequency”isthe number of solutions
within the class.
Locating capacitated facilities 1023
Table 2. Comparing DAC with simulation for the 30-node
network
Cumulative
Relative error (%) Frequency percentage (%)
[0, 0.1280] 124 68.89
(0.1280, 0.2559] 18 78.89
(0.2559, 0.3839] 12 85.56
(0.3839, 0.5118] 6 88.89
(0.5118, 0.6398] 5 91.67
(0.6398, 0.7677] 2 92.78
(0.7677, 0.8957] 2 93.89
(0.8957, 1.0236] 5 96.67
(1.0236, 1.1516] 3 98.33
(1.1516, 1.2795] 1 98.89
(1.2795, 1.4075] 2 100.00
As we can see from the tables, about 97% of the solutions
are within 1.00% of the relative error. The results are suffi-
ciently good to conclude that Procedure DAC works very
well.
In Tables 4, 5 and 6, we compare the PH, the GH and the
RH. In the RH we randomly generate 100 locations of p
facilities on the network. The service rate µ varies accord-
ing to |N| and p to avoid some extreme cases that add no
value to the experiment. Table 4 contains computational
results for |N|=30. Ten instances were generated for each
combination of p ∈{4, 7, 10}, c ∈{2, 5} and α ∈{0.15, 0.3}.
For the ten instances from each combination, we obtain the
Average Number of distinct Locations (ANL), the Average
Objective Function Value (AOFV), the average computa-
tional time (labeled as Time in the tables), the fraction of ten
instances that each heuristic gives the best solution among
the three heuristics (labeled as Best in the tables), and the
worst-case ratio (Z
U
Z
H
)/Z
U
(Ratio), where Z
U
is the
upper bound of the optimal objective function value and
Z
H
is the objective function value of heuristic H.
We note that even though ANL is very close to p in most
instances in Tables 4-6, it is not the case in general. For net-
Table 3. Comparing DAC with simulation for the 50 node
network
Cumulative
Relative error (%) Frequency percentage (%)
(0, 0.2373] 151 83.89
(0.2373, 0.3559] 6 87.15
(0.3559, 0.4745] 9 92.18
(0.4745, 0.5932] 3 93.85
(0.5932, 0.7118] 0 93.85
(0.7118, 0.8304] 3 95.53
(0.8304, 0.9491] 1 96.09
(0.9491, 1.0677] 3 97.77
(1.0677, 1.1863] 1 98.32
(1.1863, 1.3050] 3 100.00
works with large variability of weights, ANL can be much
smaller than p (Proposition 1 shows an extreme case of this).
Therefore, using (MP) as a heuristic might be dangerous.
Moreover, solving (MP) is the starting point of the PH.
We repeat similar experiments for |N|=50 (Table 5)
with combinations of p ∈{5, 10, 15}, c ∈{2, 5} and α
{0.1, 0.25}.For|N|=80 (Table 6), we limit p to p = 8 with
combinations of c ∈{2, 5} and α ∈{0.05, 0.1}.
Overall, the GH and the PH outperform the RH in terms
of worst-case ratio, which is the main performance criterion
of our interest. The average worst-case ratio of the RH is 0.2,
0.23 and 0.26 respectively for |N|=30, 50, 80. For |N|=30
and |N|=80, the PH and GH have the same average worst-
case ratio of 0.11 and 0.06, respectively. This implies that
both heuristics perform even better for a larger network.
For |N|=50, the average worst-case ratio of the GH is 0.1,
which is slightly better than the value of 0.11, recorded for
the PH.
Note that when µ increases the worst-case ratios of the
PH and the GH get smaller. When µ increases, the system
is overall less congested and thus tends to accept more cus-
tomers who are also accepted by the M/M/p/cp system. In
this case, it would yield a better objective function value by
spreading facilities over different nodes and thus accepting
more customers who might be sensitive to distances.
The RH takes the least computational time with smallest
objective function values. The GH outperforms PH and is
highly efficient in terms of computational time. It is inter-
esting to observe that for |N|=50 and p = 15, GH takes
more time than for |N|=80 and p = 8, which shows that
computational time is primarily affected by p.For|N|=80,
the GH is much more efficient than the PH.
Table7address the final question: what is the relative
error between the GH and PH solution values (using DAC)
and the actual optimal solution values, derived by complete
enumeration where the Objective Function Value (OFV) is
calculated by simulation? The experiments were done on
a small random network of 12 nodes which enables us to
solve problems by complete enumeration. The results are
encouraging, with the GH returning the optimal solution
in more than 50% of the cases and the Relative Errors (RE)
are small. The PH returns the optimal solutions about 20%
of the time with small relative errors (average is even smaller
than for the GH).
6. Additional formulations
We now change our focus from (P1) to problems that ad-
dress in some sense the same objective of capturing demand,
but assuming a different customer behavior. By looking at
different customer behavior we can obtain several interest-
ing insights.
In (P2), which has a similar formulation as (P1), we as-
sume that for a customer from i whoiscurrently visiting
facility k the probability that facility j will be visited next is
Table 4. Computational results for |N|=30
PH GH RH
# |N| p µ c α ANL AOFV Time Best Ratio ANL AOFV Time Best Ratio ANL AOFV Time Best Ratio
130 40.1320.15 4 0.26 5.53 0.7 0.20 4 0.26 0.21 1 0.20 3.9 0.24 0.19 0 0.26
230 40.1320.33.4 0.34 2.94 0.2 0.23 4 0.35 0.20 0.9 0.21 3.9 0.34 0.17 0.1 0.24
330 40.1350.15 4 0.32 4.72 0.7 0.08 4 0.33 0.19 1 0.07 4 0.29 0.16 0 0.19
430 40.1350.33.9 0.41 3.90 0.3 0.14 4 0.42 0.19 0.9 0.13 4 0.39 0.16 0 0.17
530 40.2520.15 4 0.33 4.23 0.9 0.10 4 0.33 0.18 0.9 0.10 4 0.30 0.16 0 0.19
630 40.2520.33.9 0.44 2.98 0.4 0.15 4 0.45 0.18 0.6 0.14 4 0.41 0.16 0 0.20
730 40.2550.15 4 0.36 3.68 0.9 0.01 4 0.36 0.18 1 0.01 4 0.31 0.16 0 0.14
830 40.2550.34 0.50 3.00 0.9 0.03 4 0.50 0.19 0.6 0.04 4 0.46 0.16 0 0.12
930 40.3820.15 4 0.32 4.56 0.9 0.04 4 0.32 0.19 1 0.04 4 0.29 0.16 0 0.14
10 30 4 0.3820.34 0.46 3.38 0.7 0.09 4 0.46 0.18 0.7 0.09 4 0.42 0.16 0.1 0.15
11 30 4 0.3850.15 4 0.38 4.51 1 0.00 4 0.38 0.19 1 0.00 4 0.33 0.16 0 0.12
12 30 4 0.3850.34 0.52 3.39 1 0.01 4 0.52 0.19 0.8 0.01 4 0.47 0.16 0 0.10
13 30 7 0.0720.15 6.8 0.34 11.25 0.4 0.27 7 0.34 1.06 1 0.27 6.9 0.31 0.60 0 0.34
14 30 7 0.0720.36 0.38 7.94 0 0.22 6.9 0.39 1.05 1 0.20 6.8 0.37 0.60 0 0.25
15 30 7 0.0750.15 6.7 0.42 10.44 0.4 0.15 7 0.42 1.06 1 0.14 7 0.37 0.59 0 0.25
16 30 7 0.0750.35.7 0.46 7.61 0 0.08 6.6 0.47 1.06 1 0.06 6.9 0.44 0.60 0 0.11
17 30 7 0.1420.15 7 0.45 10.95 0.7 0.17 7 0.45 1.06 0.9 0.17 6.9 0.38 0.60 0 0.29
18 30 7 0.1420.36.4 0.54 7.73 0 0.22 6.8 0.56 1.04 1 0.20 7 0.52 0.59 0 0.25
19 30 7 0.1450.15 7 0.51 10.39 0.6 0.03 7 0.51 1.06 0.9 0.03 6.9 0.42 0.59 0 0.20
20 30 7 0.1450.36.9 0.63 8.10 0.5 0.08 7 0.63 1.05 0.6 0.08 6.9 0.57 0.59 0 0.17
21 30 7 0.2120.15 7 0.47 10.43 0.8 0.09 7 0.47 1.05 0.7 0.09 7 0.41 0.60 0 0.22
22 30 7 0.2120.36.8 0.59 6.66 0.4 0.14 7 0.60 1.05 0.7 0.13 6.9 0.54 0.60 0 0.21
23 30 7 0.2150.15 7 0.51 10.35 1 0.01 7 0.51 1.06 1 0.01 6.9 0.43 0.60 0 0.16
24 30 7 0.2150.37 0.67 7.31 0.8 0.03 7 0.67 1.06 0.3 0.03 6.9 0.59 0.60 0 0.14
25 30 10 0.0520.15 9.3 0.38 18.56 0 0.24 9.9 0.38 4.45 1 0.24 9.2 0.34 1.93 0 0.32
26 30 10 0.0520.38.4 0.42 11.72 0 0.17 8.6 0.43 4.36 1 0.14 9 0.40 1.94 0 0.19
27 30 10 0.0550.15 9.3 0.45 18.17 0 0.10 10 0.46 4.49 1 0.08 9.5 0.40 1.94 0 0.19
28 30 10 0.0550.38.7 0.48 13.76 0 0.05 9.1 0.49 4.40 0.9 0.03 8.9 0.47 1.94 0.1 0.06
29 30 10 0.120.15 9.7 0.51 18.67 0 0.22 10 0.52 4.68 1 0.21 9.7 0.44 2.02 0 0.33
30 30 10 0.120.39.2 0.60 8.97 0.2 0.25 9.9 0.62 4.56 0.8 0.23 9.5 0.57 1.97 0 0.29
31 30 10 0.150.15 9.9 0.62 18.85 0.6 0.07 10 0.62 4.60 1 0.07 9.7 0.52 2.00 0 0.23
32 30 10 0.150.39.4 0.72 10.54 0.2 0.11 10 0.73 4.62 0.8 0.10 9.8 0.65 1.98 0 0.20
33 30 10 0.1520.15 10 0.57 18.20 0.7 0.13 10 0.57 4.60 0.8 0.13 9.7 0.48 1.99 0 0.26
34 30 10 0.1520.39.7 0.67 13.76 0 0.15 9.9 0.68 4.61 1 0.14 9.7 0.63 1.99 0 0.21
35 30 10 0.1550.15 10 0.66 18.47 0.9 0.02 10 0.66 4.51 1 0.02 9.6 0.53 1.95 0 0.20
36 30 10 0.1550.39.9 0.77 12.94 0.5 0.04 10 0.77 4.52 0.5 0.04 9.5 0.67 1.97 0 0.16
9.41 0.48 0.11 1.93 0.87 0.11 0.91 0.008 0.20
1024
Table 5. Computational results for |N|=50
PH GH RH
# |N| p µ c α ANL AOFV Time Best Ratio ANL AOFV Time Best Ratio ANL AOFV Time Best Ratio
150 50.120.15 0.22 49.07 0.9 0.16 5 0.22 2.18 1 0.16 5 0.19 0.91 0 0.27
250 50.120.25 4.5 0.33 50.39 0.2 0.25 5 0.33 2.24 0.8 0.23 5 0.31 0.94 0 0.30
350 50.150.15 0.27 53.74 0.9 0.04 5 0.27 2.23 0.9 0.04 5 0.23 0.93 0 0.17
450 50.150.25 4.8 0.40 49.75 0.1 0.16 5 0.41 2.18 1 0.14 5 0.37 0.95 0 0.22
550 50.220.15 0.26 52.49 1 0.06 5 0.26 2.38 1 0.06 4.9 0.22 0.97 0 0.21
650 50.220.25 5 0.41 40.47 0.6 0.14 5 0.41 2.12 0.4 0.14 5 0.37 0.89 0 0.22
750 50.250.15 0.25 48.57 1 0.00 5 0.25 2.12 1 0.00 5 0.22 0.90 0 0.15
850 50.250.25 5 0.47 41.89 0.9 0.02 5 0.47 2.12 0.6 0.03 5 0.42 0.89 0 0.14
950 50.320.15 0.26 48.07 1 0.03 5 0.26 2.11 1 0.03 5 0.21 0.89 0 0.20
10 50 5 0.320.25 5 0.46 41.39 0.7 0.08 5 0.46 2.11 0.6 0.08 5 0.40 0.89 0 0.20
11 50 5 0.350.15 0.29 46.12 0.9 0.00 5 0.29 2.11 1 0.00 5 0.23 0.89 0 0.18
12 50 5 0.350.25 5 0.48 39.85 1 0.01 5 0.48 2.13 0.4 0.01 5 0.42 0.89 0 0.14
13 50 10 0.05 2 0.19.8 0.31 123.57 0.6 0.28 9.9 0.31 22.05 1 0.28 9.4 0.26 5.82 0 0.40
14 50 10 0.05 2 0.25 9.1 0.39 109.26 0 0.22 9.7 0.40 21.83 1 0.20 9.4 0.37 5.85 0 0.26
15 50 10 0.05 5 0.19.9 0.39 120.27 0.3 0.15 10 0.39 22.07 0.9 0.14 10 0.31 5.84 0 0.30
16 50 10 0.05 5 0.25 8.9 0.46 113.99 0 0.08 9.5 0.47 22.04 1 0.05 9.7 0.45 5.82 0 0.11
17 50 10 0.120.110 0.39 122.48 0.8 0.13 10 0.39 22.04 1 0.13 9.9 0.31 5.86 0 0.31
18 50 10 0.120.25 9.2 0.54 107.42 0.2 0.22 10 0.56 22.04 0.8 0.20 9.8 0.50 5.83 0 0.29
19 50 10 0.150.110 0.45 123.84 0.9 0.02 10 0.45 22.05 0.9 0.02 9.8 0.35 5.86 0 0.25
20 50 10 0.150.25 10 0.64 105.02 0.4 0.08 10 0.64 22.04 0.6 0.08 9.8 0.56 5.83 0 0.20
21 50 10 0.15 2 0.110 0.43 122.34 0.8 0.08 10 0.43 22.06 0.9 0.08 9.9 0.34 5.84 0 0.26
22 50 10 0.15 2 0.25 9.7 0.60 113.89 0.4 0.13 10 0.61 22.07 0.6 0.12 10 0.54 5.84 0 0.22
23 50 10 0.15 5 0.110 0.45 123.14 0.9 0.00 10 0.45 22.05 0.9 0.00 9.8 0.35 5.86 0 0.22
24 50 10 0.15 5 0.25 10 0.68 103.57 0.3 0.04 10 0.68 22.13 0.7 0.04 9.8 0.59 5.84 0 0.16
25 50 15 0.03 2 0.114.3 0.36 213.95 0 0.28 14.9 0.36 129.90 1 0.27 14 0.31 24.88 0 0.38
26 50 15 0.03 2 0.25 12.8 0.41 170.68 0 0.17 12.2 0.43 119.70 1 0.13 13.3 0.40 24.83 0 0.19
27 50 15 0.03 5 0.114.5 0.44 217.79 0 0.12 14.9 0.44 130.59 1 0.11 14.7 0.37 25.47 0 0.27
28 50 15 0.03 5 0.25 13.3 0.48 171.63 0 0.05 13.9 0.49 127.48 1 0.02 14.3 0.46 24.87 0 0.08
29 50 15 0.07 2 0.114.9 0.49 210.71 0.3 0.20 15 0.49 130.28 0.8 0.20 14.3 0.39 24.98 0 0.36
30 50 15 0.07 2 0.25 14.3 0.61 162.70 0 0.25 14.8 0.62 131.23 1 0.23 14.2 0.56 25.30 0 0.30
31 50 15 0.07 5 0.114.9 0.58 231.75 0.4 0.05 15 0.58 144.24 1 0.05 14.3 0.44 27.38 0 0.27
32 50 15 0.07 5 0.25 14.5 0.72 186.99 0.1 0.11 15 0.73 147.60 0.9 0.10 14.4 0.64 28.13 0 0.21
33 50 15 0.120.115 0.53 235.51 0.7 0.11 15 0.54 148.75 0.8 0.11 13.9 0.42 28.24 0 0.31
34 50 15 0.120.25 14.7 0.67 191.52 0.1 0.15 15 0.68 155.79 0.9 0.13 14.1 0.60 29.15 0 0.24
35 50 15 0.150.114.9 0.61 1187.28 0.8 0.02 15 0.61 822.02 0.9 0.02 14.4 0.46 165.34 0 0.27
36 50 15 0.150.25 15 0.78 1574.69 0.7 0.04 15 0.77 936.43 0.3 0.04 14.4 0.67 175.75 0 0.17
186.27 0.50 0.11 94.85 0.85 0.10 19.04 0.00 0.23
1025
Table 6. Computational results for |N|=80
PH GH RH
# |N| p µ c α ANL AOFV Time Best Ratio ANL AOFV Time Best Ratio ANL AOFV Time Best Ratio
18080.06 2 0.05 8 0.19 1078.38 0.7 0.12 8 0.19 87.24 1 0.12 8 0.15 16.35 0 0.30
28080.06 2 0.180.24 1057.98 0.4 0.20 8 0.24 87.19 0.9 0.19 7.9 0.20 16.33 0 0.35
38080.06 5 0.05 8 0.23 1075.90 0.8 0.02 8 0.23 87.13 1 0.02 8 0.18 16.30 0 0.23
48080.06 5 0.180.30 1052.43 0.9 0.05 8 0.30 87.25 0.9 0.06 8 0.24 16.45 0 0.25
58080.19 2 0.05 8 0.22 1075.45 0.9 0.02 8 0.22 87.20 1 0.02 8 0.17 16.35 0 0.25
68080.19 2 0.180.29 1054.16 1 0.04 8 0.29 87.19 0.8 0.04 8 0.24 16.36 0 0.23
78080.19 5 0.05 8 0.23 1076.94 0.9 0.00 8 0.23 87.20 0.9 0.00 8 0.18 16.33 0 0.22
88080.19 5 0.180.31 1076.29 1 0.00 8 0.31 87.11 0.5 0.00 8 0.23 16.29 0 0.25
1068.44 0.83 0.06 87.19 0.88 0.06 16.35 0.00 0.26
1026
Locating capacitated facilities 1027
Table 7. Comparing the GH and PH with complete enumeration using simulation for 12-node network with p = 3, µ = 0.3 and c = 1
OFV of OFV of RE of GH found OFV of RE of Parametric found
α simulation GH GH (%) optimal solution? PH PH (%) optimal solution?
0.25 0.3578 0.3546 0.92 Yes 0.3546 0.92 Yes
0.75 0.4932 0.5002 1.43 No 0.4991 1.20 No
1.25 0.5466 0.5609 2.62 No 0.5609 2.62 No
0.25 0.3465 0.3500 1.00 Yes 0.3376 2.57 No
0.75 0.5028 0.5131 2.04 No 0.4938 1.78 No
1.25 0.5511 0.5696 3.37 No 0.5627 2.11 No
0.25 0.3379 0.3361 0.52 No 0.3367 0.36 Yes
0.75 0.4868 0.4933 1.32 Yes 0.4673 4.01 No
1.25 0.5403 0.5577 3.21 No 0.5485 1.50 No
0.25 0.3580 0.3653 2.04 Yes 0.3589 0.28 No
0.75 0.5083 0.5165 1.63 Yes 0.4953 2.55 No
1.25 0.5511 0.5691 3.26 Yes 0.5596 1.53 No
0.25 0.3686 0.3688 0.06 Yes 0.3688 0.06 Yes
0.75 0.5116 0.5225 2.14 Yes 0.4946 3.31 No
1.25 0.5567 0.5726 2.85 No 0.5660 1.67 No
0.25 0.3495 0.3533 1.07 Yes 0.3497 0.05 No
0.75 0.4862 0.4953 1.88 Yes 0.4594 5.51 No
1.25 0.5423 0.5592 3.12 No 0.5477 1.00 No
0.25 0.3535 0.3513 0.60 Yes 0.3513 0.60 Yes
0.75 0.4753 0.4887 2.82 No 0.4809 1.18 No
1.25 0.5379 0.5617 4.42 No 0.5617 4.42 No
Average 2.01 1.87
Percentage of finding optimal solution (%) 52.38% 19.05%
a function of the total distance traveled from i to k plus the
leg from k to facility j. Thus, the contribution that node i
makes to the objective function value in (P2) is
V
P2
i
(L) = w
i
p
j=1
1 ρ
S
(i)
j

j
k=1
ρ
S
(i)
k1
f
k
t=1
d
L
(i)
t1
, L
(i)
t
. (13)
The problem is then to maximize the total expected de-
mand captured by p facilities as follows:
(P2) V
P2
(L
) = max
LN
{V
P2
(L) =
iN
V
P2
i
(L):|L|=p}.
(14)
Note that the customer behavior in (P2) is not memo-
ryless as it was in (P1). Here, customers having the same
search path as in (P1) will have a smaller expected number
of facilities visited. Hence, the demand captured in (P2),
which we call P2[L], is always smaller than or equal to the
demand captured in (P1), which we call P1[L], as is stated
in observation 1.
Observation 1. If the distance tolerance function f is non-
increasing, then for any set of facility locations L, P1[L]
P2[L].
Anatural problem to investigate is the problem when
customers receive information so they can travel directly to
an available facility. We call this (P3). Let
V
P3
i
(L) = w
i
p
j=1
1 ρ
S
(i)
j

f
d
i, L
(i)
j

j
k=1
ρ
S
(i)
k1
,
(15)
(P3) V
P3
(L
) = max
LN
V
P3
(L) =
iN
V
P3
i
(L):|L|=p
.
(16)
Obviously due to the triangle inequality property, cus-
tomers will always travel less in the setting of (P3) than in
the setting of (P2).
Observation 2. If the distance tolerance function f is non-
increasing, then for any set of facility locations L, P3[L]
P2[L].
The relationship between P1[L] and P3[L]isnot clear.
On the one hand, by knowing the total distance traveled to
obtain service, customers are less likely to begin to travel at
all in the setting of (P3). On the other hand, the distance
traveled (search path) to obtain service from the jth facility
in (P1) may be much larger than directly traveling from node
i to facility j in (P3).
Comparing the last problem with the other two give us
a sense of how much is lost (or gained) due to customers’
lack of information about facility congestion.
1028 Berman et al.
Table 8. Comparing (P1), (P2), (P3) and (BKW)
OFV of OFV of OFV of OFV of
α Solution (P1) (P2) (BKW) (P3)
0.5 S
1
={1, 7, 9} 0.626 0.620 0.607 0.631
S
2
={1, 7, 9} 0.626 0.620 0.607 0.631
SB ={1, 7, 9} 0.626 0.620 0.607 0.631
1.0 S
1
={6, 7, 9} 0.722 0.698 0.660 0.742
S
2
={1, 7, 9} 0.720 0.699 0.662 0.740
SB ={1, 7, 9} 0.720 0.699 0.662 0.740
1.5 S
1
={3, 7, 9} 0.765 0.732 0.669 0.776
S
2
={6, 7, 9} 0.764 0.735 0.677 0.769
SB ={1, 7, 9} 0.762 0.735 0.678 0.782
2.0 S
1
={7, 9, 9} 0.791 0.738 0.510 0.782
S
2
={6, 7, 9} 0.787 0.760 0.685 0.780
SB ={1, 7, 9} 0.785 0.759 0.686 0.779
Finally, we consider (BKW), introduced by Berman,
Krass and Wang (2006). They use the probability of vis-
iting a facility as a function of the distance traveled, but
customers only travel to the closest facility and are either
served or lost. Clearly, this problem cannot capture more
demand than any of the problems above. Hence,
Observation 3. If the distance tolerance function f is non-
increasing, then for any set of facility locations L, BKW[L]
min{P1[L], P2[L], P3[L]} = P2[L].
To shed more light on this discussion we present an exam-
ple for which we demonstrate the differences in the objective
function values P1[L], P2[L], P3[L] and BKW[L]. Let S1,
S2, S3 and SB be the corresponding optimal solutions of
(P1), (P2), (P3) and (BKW).
Although Procedure DAC works very well for (P1) and
(P2), it does not deliver good results for (P3). Therefore, we
will not attempt to optimize (P3) but compute the objec-
tive function value using simulation for any given facility
location set.
We use Fig. 1 with all parameters having the same values
stated earlier except for α that varies in {0.5, 1.0, 1.5, 2.0}.
The results are summarized in Table 8. For each problem we
state the optimal solution and the OFV. The table compares
the objective function values of the optimal solutions to the
three problems.
For instance, when α = 1.0, S2, the optimal solution
to (P2), is {1, 7, 9} and the objective function value of
S
2
, P2[S2] = 0.699. Also, P1[S1] = 0.722 and BKW[SB] =
Table 9. Comparing (P1), (P2), (P3) and (BKW) for the 25-node
case with µ = 0.5, α = 1.25 and c = 1
OFV of OFV of OFV of OFV of
Solution (P1) (P2) (BKW) (P3)
S
1
={5, 21, 23} 0.670 0.608 0.504 0.709
S
2
={5, 16, 18} 0.662 0.619 0.541 0.701
SB ={4, 8, 22} 0.640 0.600 0.547 0.704
0.662. From the same table, if we compute the objective
function value of (BKW) using the solution for (P1) we
have that BKW[S1] = 0.660 and P3[S1] = 0.742. Note that,
when α = 2.0, P3[S1] < P1[S1], implying that providing
customers with information about facility congestion does
not help to capture more demand. However, when α = 1.0,
P3[S1] > P1[S1]. Hence, as customers are more prone to
travel an extra leg in search of service the decision maker
is less willing to provide that information. If customers, on
the other hand, are very sensitive to distance, giving the
information is beneficial.
This change in the relation of objective function values
between (P1) and (P3) does not occur between (P2) and
(P3). Note that it is always better to provide information
about facility congestion if customers behave as assumed
in (P2).
Another interesting question is how much is lost when
using the solution to (BKW), as a proxy for the solution to
(P1). In this particular example, it can be easily seen from
the tables above that there is no major loss in doing so. How-
ever, in general this may be dangerous. For example, for an
instance with 25 nodes and µ = 0.5, α = 1.25 and c = 1,
as shown in Table 9, the relative difference between P1[SB]
and P1[S1] is close to 4.5% but even more relevant, the ab-
solute difference is 3%, a percentage of demand that goes
directly to the company’s bottom line. We omit the results
for other values of α for that same instance but similar val-
ues in the differences were found. In summary, instances
returning up to 6% in the absolute differences are not
uncommon.
7. Conclusions
In this paper we extended the model presented by Berman,
Krass and Wang (2006), where distance-sensitive customers
only obtain service from the closest facility. We presented
three alternatives to that model.
In the first model, customers travel from one facility to
another until service is obtained or their sensitivity to dis-
tance traveled forces them to abandon the search. In this
model we consider that the sensitivity is a function of the
distance between the current location and the next-closest
facility; the distance traveled up to that moment has no
impact in the upcoming decision.
In the second model, customers take into account the to-
tal distance traveled from the home nodes. The third model
considers the case where the decision maker gives full infor-
mation to customers about congestion. Customers can then
travel directly to the closest facility that is not congested.
We showed the relationship between the objective func-
tions of these three models, and discussed some bounds. The
choice of models should be done according to customers’
behavior. Alternatively, the decision maker may be capable
of forcing the customers to behave in a way that impacts the
objective function value better. Note that the first and third
Locating capacitated facilities 1029
models do not have a fixed relationship in terms of objec-
tive function values. Thus, if customers behave according to
the first model then they can behave differently if informa-
tion about congestion is given. Alternatively, by omitting
that information the behavior can shift from the one de-
scribed by the third model to that considered in the first
model.
Another contribution of this paper is the approximation
algorithm (DAC) to calculate objective function values. The
DACprocedure is a valuable tool to get approximate val-
ues given that the values obtained by simulation are time
expensive. Even small problems can take close to an hour
to obtain the objective function value through simulation.
However, for any one of the heuristics presented the objec-
tive function value has to be computed many times.
Future research can be done in the direction of finding
ways to optimize the second and third models which we just
touched on. We believe that while the methodology to opti-
mize (P2) will be quite similar to that of (P1), (P3) requires
a completely different approach since for each point there is
a different ordering of preference for the visiting facilities.
A hint of the complexity of (P3) can be seen in the working
paper by Berman, Krass and Menezes (2006b), where the
probability of finding an operating facility is not a function
of facility location but a fixed exogenous parameter.
Finally, empirical research using our models with the
objective of addressing the issues of customers’ behavior
could be interesting. The real impact of locating facilities
under congestion is just starting to be understood from
the theoretical perspective but very little has been done to
evaluate the validity and relevance of our models from the
empirical side.
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Biographies
Oded Berman is the endowed Sydney Cooper Chair in Business and
Technology and the former Associate Dean of Programs at the Joseph
L. Rotman School of Management at the University of Toronto. He re-
ceived his Ph.D. (1978) in Operations Research from the Massachusetts
Institute of Technology. He had been with the Electronic Systems Lab at
MIT, the University of Calgary, and the University of Massachusetts at
Boston, where he was also the Chairman of the Department of Manage-
ment Sciences. He has published over 170 articles and has contributed to
several books in his field. His main research interests include operations
management in the service industry, location theory, network models,
and software reliability. He is an Associate Editor for Management Sci-
ence and Transportation Science, and a member of the editorial board for
Computers and Operations Research.
Rongbing Huang is an Assistant Professor at the School of Administra-
tive Studies, York University, Canada. He holds a Ph.D. degree in Oper-
ations Management from the Joseph L. Rotman School of Management,
University of Toronto. His research interests include facility location the-
ory and combinatorial auctions. He teaches courses in business statistics,
quantitative methods, decision analysis and introduction to operations
research.
Seokjin Kim is an Assistant Professor of Management at the Depart-
ment of Business Administration, Millersville University. His teaching
assignments include research methods in business, quantitative methods
forbusiness, and production and operations management. His research
deals with stochastic optimization models in facility location and work-
force scheduling. He received a Ph.D. degree in Operations Management
at the University of Toronto’s Rotman School of Management and an
M.S. degree in Engineering-Economic Systems (currently, Management
Science and Engineering) at the Stanford University. He also received
M.B.A. and B.B.A. degrees in Business Administration at the Yonsei
University, South Korea.
Mozart B. C. Menezes is an Assistant Professor at the Department of Op-
erations Management and Information Technology at the HEC School
of Management, Paris. His teaching assignments include supply chain
management, inventory management and combinatorial optimization.
His research interests include supply chain management, facility location
problems, inventory management and services operations management.
His B.Sc. degree in Civil Engineering was obtained at the Universidade
Federal do Para, Brazil. He holds a M.Sc. degree in Industrial Admin-
istration and another in Project Management from Clemson University
and a Ph.D. degree in Operations from the Joseph L. Rotman School of
Management, University of Toronto.
... Generally, Covering models are of two types, namely, Location Set Covering Problem (LSCP) and Maximal Covering Location Problem (MCLP) (Ayeni, 1992;Berman, Huang, Kim, and Menezes, 2007;Jia et al., 2005;Marianov and Serra, 2002). Location Set Covering Problems (LSCP) seek to determine the minimum number of facilities and the optimal locations that will place all demand points within a critical distance of a service centre. ...
... Modelling location of infrastructure within a GIS framework may not be new as previous endeavours have employed optimization software coupled (tightly or loosely) with GIS software to achieve reliable location models that are somewhat realistic (Berman et al., 2007;Chevalier et al., 2010;Church and Sorensen, 1995). Nonetheless, the present effort wholly used GIS based optimization model to evaluate and prescribe locational configurations of ICT service infrastructures. ...
Thesis
Full-text available
Availability and access to efficient and affordable telecommunication systems are central to promoting rapid socio-economic development in any society. However, the goal of universal access to information and communication technologies (ICTs) has remained unrealised partially due to the uneven distribution of network (Base Transceiver Stations (BTS)) and service infrastructures (cybercafés and call centres) in Nigeria. Previous studies on disparity in access to ICT services emphasised socioeconomic barriers, often ignoring the role of infrastructure location and its impact on access to, and use of ICT services. This study, therefore, analysed the location and predictors of access to, and use of ICT infrastructures in the Ibarapa area of Oyo State. Central Place Theory, location-allocation models, and the concept of universal access guided the study and survey design was adopted. Using a purposive sampling technique, 710(2.0%) out of 35,500 households were proportionately drawn from 32 settlements in the study area. A structured questionnaire containing socio-demographic characteristics (age, sex, income and level of education), accessibility to ICT infrastructures (call centre and cybercafé availability, distance to call centre and cybercafé, cost of service) and utilisation (frequency of calls, number of hours online, perceived usefulness of phone and internet services) were administered to the respondents. Data on settlements' location and their characteristics were sourced from maps and records obtained from Oyo State Ministries of Lands and Physical Planning and Agriculture and Rural Development. Descriptive statistics, multiple and logistic regressions were used to determine the determinants of ICT infrastructure location and service utilisation, while analysis of variance was employed to analyse differentials in access to ICT services between rural and urban households. P-median and maximal covering location models were used to evaluate the locational optimality of existing ICT infrastructures. All hypotheses were tested at a 0.05 level of significance. Respondents' average age and income were 46±15 years and N23,599±16,803 respectively, while 36.0% had secondary education. Household heads' income (β=0.71; t=7.54), level of education (β=0.54; t=4.24), distance from a call centre (β=-0.64; t=-8.52), cost of using phone service (β=-0.433; t=-7.67) and perceived usefulness of phone service (β=0.33; t=2.98) were important predictors of telephone usage. Monthly income (β=1.75, t=2.96), cybercafé availability (β=4.51; t=7.91) and distance of cybercafé from households (β=-0.56; t=-2.05) explained internet utilisation. The likelihood of locating a BTS in a settlement increased with the settlement's population (℮(β)=6.01), size (℮(β)=10.61) and presence of cottage industries (℮(β)=17.74). The presence of markets (℮(β)=59.78) and secondary schools (℮(β)=15.57) increased the likelihood for call centre location; high population (℮(β)=4.20), presence of markets (℮(β)=5.64) and cottage industries (℮(β)=5.57) increased the probability for cybercafé location. Significant differences in accessibility to call centres (F1,641=283.26) and cybercafés (F1,405=41.1) existed between rural and urban dwellers. All the cybercafés, 56.0% of BTS, and 21.0% of call centre locations were optimal, indicating problems of inadequacy and locational redundancy. Inequalities existed between urban and rural areas in access to, and use of Information and Communication Technology. Policies that address deficiency between urban and rural areas in the location of infrastructures and socio-demographic divide should be formulated and implemented by relevant government agencies.
... Alternatively, the p-median problem likewise locates facilities and allocates demands to them, but it minimizes the total (or equivalently, the average) facility-to-assigned-demand distance. Extensions to these early models consider facilities having capacities [32], deterministic and stochastic demands [33,34], multiple covering of demands [35,36], hierarchical facility structures [37], and other variants [38,39]; an interested reader is referred to recent reviews by Jia et al. [40] and Farahani et al. [41], or to any of four excellent books by Drezner and Hamacher [42], Daskin [43], Laporte et al. [44], or Church and Murray [45], each of which provides a thorough treatment of the subject. Most closely related to our problem is the p-median location problem, which we adapt for the heterogeneous District 14 fleet of assets, as well as the multiple objectives entailed by locating assets effectively yet minimizing changes to the existing enterprise. ...
Preprint
The United States Coast Guard is charged with the coordination of all search and rescue missions in maritime regions within the United States purview. Given the size of the Pacific Ocean and the limited resources available to respond to search and rescue missions in this region, the service seeks to posture its aligned fleet of maritime and aeronautical assets to reduce the expected response time for such missions. Leveraging historic event records for the region of interest, we propose and demonstrate a two-stage solution approach. In the first stage, we develop and apply a stochastic zonal distribution model to evaluate spatiotemporal trends for emergency event rates and corresponding response strategies to inform the probabilistic modeling of future rescue events respective locations, frequencies, and demands for support. In the second stage, the results from the aforementioned analysis enable the parameterization and solution of a integer linear programming formulation to identify the best locations at which to station limited heterogeneous search and rescue assets. Considering both the 50th and 75th percentile levels of forecast event and asset demand distributions using 7.5 years of historical event data, our models identify asset location strategies that respectively yield a 9.6 percent and 17.6 percent increase in coverage over current asset basing when allowing locations among current homeports and airports, as well as respective 67.3 percent and 57.4 percent increases in coverage when considering a larger set of feasible basing locations. Keywords: search and rescue, spatiotemporal forecasting, location-allocation modeling, p-median location problem, multi-objective optimization
... The objective was to find the number of facilities in the network in which the facilities could not serve more than a given number of customers. For the same problem, Berman et al. (2007) considered the problem of locating a set of service facilities in a network with the demand for service being stochastic and congestion at facilities being allowed. They proposed heuristic-based solution procedures to maximize the expected amount of captured demands. ...
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... The objective was to find the number of facilities in the network in which the facilities could not serve more than a given number of customers. For the same problem, Berman et al. (2007) considered the problem of locating a set of service facilities in a network with the demand for service being stochastic and congestion at facilities being allowed. They proposed heuristic-based solution procedures to maximise the expected amount of captured demands. ...
Article
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Article
The United States Coast Guard is charged with coordinating all search and rescue missions in maritime regions within the United States’ purview. Given the size of the Pacific Ocean and limited available resources, the service seeks to posture its fleet of organic assets to reduce the expected response time for such missions. Leveraging 7.5 years of historic event records for the region of interest, we demonstrate a two-stage solution approach. In the first stage, we develop a stochastic zonal distribution model to evaluate spatiotemporal trends for emergency event rates and response strategies. In the second stage, results from the aforementioned analysis enable parameterization of a bi-objective MILP to identify the best locations to station limited heterogeneous search and rescue assets. This research models both 50th and 75th percentile forecast demands across both the set of current homeports, and a larger set of feasible basing locations. Results provide a minimum 9.6% decrease in expected response time over current asset basing. Our analysis also reveals that positioning assets to respond to 75th percentile demands sacrifices, at most, 2.5% in response time during median demand months, whereas positioning for median demand results in operationally inadequate response capability when 75th percentile demands are encountered.
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