A Mixed Linear Integer Programming Formulation and a Simulated
Annealing Algorithm for the Mammography Unit Location Problem
Marcos Vin
´
ıcius Andrade de Campos
1 a
, Manoel Victor Stilpen Moreira de S
´
a
2 b
,
Patrick Moreira Rosa
2 c
, Puca Huachi Vaz Penna
2 d
, S
´
ergio Ricardo de Souza
1
and Marcone Jamilson Freitas Souza
2 e
1
Departamento de Computac¸
˜
ao, Centro Federal de Educac¸
˜
ao Tecnol
´
ogica,
Avenida Amazonas, 7675, 30.510-000 Belo Horizonte, Brazil
2
Departamento de Computac¸
˜
ao, Universidade Federal de Ouro Preto, 35.400-000 Ouro Preto, Brazil
Keywords:
Mammography Unit Location, Maximal Covering Location, Facility Location, Simulated Annealing.
Abstract:
Breast cancer is the most commonly occurring one in the female population. Early diagnosis of this disease,
through mammography screening, can increase the chances of cure to 95%. Studies show that Brazil has a
relatively satisfactory number of mammography units, but this equipment is poorly geographically distributed.
This paper focuses on the Mammography Unit Location Problem (MULP), which aims an efficient distribution
of mammography units, in order to increase the covered demand. Focusing on the State of Minas Gerais,
Brazil, an analysis is made considering that, in the real world, there is a difficulty in relocating equipment
already installed. Therefore, it would be interesting to optimize the location of new equipment purchases.
Since MULP is NP-hard, an algorithm based on the Simulated Annealing meta-heuristic is also developed to
handle large instances of the problem.
1 INTRODUCTION
Breast cancer is the most common cause of death
among women (Bray et al., 2018). Only in Brazil,
this type of cancer led to obit 16724 women in
2017, which represents 2.9% of female deaths (INCA,
2017).
In turn, mammography screening is the primary
method of early detection for the diagnosis of breast-
related malignancies (Xavier et al., 2016). Combined
with proper treatment, early detection via mammog-
raphy screening contributes to the reduction in the
mortality rate of women with breast cancer (Berry
et al., 2005). When a tumor is detected in the early
stages, the chance of cure is greater than 95% (Witten
and Parker, 2018).
The Brazilian Ministry of Health recommends
that women between the ages of 50 and 69 perform
a
https://orcid.org/0000-0002-5599-8889
b
https://orcid.org/0000-0003-2635-3157
c
https://orcid.org/0000-0003-3897-0961
d
https://orcid.org/0000-0001-5414-1405
e
https://orcid.org/0000-0002-7141-357X
the mammography screening at least once every two
years (Brasil, 2017). In addition, it is estimated that,
by diagnostic indication, 8.9% women of this age
group and 20% of the female population between 40
and 49 years of age need to perform the screening an-
nually. Thus, 58,9% of women, between 50 and 69
years old, need mammography screenings annually,
besides 20% of those between 40 and 49 years old.
The Brazilian healthcare system has a significant
amount of mammography units to serve the popula-
tion (Amaral et al., 2017). In December 2015, these
authors estimated that there were 4647 units in use,
of which 2083 were available for use in the Brazil-
ian public healthcare system. This amount of equip-
ment would be sufficient to cover the estimated de-
mand of 12.7 million mammography screenings per
year. On the other hand, also according to these au-
thors, the Brazilian Ministry of Health determines that
the maximum acceptable distance between patients
and equipment is 60 km. Given this guideline, 4.5%
of the required screenings would not be performed.
This situation would be even more grave if only pub-
lic mammography equipment were considered. Thus,
the number would go from 4.5% to 17% of uncovered
428
de Campos, M., Moreira de Sá, M., Rosa, P., Penna, P., de Souza, S. and Souza, M.
A Mixed Linear Integer Programming Formulation and a Simulated Annealing Algorithm for the Mammography Unit Location Problem.
DOI: 10.5220/0009420704280439
In Proceedings of the 22nd International Conference on Enterprise Information Systems (ICEIS 2020) - Volume 1, pages 428-439
ISBN: 978-989-758-423-7
Copyright
c
2020 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
women, which would represent 2.17 million screen-
ings not performed due to the lack of available equip-
ment within a maximum radius of 60 km.
In the current scenario, there is an inefficient ge-
ographical distribution of mammography equipment
as detected by several authors, in addition to the one
previously mentioned (Miranda and Patrocinio, 2018;
Silva et al., 2018). Each of these papers has a different
focus, however, all of them agree that, although the
amount of equipment is relatively adequate, the poor
distribution of mammography equipment still causes
difficulty in accessing the mammography screening.
In this paper, we address the Mammography Unit
Location Problem (MULP), considering the Brazilian
reality. The objective is to improve the geographic
distribution of mammography units available in the
public healthcare system of Brazil. Initially, we pro-
pose a Mixed-Integer Linear Programming (MILP)
model for the MULP. A case study of the Minas
Gerais state is introduced, and two scenarios are ana-
lyzed. In the first one, we considered the impossibil-
ity of changing the location of the equipment instal-
lation, and the goal is to determine the locations of
new mammography units to be acquired; in the sec-
ond one, there is the freedom to move the equipment,
which enables better results with the same amount of
mammography units.
The problem under study can be formulated as
the Maximal Covering Location Problem (Church and
ReVelle, 1974) with additional constraints. It is an
NP-Hard problem (Garey and Johnson, 1979), and
this issue limits the use of exact methods, since they
may require an unacceptable time for decision making
in real instances. Thus, we developed an algorithm
based on the Simulated Annealing metaheuristic to al-
low the treatment of large instances of the problem.
The rest of this paper is organized as follows. Sec-
tion 2 shows a brief literature review dealing with
Facility Locating Problems, especially concerning to
health prevention equipment, such as mammography
units. Section 3 describes the Mammography Unit
Location Problem, considering the characteristics es-
tablished by the Brazilian Ministry of Health. Sec-
tion 4 introduces a MILP formulation for this prob-
lem, while Section 5 presents the proposed Simu-
lated Annealing based-algorithm. The results of the
computational experiments with these methods are re-
ported in Section 6. Finally, the conclusions and the
perspectives of future work are presented in Section 7.
2 LITERATURE REVIEW
The facility location is a critical issue, either for in-
dustry or for a healthcare facility (Daskin and Dean,
2005). In this kind of problem, there are customers,
with their respective demands, and locations, where
potentially the facilities will be installed. The aim is
to determine in which locations the facilities will be
open and, consequently, in which of those open facil-
ities each customer will be associated with.
The first facility location study dates from 1909.
The objective was to determine the location of a ware-
house so that its distance to its customers was mini-
mized (Weber, 1929). Since then, several studies have
been carried out in the theme (Ahmadi-Javid et al.,
2017).
A facility location model can be continuous or
discrete in nature. Continuous models are those
where facilities can be located anywhere in the fea-
sible region, while discrete models only allow them
to be established at predetermined locations, which
eventually may also be a point of demand. Dis-
crete models can be classified into three categories:
(i) median-based problems; (ii) covering-based prob-
lems; and (iii) other problems (Ahmadi-Javid et al.,
2017). Median-based models are characterized by
locating facilities at candidate points to minimize
the weighted average distance costs between demand
points and the facilities to which they are assigned.
In turn, in the covering-based location problem, given
a specified level of demand coverage, which must be
achieved, the goal is to find the number and location
of facilities such that all demand points are within a
specified travel distance (or time) of the facility that
serves them. The third category includes problems
that are not in either of the above two categories, for
example, p-dispersion problem, maximum dispersion
problem, among others.
Problems dealing with health services, such as the
MULP, are often coverage type. This category is
subdivided into p-center location, set covering loca-
tion, and maximal covering location problems. The
p-center location problem consists of locating a fa-
cility, such as a school or a hospital, so that the dis-
tance of the customer farthest from it is minimized
(Hakimi, 1964). The set covering location problem
aims to minimize the number of open facilities, ensur-
ing that all demand is met and respecting a maximum
distance/time between any customer and the facility
that covers it (Toregas et al., 1971). Finally, given the
number of facilities to be opened, the Maximal Cov-
ering Location Problem (MCLP) intends to determine
the best location of them, so the covered demand is
maximized.
A Mixed Linear Integer Programming Formulation and a Simulated Annealing Algorithm for the Mammography Unit Location Problem
429
The MCLP-based models are widely used in
healthcare, especially in the public sector, due to bud-
getary constraints (Sathler et al., 2017).
The location of preventive healthcare facilities
is addressed in several papers (Verter and Lapierre,
2002; Zhang et al., 2009; Zhang et al., 2010; Gu et al.,
2010; Zhang et al., 2012; Davari et al., 2016; Dogan
et al., 2019).
Verter and Lapierre (2002) proposed a model for
determining the optimal configuration of a network
of preventive healthcare facilities. The goal was to
maximize the participation in prevention programs,
including mammography screening. The model pre-
sented was based on three premises: (i) each individ-
ual would look for the nearest facility; (ii) the prob-
ability of a person participating in a prevention pro-
gram decreases as his/her distance to the facility in-
creases; and, finally, (iii) each opened facility would
need to have a minimum number of customers. To
model the probability of participation, a decay func-
tion was used, where the value would be 0 (zero)
for distances greater than the maximum allowed, and
would progressively increase until it reached 1 as the
distance approached 0. The model was applied to lo-
cate mammography centers in the city of Montreal,
Canada. Zhang et al. (2009) added the concept of
congestion. Contrary to the previous study, where
only time or distance was the factor that defined ac-
cessibility to the facility, the authors consider the to-
tal time spent, which is composed of the travel time,
the waiting time to receive the service, and the ser-
vice time. To solve it, an allocation heuristic and
four location heuristics were developed. In Zhang et
al. (2010), the authors add a decision variable in re-
lation to the previous work. In addition, to decide
whether a facility will open or not, there is concern
about the number of services that each facility will
offer. The model was built as a bi-level nonlinear opti-
mization model. A Tabu Search based algorithm was
developed to treat the problem. A new measure of ac-
cessibility was presented in Gu et al. (2010). Unlike
previous work, where distance/time was the main fac-
tor of accessibility, the proposed measure combined
regional availability, the distance between customers
and facilities, and the number of customers that the
facility attracts. The Huff-based competitive location
model (Huff, 1964) was used to estimate the workload
of facilities. The problem is solved using both exact
methods and a local search-based heuristic algorithm
using relocate moves. In Zhang et al. (2012), the
probability of a customer to choose a feature was con-
sidered. The authors presented two models were pre-
sented. The first considers that the customer chooses
a facility with a certain probability. The probability
increases according to the attractiveness of the facil-
ity. The other model defines that the customer will
choose the most attractive facility. The closer a fa-
cility is to the customer, the more attractive it will be
to the facility. To solve the problem, a probabilistic-
search based heuristic and a genetic algorithm were
presented. Davari et al. (2016) enhanced the model
by adding budget constraints. As a solution method,
an algorithm based on the Variable Neighborhood
Search metaheuristic was used. Dogan et al. (2019)
presented a multi-objective model consisting of three
terms: (i) minimization of the average total weighted
deviation between the realized and maximum possible
participation incurred over the planning period; (ii)
minimization of the total deviation resulting from ex-
ceeding the expected acceptable waiting time at facil-
ities averaged over the planning period; and (iii) mini-
mization of the deviation resulting from exceeding the
budget. The model was applied to locate Cancer Di-
agnostic, Screening and Training Centers in Istanbul,
Turkey.
For the Brazilian reality, there are few proposals
for a more efficient geographical distribution of mam-
mography units (Corr
ˆ
ea et al., 2018; Souza et al.,
2019).
Corr
ˆ
ea et al. (2018) analyzed if a more rational
distribution of existing mammography units was pos-
sible, taking, as a case study, a set of 12 health regions
of the Minas Gerais State, Brazil, involving 142 cities.
The authors developed four formulations of integer
linear programming, based on the classical p-median
problem, whose objective was to minimize the sum
of the distances between served locality and serving
locality. The first formulation adds a maximum dis-
tance constraint between each city and the location
that provides it with care. In the second formulation,
the distance restriction is allowed to be violated, pe-
nalizing the positive distance deviation in the objec-
tive function. The third and fourth formulations re-
produce the first and second ones, respectively. How-
ever, considering in the objective function, in addition
to the distance between the city that serves and the
city served, also the demand of screenings in the city
that needs care. The authors concluded that, even re-
specting the distance restriction, the amount of mam-
mography units in the studied region is sufficient to
meet all the demand for screenings.
Souza et al. (2019) considered, as a case study,
the distribution of the mammography units in the
Rond
ˆ
onia State, Brazil. The authors proposed two in-
teger linear programming formulations, both aiming
to maximize the number of women attended, respect-
ing the minimum demand restriction and the maxi-
mum distance restriction between each city and the
ICEIS 2020 - 22nd International Conference on Enterprise Information Systems
430
city that hosts the mammography units. The two for-
mulations differ with respect to the form of attending
each city. In the first formulation, the demand of a city
should be fully met, either by itself or by a neighbor-
ing city located within the service radius. In the sec-
ond formulation, a partial service is possible, that is,
a city could have fractions of its demand met by dif-
ferent cities. In order to solve the problem, a prepro-
cessing is applied, in which every city with demand
higher than the mammography attendance receives p
mammography units, where p is the integer part of the
division of the city demand for equipment screening
capacity.
3 PROBLEM STATEMENT
The Mammography Unit Location Problem (MULP)
addressed here has the following characteristics:
There is a set S of n candidate cities to host p
mammography units;
Each mammography unit has an annual capacity
of cap screenings;
Each city i has an annual demand for screenings.
According to the recommendations of the Brazil-
ian Ministry of Health (Brasil, 2017), this number
is composed by 58.9% of the female population
aged from 50 to 69 years old and 20% of women
aged from 40 to 49 years old;
Women from a city j may be covered by mam-
mography units installed in another city i. How-
ever, the distance between i and j can not exceed
R km. In Brazil, the Ministry of Health (Brasil,
2017) recommends that R be equal to 60 km;
In order to host a mammography unit, a city i must
have the infrastructure to do so. In this paper, we
consider that a city is eligible for hosting an equip-
ment if it has an annual demand greater than or
equal to demMin screenings;
A city j can be covered by one or more cities;
A city i that hosts an equipment can serve one or
more cities, provided that its own demand is fully
met.
The objective of MULP is to maximize the total num-
ber of women covered by the existing mammography
units. In short, it is necessary to define in which cities
the equipment will be installed and the number of
equipment installed. Besides, it is necessary to define
which cities will be served by these mammography
units.
4 MILP FORMULATION
The mathematical programming formulation in the
current paper is based on the work of Souza et
al. (2019). In the mentioned study, it is necessary
that the demand for mammography screening in each
city be less than the capacity of a mammography unit.
When this condition is not met, a preprocessing pro-
cedure is required. For each city with demand greater
than the equipment’s capacity, as many mammogra-
phy units as necessary are allocated until residual de-
mand is less than the capacity for screenings of the
equipment. At the end of the procedure, all cities had
lower demand than the service capacity, and the num-
ber of mammography units used for this is decreased
from the total available. In this new work, we add
variables and constraints that make the preprocessing
unnecessary. Table 1 presents the formulation param-
eters and decision variables.
The proposed MILP formulation is given by the
Equations (1)-(15):
max
iN
jS
i
dem
j
· x
i j
(1)
s.t.
iS
j
x
i j
1 j N (2)
i N
y
i
= p (3)
jS
i
dem
j
· x
i j
cap.y
i
i N (4)
z
i
y
i
/p i N (5)
z
i
y
i
i N (6)
t
i
z
i
i N (7)
t
i
dem
i
· x
ii
dem
i
+ 1 i N (8)
t
i
x
ii
i N (9)
t
i
x
i j
i, j N | i 6= j (10)
y
i
= 0 i N | dem
i
< demMin (11)
x
i j
[0,1] i, j N (12)
y
i
Z i N (13)
z
i
{0,1} ∀i N (14)
t
i
{0,1} ∀i N (15)
The objective function (1) aims to maximize the
total demand for mammography screenings. Con-
straints (2) ensure that the city j can only be served
at most 100% of your demand. Constraint (3) forces
that are used exactly p mammography units, allowing
a city i hosts from 0 to p equipment. Constraints (4)
ensure that the equipment’s service capacity is re-
spected. Constraints (5) and (6) make z
i
to be 1 if
there is at least one equipment installed in city i and
A Mixed Linear Integer Programming Formulation and a Simulated Annealing Algorithm for the Mammography Unit Location Problem
431
Table 1: Description of Parameters and Decision Variables.
Problem Parameters
N Set of candidate cities
S
i
Set of cities whose distance from city i is less or equal to R km, that is,
S
j
= { j N | d
i j
R and d
ji
R}
d
i j
Distance from city i to city j
dem
j
Demand for mammography screenings from city j
cap Annual screening capacity of each mammography unit
p Number of mammography units to be allocated
R Maximum distance women should travel to perform a mammography screening
demMin Minimum annual screening demand that a city must have in order to host an equipment
Decision variables
x
i j
Continuous variable into interval [0, 1] that indicates the fraction of demand
from city j which is served by mammography units in the city i.
y
i
Integer variable that represents the amount of equipment installed in the city i.
z
i
Binary variable that assumes value 1 if i hosts an equipment and 0, otherwise.
t
i
Binary variable that assumes value 1 if demand of city i is fully served by the
mammography units themselves and 0, otherwise.
0, otherwise. Constraints (7), (8), and (9) force t
i
to
be 1 if the demand of city i is fully met by its mam-
mography units and 0, otherwise. Constraints (10)
stipulate that a city i will only serve another city j if
its demand just has been fully served by itself. Con-
straints (11) prevent a local with a demand less than
the minimum demand hosts an equipment. Finally,
Constraints (12), (13), (14), and (15) impose the do-
main of the decision variables.
The formulation earlier presented works in a sce-
nario where there is freedom to relocate equipment.
Although this strategy improves equipment distribu-
tion, in the real world it would be difficult to perform
this operation. Hence, it makes sense to maintain the
current distribution of already installed mammogra-
phy units and to optimize the location of future equip-
ment. In this alternative formulation, it is necessary to
add the following parameter:
c
i
= Number of mammography units currently in-
stalled in the city i
Finally, to conclude this new formulation, just add the
following set of constraints:
y
i
c
i
i N (16)
Constraints (16) determine that the number of mam-
mography units installed in each city must be greater
than or equal to the number currently installed in that
one.
5 THE PROPOSED SIMULATED
ANNEALING ALGORITM
This Section presents the proposed Simulated An-
nealing algorithm, developed for solving realistic in-
stances of the MULP. It is organized as follows. Sub-
section 5.1 shows the representation of a solution for
the MULP. Subsection 5.3 describes how the idle
capacity of the equipment is distributed among the
cities of a region. In Subsection 5.2, the procedure
for creating an initial solution is presented. Subsec-
tion 5.4 shows as a solution is evaluated. The neigh-
borhood structure for exploring the solution space of
the MULP is discussed in Subsection 5.5. The details
of the Simulated Annealing algorithm are presented
in Subsection 5.6.
5.1 Solution Representation
A MULP solution s is represented by a pair (y,x), in
which y = (y
1
,y
2
,··· , y
n
) is a n-vector, where y
i
rep-
resents the number of mammography units installed
at city i, and x = (x
i j
) is a (n × n)-matrix, where x
i j
represents the fraction of the demand of city j that is
covered by the city i. As an example, Figure 1 illus-
trates a solution s = (y,x) to a problem with 4 cities
and 2 mammography units.
Figure 1: Representation of a Solution with Two Mammog-
raphy Units and Four Cities.
In Figure 1, positions 1 and 3 of the vector y indi-
cate that the corresponding cities have one equipment
ICEIS 2020 - 22nd International Conference on Enterprise Information Systems
432
each, while the others host none. In the matrix x,
the cells x
11
and x
33
indicate that mammography units
fully meet the demand of the locations where they are
installed. Finally, the cells x
14
, x
12
and x
32
indicate
that city 1 covers 90% of the demand of city 4 and
30% of the demand of city 2. On the other hand, city 3
covers 60% of the demand of city 2.
5.2 Initial Solution
Simulated Annealing (SA) is a metaheuristic that does
not depend on a good initial solution to generate good
results. Therefore, we use a random initial solution
for solving MULP with this metaheuristic. The pro-
cess starts by generating a subset of candidate loca-
tions to host equipment, which consists of all cities
that have the demand equal to or greater than the min-
imum demand. The next step is to choose at random
one of these cities to host the equipment. Once this is
done, all mammography units are assigned to this lo-
cation. Finally, the idle capacity of these mammogra-
phy units should be used to serve other cities within a
radius of R kilometers, as described in Subsection 5.3.
5.3 Idle Screening Capacity
The idle capacity of the equipment in a given city i
can be used to cover the demand of one or more cities.
All cities that are no more than R kilometers from
city i are candidates to have their demand, or part of
it, attended by this city. Among the candidate cities,
the priority will be given to those who have no other
cover option than the city i. For any city to be a po-
tential choice for another, it must be within a radius
of R kilometers and have a minimum demand to host
a mammography unit.
If the sum of residual demands from priority cities
surpass the idle capacity of city i, the allocation pref-
erence is given to cities with lower residual demand.
The residual demand of a city is the difference be-
tween the total number of examinations required by
it and the portion of this demand that is already cov-
ered. Finally, if even after meeting all residual de-
mand from priority cities, there is still idle capacity at
city i, so the other candidate cities will be analyzed,
giving preference again to those with lower residual
demand.
5.4 Solution Evaluation
A MULP solution is evaluated according to Equa-
tion (1), where x
i j
represents the fraction of the de-
mand from city j covered by city i.
5.5 Neighborhood Structure
The solution space of the MULP is explored by re-
location moves. This move consists of removing the
equipment from a city i and installs it in a different
city j, assuming that j has demand equal to or greater
than demMin. Eventual idle screening capacity in
city j is handled as described in Subsection 5.3. The
set of neighbors s
0
of a solution s generated by this
type of move forms the neighborhood N (s).
Relocating the equipment requires attention, since
the demand of the city i, in which the equipment was
removed, may be fully or partially uncovered. Con-
sequently, the care service for its neighboring cities
may be compromised. On the other hand, the city
that receives the equipment may have all or part of its
demand attended, and it may serve other neighboring
cities, both partially and fully.
Algorithm 1 shows a relocation move from a
city i to a city j in a given solution. Initially, at
lines 2 and 3, the RetHubs(i, x) and RetHubs( j, x)
functions are called, returning lists lst
i
and lst
j
,
which represent the cities that serve the cities i
and j, respectively. In line 4, the number of equip-
ment installed in the city i is decreased by one
unit. Subsequently, the UnserveNeighborhood(i,x)
and UnserveNeighborhood( j,x) functions are called
for eliminating the links of the cities i and j with cities
in their regions, respectively. In line 7, the number
of equipment in city j is increased by one unit. In
line 8, it is verified if the list of locations that serves
city i is empty. For each city in the list lst
i
, the
ServeNeighborhood(i,x) function is called at line 11
to update the links of this city to cities in its region.
In line 15, the links of city i to cities in its region
are updated. The same procedure for city i started
at line 8 also is applied for city j. In this case, the
number of elements of lst
j
is checked at line 16, the
ServeNeighborhood( j,x) function is called at line 19,
and the links of city j to cities in its region are updated
at line 23.
5.6 Simulated Annealing
The Simulated Annealing (SA) metaheuristic (Kirk-
patrick et al., 1983) makes an analogy with the metal
annealing process. In this process, the metal is heated
to a high temperature, and then cooled slowly so that
the final product is a homogeneous mass (Haeser and
Ruggiero, 2008).
Unlike traditional descent methods, where neigh-
boring solutions are accepted only if they generate an
improvement in the objective function, SA also ac-
cepts worsening solutions, according to the probabil-
A Mixed Linear Integer Programming Formulation and a Simulated Annealing Algorithm for the Mammography Unit Location Problem
433
Algorithm 1: Relocation Move.
1: procedure RELOCATE(i, j, y,x)
2: lst i RetHubs(i, x);
3: lst j RetHubs( j,x);
4: y[i] y[i] 1;
5: UnserveNeighborhood(i,x);
6: UnserveNeighborhood( j,x);
7: y[ j] y[ j] + 1;
8: if (lst i > 0) then
9: it lst i.begin();
10: while (it 6= lst i.end()) do
11: ServeNeighborhood(it,x);
12: it + +;
13: end while
14: end if
15: ServeNeighborhood(i,x);
16: if (lst j > 0) then
17: it lst j.begin();
18: while (it 6= lst j.end()) do
19: ServeNeighborhood(it,x);
20: it + +;
21: end while
22: end if
23: ServeNeighborhood( j,x);
24: end procedure
ity function (Dowsland, 1993) given by Eq. (17):
P(,T ) = e
/T
(17)
where P(,T ) is the probability of accepting a move,
is the variation in the value of the objective function
(in this case, a decrease variation), and T is the current
temperature.
The idea of SA is to start from a high initial tem-
perature, and, as the method progresses, it will cool
until it reaches a freezing value at the end of the pro-
cedure. The pseudocode of the Simulated Annealing
metaheuristic is described in Algorithm 2.
In line 2 of Algorithm 2, the current solution
is saved in a variable s
?
, which represents the best
solution obtained so far. In line 4, it is checked
if the temperature has reached the freezing value.
If this value has not been reached, there will be
SAmax (input parameter) iterations (line 6), where
neighbors of the current solution (line 8) will be
generated at random. Each neighbor is evaluated
at line 9 according to Eq. (1). In line 10, it is
checked if there was an improvement and, if so,
the neighboring solution becomes the current so-
lution (line 12). If the new generated solution is
the best solution obtained so far, it is stored in
the variable s
?
(line 13). In the case of the neigh-
boring solution has not generated improvement, the
Algorithm 2: Simulated Annealing.
1: procedure SA( f (s), N (s), SAmax,α, T
i
,T
zero
,s)
2: s
?
s;
3: T T
i
;
4: while (T > T
zero
) do
5: IterT 0;
6: while (IterT < SAmax) do
7: IterT + +;
8: Choose s
0
N (s) at random
9: = f (s) f (s
0
);
10: if < 0 then
11: s s
0
12: if f (s) > f (s
?
) then
13: s
?
s
14: end if
15: else
16: Generate x [0,1] at random;
17: if (x < e
/T
) then
18: s s
0
19: end if
20: end if
21: end while
22: T α · T ;
23: end while
24: Return s
?
;
25: end procedure
probability test is performed at line 17. If this wors-
ening solution is accepted, it becomes the new cur-
rent solution. In line 22, the current temperature is
cooled based on a factor α, received as an input pa-
rameter. Finally, the method returns the best solution
s
?
(line 24) found during the search.
The initial temperature (T
i
) of Algorithm 2 is de-
termined by simulation (Gomes J
´
unior et al., 2005).
Their procedure works as follows. Given a random
initial solution and a low temperature (T
i
) as the ini-
tial temperature, the procedure verifies how many so-
lutions are accepted in SAmax iterations. If γ×SAmax
solutions are accepted, then this current temperature
is the starting temperature; otherwise, the temperature
is increased by a rate β > 1. Thus, the initial temper-
ature is that at which γ% of the solutions are accepted
at the beginning of the cooling process.
6 COMPUTATIONAL
EXPERIMENTS
The mathematical programming model was imple-
mented using CPLEX solver, version 12.7.1. The pro-
posed Simulated Annealing algorithm was developed
in C++ language. All tests were performed on a Intel
ICEIS 2020 - 22nd International Conference on Enterprise Information Systems
434
Core i5-4200U 1.6 GHz CPU computer with 6 GB of
RAM under the Ubuntu 16.04.2 operating system.
Subsection 6.1 shows the characteristics of the
instances used for testing the methods. In Subsec-
tion 6.2, a comparison between the results obtained by
the Simulated Annealing algorithm and the CPLEX
solver is introduced. In the following, Subsection 6.3
presents an analysis of the best location of new mam-
mography units to be acquired. A comparison is made
between a scenario where there is freedom to relo-
cate all existing mammography units and another one
where only new equipment to be acquired can have
their locations optimized.
All solutions obtained through both CPLEX
solver and Simulated Annealing algorithm, and the
complete results are available at http://www.decom.
ufop.br/prof/marcone/projects/MULP/MULP.html.
6.1 Instance Characteristics
For testing the methods, 8 instances referring to the
projection of the female population in the age group
indicated for mammography screenings in 2020 were
used. Table 2 shows the characteristics of them.
Table 2: Instance Characteristics.
Instance Name
#
Equip.
Min.
Demand
Screening
Capacity
RO-8-1800-5069 8 1800 5069
RO-8-1800-6758 8 1800 6758
MG-344-375-5069 344 375 5069
MG-258-375-6758 258 375 6758
MG-324-375-5069 324 375 5069
MG-324-375-6758 324 375 6758
MG-324-0-5069 324 0 5069
MG-324-0-6758 324 0 6758
In the first column of Table 2, the name of
each instance indicates its properties in the form
“State-Quantity of Mammography Units-Minimum
Demand-Productivity”. The second, third, and fourth
columns represent, respectively, the number of equip-
ment, the minimum demand (demMin), and the an-
nual screening capacity of the equipment. For exam-
ple, instances “RO-8-1800-5069” and “RO-8-1800-
6758” refer to the Rond
ˆ
onia State, Brazil. In these
two instances, the number of mammography units is
equal to 8, and the minimum demand for screenings
that a city needs to have to host equipment is 1800.
The difference between them is due to the productiv-
ity of the equipment. In the first one, a mammography
unit is able to perform 5069 mammography screen-
ings per year, while in the other one, this number is
6758.
The other 6 instances refer to the Minas Gerais
State. The “MG-344-375-5069” and “MG-258-375-
6758” instances have dummy data on the number of
existing mammography units. In these two instances,
the number of mammography units varies according
to the screening capacity of them. The quantities of
344 and 258 equipment are obtained by dividing the
total demand, equivalent to 1739432 screenings, by
the productivity of the equipment. In the last 4 in-
stances, the number of equipment corresponds to the
current scenario, totaling 324 mammography units.
The difference between them is due to the screening
capacity and the minimum demand considered.
Instances where the minimum demand is 0 (zero)
have been created to analyze scenarios containing
cities with low demand and located in remote regions,
without coverage. In these cases, we simulated a sce-
nario in which any city could host the equipment.
We carry out the projection to the female popu-
lation indicated for the mammography screenings in
2020 as follows. The female population of each city
was determined from Census 2010 (Brasil, 2010). We
assume that the population aged from 30 to 39 years
old in 2010 represents the population aged from 40 to
49 years old in 2020. The same holds true for the pop-
ulation aged from 50 to 69 years old, which, in 2020,
is represented by the population aged from 40 to 59
years old in 2010. Hence, to estimate the demand
of each city is possible, according to the percentages
recommended for the two age groups of the female
population (see Section 3). We observed that the
first two instances refer to the database of Rond
ˆ
onia
State, Brazil, and the last six to the database of Minas
Gerais State, Brazil. In Rond
ˆ
onia, the estimated de-
mand for screenings is 73900, while in Minas Gerais
is 1739432 screenings.
The existing number of mammography units of
the real instances was calculated using information
from the DATASUS website (Brasil, 2019) for August
2019. The annual screening capacity for each equip-
ment was set to 5069 screenings by the Brazilian Na-
tional Cancer Institute (INCA), on November 1, 2015
(INCA, 2015). However, in 2017, this capacity was
changed to 6758 ones (Brasil, 2017). The difference
between these two estimates is related to the screen-
ing amount that can be performed per hour. To reach
5069 screenings per year, a mammography unit must
be available 80% of the time for 22 working days in
each of the 12 months of the year, with 8 working
hours a day and 20 minutes for each screening. On
the other hand, if we assumed that a mammography
screening is done in 15 minutes, then 6758 mammog-
raphy screenings can be performed annually.
The parameter R, which aims to set the maximum
A Mixed Linear Integer Programming Formulation and a Simulated Annealing Algorithm for the Mammography Unit Location Problem
435
distance between two cities that have a service rela-
tionship, was set to 60 km, following recommenda-
tions from the Brazilian Ministry of Health (Brasil,
2017). The distances between cities, which are a key
issue for defining eligibility for a service, were taken
from the Google Maps API.
Finally, was established that a city needs to have a
minimum demand for hosting an equipment. Analyz-
ing the current distribution of mammography units in
Rond
ˆ
onia and Minas Gerais, we defined, as the min-
imum demand parameter, the demand from the city
that, among those that already have the equipment
installed, has the smallest female population within
the age range indicated for mammography screenings.
In Minas Gerais, this value was set to 375 and in
Rond
ˆ
onia to 1800.
6.2 Simulated Annealing based Solution
In this Subsection, we report the results of applying
the CPLEX solver and the Simulated Annealing algo-
rithm for solving the instances shown in Table 2. The
CPLEX solver ran with a timeout of 3600 seconds for
each instance. The results of the Simulated Anneal-
ing algorithm were based on 30 executions for each
instance.
The value of the parameter SAmax was set to a
number proportional to the instance dimension. After
preliminary tests, we set SAmax = 20 × n, where n is
the number of cities of the instance. The other SA pa-
rameters are the cooling rate α and the freezing tem-
perature T
zero
. After testing, α = 0.99 and T
zero
= 0.1
were used. Finally, the initial temperature was ob-
tained by the simulation method, in which the low
temperature to start the process was set to 500. For
the other two parameters of the simulation method,
we assign the values γ = 0.95 and β = 2.
Table 3 presents part of the results of the experi-
ments comparing CLPEX and Simulated Annealing.
The first column indicates the name of the instance.
In the sequel, the column “CPLEX” is divided into
four other columns. The first one indicates the execu-
tion time, in seconds. The results obtained, the upper
bound and the GAP to the upper bound, are shown in
sequence. The Simulated Annealing column also has
four sub-columns. The column “Best” brings the best
result found in 30 runs, while the column Average”
shows the average result of these runs. The average
execution time is displayed in the column Average
Time”. In column “GAP (Average)”, it is presented
the GAP of the SA average result regarding the upper
bound of each instance, obtained via CPLEX.
Regarding the instances of the Rond
ˆ
onia State
(“RO-8-1800-5069” and “RO-8-1800-6758”), the
Simulated Annealing algorithm obtained the optimal
solution in all 30 executions. For the Minas Gerais
instances, the Simulated Annealing algorithm found
values very close to those generated by CPLEX. In
none of these instances, the gap concerning the up-
per bound reached 1%. For the “MG-324-0-6758”,
the optimal value was found. Regarding the instance
“MG-324-375-6758”, the average GAP is 0.001%.
Even though this value may be considered small, it
is important to emphasize that, in the best result ob-
tained by SA, the global optimum was achieved.
Due to the timeout set to 3600 seconds, in three
instances, CPLEX did not reach the optimal value.
Comparatively, in these same instances, the Simulated
Annealing algorithm achieved similar quality results
with runtimes of less than 600 seconds.
6.3 New Equipment Acquisition
Analysis
In the real world, the relocating of equipment may
be impossible. Considering this reality, an analysis
was made regarding the destination of p additional
mammography units. Tables 4 and 5 bring the results
of the acquisition proposals, having as the difference
between them only the productivity of the equipment,
with values equal to 5069 and 6758, respectively.
In each line of these tables, p mammography units
are added to the current scenarios (instances “MG-
324-375-5069” and “MG-324-375-6758”), which
contain 324 mammography units. Column “Screen-
ing Capacity” represents the maximum mammogra-
phy screenings possible to be done using the original
number of mammography units plus those added. To
reach this number, it is necessary to multiply the to-
tal quantity of equipment by the capacity for annual
screenings of them. Column “Fixed” brings the re-
sults considering that previously installed mammog-
raphy units cannot be relocated. On the other hand,
the results in column “Free” assume that equipment
can be freely relocated. For these scenarios, the tables
show the number of covered women, the percentage
of use of each equipment, and the obtained coverage
rate.
Analyzing the results, we can note how inefficient
the current distribution of equipment is. In Table 4,
the current coverage rate does not reach 80% of the
demand. If there is freedom to relocate the equip-
ment, this coverage could reach 94.40%. If the con-
sidered productivity is equal to 6758 annual screen-
ings (see Table 5), the disparity is slightly smaller,
but 170197 (that is, 1738872 - 1568675) additional
screenings could be performed.
The percentage of equipment utilization is ob-
ICEIS 2020 - 22nd International Conference on Enterprise Information Systems
436
Table 3: Comparison of CPLEX versus Simulated Annealing Algorithm Results.
Instance
CPLEX Simulated Annealing
Time
(sec)
Result
Upper
Bound
GAP Best Average
Average
Time (sec)
GAP
(Average)
RO-8-1800-5069 <1 40552 40552 0.000% 40552 40552 8 0.000%
RO-8-1800-6758 <1 50621 50621 0.000% 50621 50621 8 0.000%
MG-344-375-5069 3600 1705656 1708758 0.182% 1698617 1695581 514 0.771%
MG-258-375-6758 356 1695966 1695966 0.000% 1684046 1680693 534 0.901%
MG-324-375-5069 3600 1642031 1642356 0.020% 1640576 1639375 537 0.182%
MG-324-375-6758 16 1738872 1738872 0.000% 1738872 1738860 414 0.001%
MG-324-0-5069 3600 1642152 1642356 0.012% 1641280 1640302 556 0.125%
MG-324-0-6758 16 1739432 1739432 0.000% 1739432 1739432 412 0.000%
Table 4: Results of the Acquisition Proposal of New Mammography Units with Productivity Equal to 5069.
Quantity
Added
Screening
Capacity
Fixed Free
Covered
Equipment
Use
Coverage
Rate
Covered
Equipment
Use
Coverage
Rate
0 1642356 1383109 84.21% 79.51% 1642031 99.98% 94.40%
1 1647425 1388178 84.26% 79.81% 1645401 99.88% 94.59%
2 1652494 1393247 84.31% 80.10% 1650650 99.89% 94.90%
3 1657563 1398316 84.36% 80.39% 1655267 99.86% 95.16%
4 1662632 1403385 84.41% 80.68% 1658962 99.78% 95.37%
5 1667701 1408454 84.45% 80.97% 1663281 99.73% 95.62%
... ... ... ... ... ... ... ...
47 1880599 1621222 86.21% 93.20% 1736474 92.34% 99.83%
48 1885668 1625997 86.23% 93.48% 1736953 92.11% 99.86%
49 1890737 1630378 86.23% 93.73% 1737639 91.90% 99.90%
50 1895806 1634478 86.22% 93.97% 1737732 91.66% 99.90%
51 1900875 1638548 86.20% 94.20% 1737849 91.42% 99.91%
... ... ... ... ... ... ... ...
112 2210084 1738468 78.66% 99.94% 1738872 78.68% 99.97%
113 2215153 1738695 78.49% 99.96% 1738872 78.50% 99.97%
114 2220222 1738872 78.32% 99.97% 1738872 78.32% 99.97%
115 2225291 1738872 78.14% 99.97% 1738872 78.14% 99.97%
tained by dividing the number of women attended
by the service capacity of all installed equipment.
This data is of great interest to public decision-makers
since efficiency is one of the objectives of public ad-
ministration. In scenarios with the freedom to relo-
cate equipment, as new equipment is inserted, the per-
centage of utilization decreases. On the other hand,
when there is a need to maintain the current dis-
tribution of equipment, it is necessary to have new
equipment so that the maximum utilization can be
obtained. Considering the productivity of 5069 an-
nual screenings, the maximum usage percentage is
86.23%, which is achieved when 48 or 49 new equip-
ment is added. For productivity equal to 6758, the
best use is obtained by adding 9 mammography units,
reaching a percentage of use of 72.34%.
We may notice there is a limit regarding the cov-
erage rate. In the first lines, as more mammography
units are added, more women are served. However,
even without covering 100% of the demand, there is
a limit in which adding equipment does not increase
coverage. In Table 4, this occurs when 115 mammog-
raphy units are added. In this case, the number of
covered women is the same when 114 equipment are
added. On the other hand, in Table 5, this limit is 78.
This fact occurs once there are cities that do not have
the infrastructure to host equipment, and they are far
from other cities that can serve them.
7 CONCLUSIONS
This paper dealt with the Mammography Unit Loca-
tion Problem considering the restrictions established
by the Brazilian Ministry of Health. Based on the
formulation presented by (Souza et al., 2019), a new
Mixed-Integer Linear Programming model was pro-
posed. The advantage of this model is that it does not
require the previous processing mentioned by these
authors. The CPLEX solver, version 12.8, was used
to implement the mathematical programming model.
Data from Rond
ˆ
onia state and Minas Gerais state,
A Mixed Linear Integer Programming Formulation and a Simulated Annealing Algorithm for the Mammography Unit Location Problem
437
Table 5: Results of the Acquisition Proposal of New Mammography Units with Productivity Equal to 6758.
Quantity
Added
Screening
Capacity
Fixed Free
Covered
Equipment
Use
Coverage
Rate
Covered
Equipment
Use
Coverage
Rate
0 2189592 1568675 71.64% 90.18% 1738872 79.42% 99.97%
1 2196350 1575433 71.73% 90.57% 1738872 79.17% 99.97%
2 2203108 1582191 71.82% 90.96% 1738872 78.93% 99.97%
3 2209866 1588949 71.90% 91.35% 1738872 78.69% 99.97%
4 2216624 1595707 71.99% 91.74% 1738872 78.45% 99.97%
5 2223382 1602465 72.07% 92.13% 1738872 78.21% 99.97%
... ... ... ... ... ... ... ...
8 2243656 1622185 72.30% 93.26% 1738872 77.50% 99.97%
9 2250414 1627885 72.34% 93.59% 1738872 77.27% 99.97%
10 2257172 1632660 72.33% 93.86% 1738872 77.04% 99.97%
... ... ... ... ... ... ... ...
75 2696442 1738433 64.47% 99.94% 1738872 64.49% 99.97%
76 2703200 1738660 64.32% 99.96% 1738872 64.33% 99.97%
77 2709958 1738872 64.17% 99.97% 1738872 64.17% 99.97%
78 2716716 1738872 64.01% 99.97% 1738872 64.01% 99.97%
Brazil, were used as a case study. Additionally, a Sim-
ulated Annealing based algorithm was also developed
for treating large instances of the problem.
Two distinct experiments were performed. In
the first one, the Simulated Annealing algorithm
was compared with the CPLEX solver. The results
showed that the proposed algorithm was able to pro-
vide good quality solutions in significantly shorter
times. In the second experiment, we considered the
acquisition of new equipment for two scenarios of the
current instance of Minas Gerais state. In the first
scenario, there is the possibility of relocating existing
equipment; on the other hand, in the second scenario,
this relocation is not allowed.
We indicate two issues that may motivate future
work. The first one concerns the costs involved with
the operation. In addition to the equipment itself,
there are expenses with physical infrastructure to host
it, as well as expenses with human resources to op-
erate it. Scenarios with low utilization of equipment
may indicate a poor use of public resources. Thus,
the use of mobile units for mammography screenings
may prove to be a solution. For this, it would be
necessary to define criteria, besides the minimum de-
mand, for a city to receive equipment. Furthermore,
the mobile unit routing would also need to be opti-
mized. The other issue to be addressed in future work
concerns the distance traveled to attendance. For this,
a multi-objective optimization formulation is neces-
sary, incorporating the objectives of maximizing the
covered demand, focus of this work, and also to min-
imize the sum of the distances traveled by women.
ACKNOWLEDGEMENTS
The authors thank Coordenac¸
˜
ao de Aperfeic¸oamento
de Pessoal de Ensino Superior (CAPES), Con-
selho Nacional de Desenvolvimento Cient
´
ıfico e Tec-
nol
´
ogico (CNPq), Fundac¸
˜
ao de Amparo
`
a Pesquisa do
Estado de Minas Gerais (FAPEMIG), Centro Federal
de Educac¸
˜
ao Tecnol
´
ogica de Minas Gerais (CEFET-
MG), Instituto Federal de Educac¸
˜
ao, Ci
ˆ
encia e Tec-
nologia de Minas Gerais (IFMG), and Universidade
Federal de Ouro Preto (UFOP) for supporting this re-
search.
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