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Yugoslav Journal of Operations Research
xx (2018), Number nn, zzz–zzz
DOI: https://doi.org/10.2298/YJOR170110021I
THE CAPACITATED SINGLE-SOURCE
P-CENTER PROBLEM IN THE PRESENCE
OF FIXED COST AND MULTILEVEL
CAPACITIES USING VNS AND
AGGREGATION TECHNIQUE
Chandra A. IRAWAN
Nottingham University Business School China, University of Nottingham Ningbo
China, 199 Taikang East Road, Ningbo 315100, China
chandra.irawan@nottingham.edu.cn
Kusmaningrum SOEMADI
Department of Industrial Engineering, Institut Teknologi Nasional, Bandung
40124, Indonesia
kusmaningrum@yahoo.com
Received: January 2017 / Accepted: July 2018
Abstract: In this study, the discrete p-center problem with the presence of multilevel
capacities and fixed (opening) cost of a facility under a limited budget is investigated.
A mathematical model of the problem is produced, where we seek the location of open
facilities, their corresponding capacities, and the allocation of the customers to the open
facilities in order to minimise the maximum distance between customers and their as-
signed facilities. Two matheuristic approaches are also proposed to deal with larger
instances. The first approach is a hybridisation of a clustering-based technique, an ex-
act method, while the second one is based on Variable Neighbourhood Search (VNS).
Computational experiments show that the proposed methods produce interesting and
competitive results on newly and randomly generated datasets.
Keywords: Capacitated p-center Problem, Mathematical Model, Matheuristic, VNS.
MSC: 90B85, 90C26.
2C.A. Irawan and K. Soemadi / the capacitated p-center problem using VNS
1. INTRODUCTION
The discrete p-center problem deals with finding the location of pfacilities
among Mpotential sites and assigning customers to these facilities in order to
minimise the maximum distance between customers and their nearest facility. This
problem is also known as the minimax location problem where the minimax crite-
rion aims to minimise the adverse effects of worst-case scenarios in providing ser-
vice to the customers. This problem is considered as a single source type-problem
where a customer demand is satisfied only from one facility. The application of this
problem includes the location of facilities in emergency services such as ambulance,
fire, and police stations. In the capacitated version, each customer’s demand and
capacity of a potential facility are known. The problem is NP-hard given its re-
laxed version, the uncapacitated one, to be known as NP-hard (Kariv and Hakimi
[21]), too.
The uncapacitated p-center problem was first proposed by Hakimi [13] who
investigated an absolute 1-center problem on a graph. Bar-Ilan et al. [3] intro-
duced the capacitated version of the p-center problem to address the problem of
locating centers in distributed communication networks. According to the authors,
the problem is referred to as the Balanced p-Centre Problem as the aim of the
problem is to obtain a workload balance among centers. For more description on
the p-center problem, see the recent work by Irawan et al. [19].
In many real case applications, the fixed cost of opening facilities is usually
considered when determining the best location for the opening facilities. The
fixed cost of a facility may be dependent on its location and the capacity of the
facility. For example, the cost of land in a bigger city is higher than in smaller
city. The effect of fixed cost is successfully introduced in related location problem
such as the multi-source case, see Brimberg and Salhi [6]. As the fixed cost of
opening facilities is also related to the capacity of the facilities, decision makers
need to determine an optimal capacity for each opening facility given their limited
budget. This paper attempts to address this logistical problem by developing a
new mathematical model and solution methods. Moreover, the capacity of each
facility is a decision variable that needs to be optimally chosen from a list of
possible capacities.
The main contributions of this paper are as follows:
•propose a new mathematical model for the discrete capacitated p-center
problem with the presence of fixed cost and multilevel capacities,
•propose effective matheuristic approaches based on clustering method and
VNS algorithm,
•generate a new dataset for the new problem and produce optimal and best
solutions for benchmarking purposes.
C.A. Irawan and K. Soemadi / the capacitated p-center problem using VNS 3
The paper is organised as follows. In the next section, mathematical models
for the classical uncapacitated and capacitated p-center problems along with the
new problem considering the presence of fixed cost and multilevel capacities are
presented. The section thereafter describes the proposed matheuristic methods.
Section 5 provides the computational results and analysis. In the last section,
conclusions and some highlights of future research avenues are given.
2. LITERATURE REVIEW
In this section, a review on the capacitated p-center problem is presented. The
capacitated p-center problem on tree networks was studied by Jaeger and Gold-
berg [20]. The authors reveal that the problem can be addressed in polynomial
time when the facility capacities are identical. Khuller and Sussmann [22] inves-
tigated the capacitated p-center problem where each facility can be assigned to
at most Lcustomers. A polynomial approximation algorithm is introduced for
solving the problem. Scaparra et al. [28] studied the application of very large
neighbourhood search techniques for solving the capacitated vertex p-center prob-
lem. The authors characterize a local search neighbourhood in terms of path and
cyclic exchanges of customers among facilities. They also use principles borrowed
from network optimization theory to efficiently detect cost-decreasing solutions in
such a neighbourhood. In addition, the multi-exchange methodology with a relo-
cation mechanism is designed to perform facility location adjustments. ˝
Ozsoy and
Pinar [24] proposed an exact algorithm for solving the capacitated vertex p-center
problem. They build a simple and practical exact algorithm that iteratively sets
a maximum distance value within which it tries to assign all the clients.
Albareda-Sambola et al. [1] investigated two auxiliary problems related to the
capacitated vertex p-center problem. They proposed two different Lagrangean
duals based on each of these auxiliary problems. Two different strategies for solv-
ing exactly the problems, based on binary search and sequential search, are also
introduced. Cygan et al. [10] investigated a generalization of the capacitated p-
center problem where a constant factor approximation algorithm is introduced to
address the problem. In their method, an LP rounding technique is used to tackle
this problem. Recently, a metaheuristic for solving the problem was proposed by
Quevedo-Orozco and R´ıos-Mercado ([25], [26]) where a greedy randomized adap-
tive procedure with biased sampling in its construction phase, and iterated greedy
with a variable neighbourhood descent in its local search phase are incorporated
in their method. An et al. [2] proposed a simple algorithm incorporating the
standard LP relaxation. The algorithm first reduces the problem into special tree
instances, and then the best-possible algorithm is implemented to solve such in-
stances.
Other variants on the capacitated p-center problem include Bashiri et al. [4],
along with Chechik and Peleg [7]. A fuzzy capacitated p-hub center model was
4C.A. Irawan and K. Soemadi / the capacitated p-center problem using VNS
introduced by Bashiri et al. [4]. It aims to minimise maximum travel time in net-
works by locating phubs from a set of candidate hub locations, allocating demand,
and supply nodes to hubs. A genetic algorithm solution is proposed to address
such problem. Chechik and Peleg [7] investigated the fault-tolerant capacitated k-
center problem where one or more service facilities might fail simultaneously. Two
variants of the problem are taken into account. The first problem is called the
α-fault-tolerant problem, where after the failure of some facilities, all customers
are allowed to be reassigned to alternate facilities. The second one is referred to
as the α-fault-tolerant conservative, where once a facility fails, only the customers
assigned to this facility before its failure are allowed to be reassigned to other
facilities.
3. PROBLEM FORMULATION
In this section we first present the mathematical model of the capacitated
single source p-center problem, followed by the new problem, where the presence
of multilevel capacities and fixed cost are considered.
3.1. The capacitated single source p-center problem
In the capacitated version of the p-center problem (CPCP), each customer i
has a known demand (wi) and each potential facility jhas a known capacity (bj).
Total customer demand allocated to each facility jdoes not exceed its capacity
while imposing each customer demand to be entirely assigned to one facility (single-
source location problem). In this case, due to the capacity constraint, a customer
is not necessarily served by its closest facility. The following notations are used to
describe the sets, parameters, and decision variables of the problem.
Sets
•I: set of demand points/customers with ias its index and n=|I|
•J: set of potential sites with jas its index and M=|J|
Parameters
•dij : the distance between customer i∈Iand potential site j∈J(Euclidian
distance will be used in this study)
•p: the number of open facilities
•wi: the demand of customer i∈I
•bj: the capacity of facility located at potential site j∈J
Decision Variables
•r: the maximum distance between customers and their facilities
C.A. Irawan and K. Soemadi / the capacitated p-center problem using VNS 5
•Yij =(1 if customer i∈Iis assigned to facility j∈J,
0 otherwise
•Xj=(1 if an open facility is located at site j∈J,
0 otherwise
In Scaparra et al. [28], the capacitated single source p-center problem is mod-
elled as a Mixed Integer Linear Programming (MILP) as follows:
min r(1)
Subject to:
X
j∈J
Yij = 1,∀i∈I(2)
X
j∈J
Xj=p(3)
Yij −Xj≤0,∀i∈I, j ∈J(4)
r≥X
j∈J
(dij ·Yij ),∀i∈I(5)
X
i∈I
(wi·Yij )≤bj·Xj,∀j∈J(6)
Xj∈ {0,1},∀j∈J(7)
Yij ∈ {0,1},∀i∈I, j ∈J(8)
The objective function (1) aims to minimise the maximum distance between
a customer and its associated facility. Constraints (2) ensure that each customer
iis assigned to exactly one open facility. Constraint (3) guarantees the number
of open facilities to be exactly pwhereas Constraints (4) ensure that customer i
can only be assigned to an open facility. Constraints (5) impose the variable r
to be the minimum of longest customer-facility distance. Constraints (6) impose
the total demand of customers allocated to a facility not to exceed its capacity.
Constraints (7) and (8) indicate that Xand Yare binary decision variables.
6C.A. Irawan and K. Soemadi / the capacitated p-center problem using VNS
3.2. The capacitated single source p-center problem with fixed cost and multilevel
capacities
The mathematical model of the single source capacitated p-center problem with
fixed cost and multilevel capacities (CPCPFC) is presented in this subsection.
In this study, the model treats the capacity of the chosen facility as a decision
variable. In other words, each potential facility has a set of possible facility designs
(Kj, j ∈J), where a facility design defines the capacity of a potential facility
(ˆ
bjk , j ∈J, k ∈Kj). As the fixed cost of an open facility is based on the chosen
capacity and its location, let fjk (j∈J, k ∈Kj) be the fixed cost of each potential
facility jwhen using capacity design k. The model also takes into account the
available budget ˆcfor opening the selected facilities meaning that the total fixed
cost cannot exceed the available budget. In this model, the notations used for sets,
parameters and decision variables are similar to the ones presented in subsection
3.1 with some additions described as follows:
Set
•Kj: set of capacity designs for facility j∈J
Parameters
•fjk : fixed cost of potential facility j∈Jwhen using capacity design k∈Kj
•ˆ
bjk : capacity of facility j∈Jusing design k∈Kj
•ˆc: maximum (available) budget for opening pfacilities
Decision Variables
•r: the maximum distance between customers and their facilities
•Yij =(1 if customer i∈Iis assigned to facility j∈J,
0 otherwise
•ˆ
Xjk =(1 if an open facility is located at site j∈Jwhen using design k∈Kj,
0 otherwise
The CPCPFC problem is much harder to solve than the CPCP problem as
the problem will not only determine the location of the open facilities but also
the capacity for each open facility. The problem can be modelled as a MILP as
follows:
min r(9)
Subject to:
X
j∈J
Yij = 1,∀i∈I(10)
C.A. Irawan and K. Soemadi / the capacitated p-center problem using VNS 7
X
k∈Kj
ˆ
Xjk ≤1,∀j∈J(11)
X
j∈JX
k∈Kj
ˆ
Xjk =p(12)
X
i∈I
(wi·Yij )≤X
k∈Kjˆ
bjk ·ˆ
Xjk ,∀j∈J(13)
Yij −X
k∈Kj
ˆ
Xjk ≤0,∀i∈I , j ∈J(14)
r≥X
j∈J
(dij ·Yij ),∀i∈I(15)
X
j∈JX
k∈Kjfjk ·ˆ
Xjk ≤ˆc(16)
ˆ
Xjk ∈ {0,1},∀j∈J, k ∈Kj(17)
Yij ∈ {0,1},∀i∈I, j ∈J(18)
The objective function of this model is the same as in the previous model,
whereas Constraints (10) ensure that each customer is served by one facility. Con-
straints (11) ensure that each open facility uses only one capacity design. Con-
straint (12) guarantees the number of open facilities to be exactly p. Constraints
(13) state capacity constraints of the facilities. Constraints (14) guarantee that
each customer can only be assigned to one open facility. Constraints (15) define
the maximum distance between customers and their facilities. Constraint (16)
ensures that the total fixed cost does not exceed the available budget. Constraints
(17) and (18) define the binary nature of decision variables ˆ
Xand Y.
4. THE PROPOSED MATHEURISTICS
The discrete capacitated p-center problem with the presence of fixed cost and
multilevel capacities (the CPCPFC problem) is very hard to solve using an exact
method (CPLEX) especially when the size of the problem is relatively large. To
overcome this weakness, matheuristic approaches are proposed in this study. For
more information on heuristic search and metaheuristic, see Salhi [27]. Here, we
propose two solution methods for solving the problem where the first one is referred
to as the adaptive matheuristic method (AMM) while the second is a VNS-based
matheuristic. For simplicity, we consider all potential facility sites as customer
sites (i.e. |J|=n).
8C.A. Irawan and K. Soemadi / the capacitated p-center problem using VNS
4.1. The adaptive matheuristic method (AMM)
One way to tackle the large problem is to simplify it by adapting a common
approach to aggregate/decompose the problem into a smaller one so it can be
addressed within a reasonable amount of computing time (Francis et al., [12]).
Irawan and Salhi [18] provide a review on aggregation techniques for large facility
location problems. Here, AMM that incorporates aggregation technique, an exact
method, and a local search is proposed. The main steps of this approach are
depicted in Algorithm 1.
Algorithm 1: The adaptive matheuristic method (AMM)
Initialisation: Define Tand m. Set z=∞,S=∅and U=∅.
Step 1: Construct pclusters of potential facility sites using Coopers algorithm
that aims to partition npotential facility sites into pclusters.
Step 2: Do the following steps Ttimes:
a. Aggregate nto mpotential facility sites by including the facility loca-
tions with their corresponding capacity design in the incumbent solution
(Sand U).
b. Solve the aggregated problem (model 9 - 18) using the exact method
(CPLEX). Let z0be its objective function value with S0and U0as the
obtained facility configuration and their corresponding capacity design
respectively. i0and j0are also obtained where z0=di0j0.
c. If z0< z, then set z=z0, i∗=i0and j∗=j0along with S←S0and
U←U0.
Step 3:
a. Apply the proposed local search (given in Algorithm 4) with z, i∗, j∗, S,
and Uobtained from previous stage as inputs and outputs.
b. Take z, S and Uas the best objective functions generated along with
their facility configuration and the corresponding capacity design.
In the first step, a clustering process is conducted to construct pclusters/group
of potential facility sites. The clustering procedure is used to aggregate npotential
facility sites into mpotential sites, with m << n. By using this method, we aim
to select relatively good potential facility sites and make sure that the selected
potential facilities are not located only in one specific area. We use coopers method
[8] to cluster npotential facility sites into pgroups.
The second step is an iterative process that incorporates potential facility sites
aggregation and the use of the exact method (CPLEX). When selecting the poten-
tial facility sites, the aggregation includes the facility sites with their corresponding
C.A. Irawan and K. Soemadi / the capacitated p-center problem using VNS 9
capacity design in the incumbent solution. This set of facility sites and their cor-
responding capacity designs are denoted by Sand U, respectively. The resulting
aggregated problem with ncustomers and mpotential facility sites is then solved
optimally by CPLEX. The model will find customer i∗who has the longest dis-
tance from its location to its facility (j∗). It can be written as z=di∗j∗. The
obtained solution (the location of open facilities and their capacity design) is then
fed to the next iteration as part of the set of the aggregated potential sites. The
process is repeated Ttimes and the best solution from this step will be fed to the
next step.
In the final step (Step 3), a local search is proposed to solve the original problem
(without aggregation) starting from the best solution obtained in the previous
step. The description of the clustering method, the aggregation technique, and
the proposed local search is presented in the following subsections.
A. The clustering method
To cluster potential facility sites, we implement the well-known alternate location-
allocation (ALA) method initially introduced by Cooper [8] to address the classical
location-allocation problem. The main idea of ALA is that the location-allocation
problem is alternately applied until no epsilon (ε) improvement in total cost is
found. The objective function of this method is to minimise the total distance
(minisum) which is not the same as the p-center problem (minimax). However,
this is used for an approximation to get some clusters. The main steps of ALA
are presented in Algorithm 2.
Initially, ppotential facilities are randomly selected as the cluster centers
(Cc(xc, yc),∀c= 1, . . . , p), and then all potential facilities are assigned to their
nearest cluster center. The total distance between potential facilities and their
cluster center (f) is also calculated. Let Nc, as the set of potential facilities be-
long to cluster cwith d(Cc, aj), be the Euclidean distance between center cand
potential facility jand aj=xa
j, ya
jbe the location of potential facility j. The
Weiszfeld equations (19) is iteratively carried out to get the new location of these
pcluster centers (Cc(xc, yc),∀c= 1, . . . , p).
ˆxc=Pj∈Nc
xa
j
d(Cc,aj)
Pj∈Nc
1
d(Cc,aj)
,ˆyc=Pj∈Nc
ya
j
d(Cc,aj)
Pj∈Nc
1
d(Cc,aj)
(19)
The potential facilities are then assigned to the new cluster centers ( ˆ
Cc,∀c=
1, . . . , p) resulting new total distance ( ˆ
f) and new allocation ( ˆ
Nc). This process is
repeated until there is no further changes in total distance, within some tolerance
(ε), in two successive iterations. As the initial cluster centers affect the quality
of the pconstructed clusters, we implement a multi-start approach to tackle this
problem. In other words, the ALA process is repeated ˆ
Ttimes. We take the cluster
10 C.A. Irawan and K. Soemadi / the capacitated p-center problem using VNS
configuration that gives the smallest total distance between the cluster centers and
their potential facilities (f∗) .
Algorithm 2: ALA heuristic with multi-start
Step 1: Define εand ( ˆ
T.
Step 2: Set f∗=∞along with C∗
c=∅and N∗
c=∅∀ =c= 1, . . . , p.
Step 3: Do the following steps ˆ
Ttimes:
a. Select pinitial starting locations at random from potential facilities
locations as cluster centers (Cc(xc, yc),∀c= 1, . . . , p).
b. Assign the potential facilities to the nearest cluster center (Cc,∀c=
1, . . . , p). Calculate f=Pp
c=1 Pj∈Ncd(Cc, aj).
c. Determine new cluster centers locations ( ˆ
Cc,∀c= 1, . . . , p) by using the
following procedure:
i. Set c= 1.
ii. Calculate δ=Pj∈Ncd(Cc, aj).
iii. Repeat the following steps:
•Calculate ˆ
Ccusing Weiszfelds equations (19).
•Calculate ˆ
δ=Pj∈Ncd(ˆ
Cc, aj).
•If (δ−ˆ
δ)< ε, then go to Step 3b(iv). Otherwise, set δ=ˆ
δand
Cc=ˆ
Cc.
iv. If c=p, then go to Step 3d, otherwise update c=c+ 1 and go to
Step 3c(ii).
d. Assign the potential facilities to the new nearest cluster center ( ˆ
Cc,∀c=
1, . . . , p) with ˆ
fand ( ˆ
Nc,∀c= 1, . . . , p) as the outputs.
e. If ˆ
f < f , then update f=ˆ
falong with Cc=ˆ
Ccand Nc=ˆ
Ncthen go
back to Step 3c(i).
f. Assign the potential facilities to the new nearest cluster center ( ˆ
Cc,∀c=
1, . . . , p) with ˆ
fand ( ˆ
Nc,∀c= 1, . . . , p) as the outputs.
g. If ˆ
f < f , then update f=ˆ
falong with Cc=ˆ
Ccand Nc=ˆ
Nc,∀c=
1, . . . , p then go back to Step 3c(i).
h. If f < f ∗, then update f∗=falong with C∗
c=Ccand N∗
c=Nc,∀c=
1, . . . , p.
Step 4: Return N∗
c=Nc,∀c= 1, . . . , p.
B. The aggregation method
This subsection presents the description of the procedure to aggregate npo-
tential facility sites into msites used in Step 2a of Algorithm 1. Our aggregation
method is an enhancement of the method proposed by Irawan et al. [17]. The set
of the m aggregated potential facility sites includes the followings:
C.A. Irawan and K. Soemadi / the capacitated p-center problem using VNS 11
•The best facility locations configuration (S) obtained from previous itera-
tions along with their corresponding capacity (U).
•(m−p) pseudo randomly generated points.
First, the method includes the incumbent facility locations (S) as part of the
aggregated potential facility sites. The use of these facility locations (S) will
increase the probability of obtaining a good solution. Moreover, the capacity for
these facility locations (S) in the aggregated problem is set to be fixed, based on
the capacity configuration (U) obtained from previous iterations. On the other
hand, there is a set of possible capacity designs (Kj) for the remaining aggregated
points. By doing this, the computing time to solve the aggregated problem using
CPLEX is reduced. Second, the cluster-based method, which is briefly described
below, is applied to generate the subsequent (m−p) aggregated points. This
scheme overcomes the weaknesses of a simple random process when dealing with
clustered customers. In addition, it ensures that the aggregated potential facility
sites are not too close to each other and will spread in the study area. The main
steps of the method are formally given in Algorithm 3.
Algorithm 3: The cluster-based method to aggregate potential facilities
Step 1: Set A=S.
Step 2: Calculate the total demand of customers (the location of potential facility
sites) for each cluster cas Dc, where Dc=Pi∈N0
cwi,∀c= 1, . . . , p with N0
c
is the set of customers belonging to cluster c.
Step 3: Determine the cluster probability distribution, say Pc=Dc/Pi∈Iwi,∀c=
1, . . . , p.
Step 4: Determine the customer (a potential facility site) probability distribution,
say P0
ci =wi/Dc,∀c= 1, . . . , p;i∈N0
c.
Step 5: for j= (p+ 1) to mdo the following steps:
a. Generate randomly β∈ {0,1}.
b. Choose cluster ˜cst. ˜c=F−1
(c)(β) with F(c)=Pc
a=1 Pa.
c. Generate randomly α∈ {0,1}.
d. Select customer ˜
ist. ˜
i=F−1
(˜ci)(α) with F(˜ci)=Pi
a=1 P0
˜ca.
e. If ˜
i∈A, then go back to Step 5a, otherwise A=AS˜
i.
Step 6: Return A.
Initially, all potential facilities are clustered using the ALA heuristic. Let Abe
a set of aggregated potential facilities which consists of the facility configuration
obtained from the previous iteration. The total demand of customers (which
12 C.A. Irawan and K. Soemadi / the capacitated p-center problem using VNS
are the location of potential facility sites) for each cluster, Dc, is calculated and
its corresponding probability distribution (Pc) is determined. The probability
distribution of each potential facility (located in customer site) (P0
ci) corresponding
to its cluster is also calculated. A cluster is selected in a pseudo random manner,
based on the cumulative probability distribution. Here, a random number β∈
{0,1}is generated and then, a cluster is chosen with ˜cst. ˜c=F−1
(c)(β) with
F(c)=Pc
a=1 Pa. See Figure 1 for an illustration in the case ˜c= 3. A potential
facility/customer (say ˜
i) is chosen pseudo randomly in cluster ˜cusing the same
technique as for choosing cluster ˜c. The location of customer ˜
iis then selected
into the set of aggregated potential facility sites (A). This procedure is reiterated
until |A|=m.
Figure 1: The illustration of selecting cluster
C. The proposed local search
The proposed local search is a hybridisation of the vertex substitution heuris-
tic and the exact method. We enhance the vertex substitution heuristic used by
Mladenovic et al. [23] by incorporating the exact method to allocate customers
to the open facilities and to determine the capacity design used by an open facil-
ity. The exact method is embedded into the local search to optimally solve the
customer allocation problem whenever the locations of open facilities are fixed. In
case the location of open facilities is known, the problem reduces to an assignment
problem. The model will also find the optimal capacity design for each opening
facility. We refer to this problem as the CPCPFC-FL. Let S(S⊂J) be a set of
facility locations in the solution with |S|=p. The formulation of the CPCPFC-
FL is similar to the one of the CPCPFC expressed as Equations (9) - (18) with
minor changes as follows:
•The set of potential facility sites (J) is replaced by the set S.
C.A. Irawan and K. Soemadi / the capacitated p-center problem using VNS 13
•Constraints (11) are replaced by Constraints (20) as follows:
X
k∈Kj
ˆ
Xjk= 1,∀j∈J(20)
•Constraints (12) are removed.
The main steps of the proposed local search are presented in Algorithm 4,
which is based on the interchange heuristic using a first improvement strategy (the
swapping process is conducted once there is an improvement). The algorithm aims
to find a potential facility site (located in customer i) to be swapped with a facility
used in the current solution. Moreover, the capacity of the facility candidate is
determined. The swap will be done if improvement occurs. First, the facility (in
S) that serves each customer i(ρ(i),∀i∈I) needs to be found. In Step 3 of
Algorithm 4, as we aim to reduce the maximum distance between a customer and
its facility, we only consider the potential facilities whose distance between their
locations and customer i∗(the customer that has the maximum distance to its
facility (facility j∗)) is less than z=di∗j∗, and which are to be swapped with a
facility in the current solution.
In Steps 3b of Algorithm 4, a potential facility located at customer iis inserted
into the solution, whereas the facility in the current solution that serves customer
iis removed. Let H(j) be a set of capacity designs for each facility j. Here, there
are several possible capacity designs for the new facility (located at customer isite)
which is based on set Ki. On the other hand, the capacity for the existing facilities
in the current solution is set to be fixed, based on the capacity configuration (U)
in the incumbent solution. This will reduce the computing time to solve the
CPCPFC-FL using CPLEX. Note that in Step 3b, the CPCPFC-FL using S0and
Hmay be infeasible due to capacity or budget constraints. In Step 3d, the swap
will be conducted if there is an improvement. Step 4 is an iterative step where the
local search will stop if there is no improvement after all possible swaps based on
incumbent solution have been done.
Algorithm 4: The proposed local search
Step 1: Find the facility (in S) that serves customer i, i.e. find array ρ(i),∀i∈I.
Step 2: Set θ= 0 (θis the saving occurred from swapping).
Step 3: For i= 1 to n,i /∈S, do the following:
If di∗i< z, then do the following procedure:
a. Set S0(j)←S(j) and H(j)←U(j)∀j= 1, . . . , p.
b. Set S0(ρ(i)) ←iand H(ρ(i)) ←Ki.
14 C.A. Irawan and K. Soemadi / the capacitated p-center problem using VNS
c. Solve the CPCPFC-FL optimally using CPLEX with S0and Has
inputs. Let z0be its objective function value and Ube a set of the
capacity designs for each open facility (S0). The model also obtain
i0and j0where z0=di0j0.
d. Set θ=z−z0.
e. If θ > 0, and the solution is feasible, do the followings:
•Update z=z0,i∗=i0,j∗=j0,S←S0and U←U0.
•Update array ρ(i),∀i∈I.
•Go to Step 4.
End If
Step 4: If θ≤0, then stop, otherwise go to Step 2.
Step 5: Return z,i∗,j∗,Sand U.
4.2. VNS-based Matheuristic approach
Variable Neighbourhood Search (VNS) is a powerful metaheuristic method in-
troduced by Brimberg and Mladenovic [5] for solving continuous location-allocation
problems. This metaheuristic was formally formulated by Hansen and Mladenovic
[14], who applied it to solve the p-median problem. VNS implementations and
variants of VNS are provided in Hansen and Mladenovic [15] and Hansen et al.
[16]. VNS consists of local search and neighbourhood search. The local search
finds local optimality, whereas the neighbourhood search aims to escape from
these local optima by systematically using a larger neighbourhood if there is no
improvement, and then reverts back to the smaller one otherwise. In the VNS, the
smallest neighbourhood is the one closest to the current solution, and the largest
is the one furthest from the current solution (Hansen and Mladenovic, [14]). In
this subsection, we propose a VNS-based method and present its main steps in
Algorithm 5..
Step 3 aims to construct a relatively good initial solution. Tfacility configu-
rations are generated where each facility configuration consists of popen facilities
(S0). Each facility configuration is then evaluated by solving the CPCPFC-FL
using an exact method (CPLEX). We take the facility configuration that yields
the smallest objective function value. In Step 3a, when selecting pfacility sites,
each cluster contributes an open facility site, which is chosen randomly based on
cumulative probability distribution described in Step 5c - 5d of Algorithm 3. The
best facility configuration is then to be fed into the next step, which is the VNS
algorithm.
C.A. Irawan and K. Soemadi / the capacitated p-center problem using VNS 15
Algorithm 5: The VNS-based matheuristic
Step 1: Define Tand kmax. Set z=∞.
Step 2: Apply Step 1 of Algorithm 1 to construct pclusters of potential facility
sites.
Step 3: Do the following steps Ttimes:
a. Select pfacilities (S0) from potential facility sites (J) based on cluster-
based method given in Section 3.1. Determine the possible capacity
designs of the popen facilities along with their corresponding fixed
(opening) costs.
b. Implement CPLEX to solve the CPCPFC-FL optimally using selected
open facilities (S0) and their corresponding possible capacity designs
and fixed costs as inputs. The outputs of the model are as follows: z0,
U0,i0and j0.
c. If (z0< z), then update z=z0,i∗=i0,j∗=j0,S←S0and U←U0.
Step 4: Update z0=z,i0=i∗,j0=j∗,S0←Sand U0←U.
Step 5: Set k= 1.
Step 6: Shaking procedure (Do the following step ktimes:)
a. Choose randomly a potential facility located at customer ˆ
isite, say
facility ˆ
j(ˆ
j /∈S0), where di0ˆ
j< z0.
b. Remove the facility that serves customer ˆ
ifrom the current solution
and insert facility ˆ
jinto current solution (S0).
c. Implement CPLEX to the CPCPFC-FL optimally using selected open
facilities (S0) and their corresponding possible capacity designs and
fixed costs as inputs. Decision variables z0,U0,i0, and j0are updated.
Step 7: Local search.
Implement the proposed local search (presented in Algorithm 4) with z0,
U0,i0and j0as inputs and outputs.
Step 8: Move or not.
If (z0< z), then update k= 1 along with z=z0,i∗=i0,j∗=j0,S←S0
and U←U0.
Else update k=k+ 1 along with z0=z,i0=i∗,j0=j∗,S0←Sand
U0←U.
Step 9: If k≤kmax, go back to Step 6.
Step 10: If computing time is less than cpumax, then go back to Step 5.
Step 11: Return z,i∗,j∗,S, and U.
16 C.A. Irawan and K. Soemadi / the capacitated p-center problem using VNS
The shaking process (Step 6) in the proposed VNS is conducted by inserting
a facility, say facility ˆ
j(ˆ
j∈S0), located at a customer site randomly selected
(say customer ˆ
i) and removing the facility that serves customer ˆ
ifrom the current
solution. Note that the distance value between facility ˆ
jand customer i0(the
customer who has the maximum distance to its facility) must be less than the
current objective function value or di0ˆ
j< z0. The new facility configuration
along with its possible capacity designs and its corresponding fixed costs is then
evaluated by solving the CPCPFC-FL. This procedure is repeated ktimes.
In the local search (Step 7), the proposed algorithm presented in Algorithm 4
is implemented to improve the quality of solution by finding the local optima. In
the move or not move step, if there is no improvement, a larger neighbourhood is
systematically used, otherwise the search returns to the smallest neighbourhood.
This can be conducted by updating kvalue, where k=kmax represents the largest
neighbourhood while k= 1 indicates the smallest one.
5. COMPUTATIONAL EXPERIMENTS
In this section, the computational experiments are presented. We carried out
extensive experiments to examine the performance of the proposed heuristic ap-
proaches. These were coded in C++ .Net 2012 where we also used the IBM ILOG
CPLEX version 12.6 Concert Library. The tests were run on a PC with an Intel
Core i5 CPU @ 3.20GHz processor, 8.00 GB of RAM. As there is no data available
in the literature for the proposed problem, we therefore generated a new dataset
where n= 50 to 1000. The demand of each customer is randomly generated be-
tween 1 and 10. In this study, the potential facility locations are located in the
customer sites i.e. |J|=n. Here, we set the number of possible capacities for all
potential facility sites to 3 |Kj|= 3, ∀j= 1, . . . , n. We generate randomly the
capacity of each potential facility for each design ˆ
bjk and the fixed cost of each
potential facility for using each capacity design (fjk ). The maximum (available)
budget for opening pfacilities (ˆc) is also generated, based on the average of fixed
cost of all potential facility sites and the number of open facilities (p).
To evaluate the performance of our proposed solution approaches, we compare
the solutions of the proposed method with solutions of the exact method (using
CPLEX). Here, we limit the computing time of CPLEX to 3 hours so lower bound
(LB), upper bound (UB), and %Gap are obtained. The performance of the pro-
posed approaches will also be measured by %Dev between the Z value obtained
by our proposed approaches and the best known Z (Z∗). %Dev is calculated as
follows:
%Dev =Zp−Z∗
Z∗·100 (21)
C.A. Irawan and K. Soemadi / the capacitated p-center problem using VNS 17
where Zprefers to the objective function value of a feasible solution obtained by
either the exact method (UB) or the proposed solution methods.
In the experimental study, we set parameters ε= 0.0001 and ˆ
T= 5 for cluster-
ing the potential facility sites. For the adaptive heuristic method (AMM), we set
parameters m=min (max(4p, 30), min(60,0.75n)) and T= 10, whereas for VNS
we set the parameters T= 10 and kmax = 10. In addition, parameter cpumax
for VNS is set based on the computational time required by AMM to address the
problem. By doing this, we can compare the performance between two proposed
methods within the same computational time. Those parameters were selected
based on our preliminary experiments. Using these settings, an acceptable perfor-
mance, in terms of both quality of the solution and the computational effort, is
obtained.
We divide the results of the proposed methods into two tables. Table 1 presents
the results on n= 50 to 500, whereas the results on n= 750 to 1000 are given in
Table 2. According to the results shown in Table 1, the exact method found the
optimal solution within a relatively short computing time for the small problems
(n≤125). CPLEX experienced difficulties when solving the problem with n >
125. On average, the proposed methods (AMM and VNS) yield a better deviation
than the exact method. Moreover, the proposed methods required less than 10% of
the computational time needed by the exact method. The AMM performs slightly
better than VNS, where the AMM and VNS produced an average deviation of
2.39% and 3.35% respectively. When n= 50, the AMM was able to obtain the
optimal solutions for all instances except for p= 5. The table also reveals that
for n≤250, the exact method produced a very small deviation, however the
computational time required to solve the problem is huge especially when n≥250.
Table 2 shows the computational results for relatively large problems (n≥750).
The table does not show the results from the exact method (CPLEX) as this
method was not able to obtain the upper and lower bounds values within 3 hours.
In this table we only compare the performance of the AMM and VNS. Here, VNS
is found to be the best method as it produced the smallest deviation (0.1%). For
all instances, VNS yields a smaller deviation than the AMM, except for p= 25.
Figure 2 shows the optimal solution of the CPCPFC for n= 50 with p= 5 where
a customer is assigned to the facility with the same colour.
18 C.A. Irawan and K. Soemadi / the capacitated p-center problem using VNS
Exact Method Matheuristics
n p Best Known Dev(%) UB LB Gap(%) CPU(s) AMM (%Dev) VNS (%Dev) CPU(s)
50 5 16.03 0.00 16.03 16.03 0.00 11 1.54 1.54 49
50 10 10.77 0.00 10.77 10.77 0.00 2 0.00 0.00 10
50 15 9.22 0.00 9.22 9.22 0.00 6 0.00 6.83 13
50 20 8.54 0.00 8.54 8.54 0.00 5 0.00 20.50 5
125 5 37.01 0.00 37.01 37.01 0.00 73 1.41 0.00 23
125 10 25.32 0.00 25.32 25.32 0.00 207 1.16 1.16 20
125 15 21.26 0.00 21.26 21.26 0.00 277 2.73 4.12 63
125 20 17.46 0.00 17.46 17.46 0.00 226 0.00 4.49 77
125 25 16.03 0.00 16.03 16.03 0.00 221 0.00 2.31 157
250 5 78.00 0.00 78.00 78.00 0.00 940 2.41 1.68 48
250 10 52.20 1.84 53.16 35.66 32.91 10,800 0.00 7.17 82
250 15 40.80 0.48 41.00 32.06 21.80 10,800 7.50 0.00 161
250 20 35.74 0.00 35.74 35.73 0.01 6,243 5.34 1.09 620
250 25 31.62 0.00 31.62 31.14 1.51 10,800 1.39 1.98 1,532
500 5 157.84 251.79 555.24 108.47 80.46 10,801 3.57 0.00 473
500 10 104.96 42.99 150.08 70.05 53.33 10,801 5.41 0.00 525
500 15 83.60 149.06 208.22 54.71 73.72 10,802 5.89 0.00 1,505
500 20 75.89 663.06 579.12 46.24 92.02 10,809 7.01 0.00 872
500 25 68.12 714.87 555.07 40.52 92.70 10,801 0.00 10.74 233
Average 96.00 4,731 2.39 3.35 341
Table 1: Computational Results for n= 50 to 500
C.A. Irawan and K. Soemadi / the capacitated p-center problem using VNS 19
Matheuristics
n p Best Known AMM(%Dev) VNS(%Dev) CPU(s)
750 5 232.13 1.11 0.00 665
750 10 163.17 2.41 0.00 1,581
750 15 129.77 5.08 0.00 6,566
750 20 113.32 7.95 0.00 2,650
750 25 105.12 0.00 0.46 797
1000 5 314.84 0.81 0.00 1,457
1000 10 215.23 2.21 0.00 4,223
1000 15 175.55 3.70 0.00 8,602
1000 20 155.31 4.67 0.00 4,742
1000 25 145.15 0.00 0.52 1,363
Average 2.79 0.10 3,265
Table 2: Computational Results on n= 750 to 1000
Figure 2: The optimal solution for n= 50 and p= 5
6. CONCLUSIONS and SUGGESTIONS
In this paper we investigate the single source p-center problem with the pres-
ence of limited budget available for opening facilities, considering several possi-
ble capacities for each potential facility. We refer it as to the capacitated single
20 C.A. Irawan and K. Soemadi / the capacitated p-center problem using VNS
source p-center problem with fixed cost and multilevel capacities (CPCPFC). A
mathematical model of this problem is introduced and two solution methods for
solving the problem. The first method, we refer to as the Adaptive Matheuristic
Method (AMM), whereas the second one is a matheuristic technique based on
Variable Neighbourhood Search (VNS). The proposed approaches were assessed
on randomly generated datasets. For relatively small problems, the solutions of
the proposed methods are compared with the solutions obtained by the exact
method executed within a limited computing time (3 hour). According to the
computational experiments, the proposed methods run very well and produced
small deviations within a short computational time. In this case, the AMM per-
forms better than VNS as it yields a smaller average deviation. For large problems
(n > 500), the exact method was not able to obtain the lower and upper bounds.
Therefore, we compared only the performance between the AMM and VNS. Based
on the results, for relatively large problems, it is found that VNS performs slightly
better than the AMM.
This research could be worthwhile expanding and adapting to its counterpart
problem namely the continuous capacitated p-center problem with the presence of
fixed cost and multilevel capacities.
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