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

2 b

Patrick Moreira Rosa

2 c

, Puca Huachi Vaz Penna

2 d

, S

ergio Ricardo de Souza

and Marcone Jamilson Freitas Souza

2 e

Departamento de Computac¸

ao, Centro Federal de Educac¸

ao Tecnol

ogica,

Avenida Amazonas, 7675, 30.510-000 Belo Horizonte, Brazil

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

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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

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

ea et al., 2018; Souza et al.,

2019).

Corr

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

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

∑

i∈N

∑

j∈S

dem

· x

i j

(1)

s.t.

∑

i∈S

i j

≤ 1 ∀ j ∈ N (2)

∑

i∈ N

= p (3)

∑

j∈S

dem

· x

i j

≤ cap.y

∀i ∈ N (4)

≥ y

/p ∀i ∈ N (5)

≤ y

∀i ∈ N (6)

≤ z

∀i ∈ N (7)

≥ dem

· x

− dem

+ 1 ∀i ∈ N (8)

≤ x

∀i ∈ N (9)

≥ x

i j

∀i,∀ j ∈ N | i 6= j (10)

= 0 ∀i ∈ N | dem

< demMin (11)

i j

∈ [0,1] ∀i,∀ j ∈ N (12)

∈ Z ∀i ∈ N (13)

∈ {0,1} ∀i ∈ N (14)

∈ {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

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

Set of cities whose distance from city i is less or equal to R km, that is,

= { j ∈ N | d

i j

≤ R and d

≤ R}

i j

Distance from city i to city j

dem

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

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.

Integer variable that represents the amount of equipment installed in the city i.

Binary variable that assumes value 1 if i hosts an equipment and 0, otherwise.

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

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:

= Number of mammography units currently in-

stalled in the city i

Finally, to conclude this new formulation, just add the

following set of constraints:

≥ c

∀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

,··· , y

) is a n-vector, where y

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

and x

indicate that mammography units

fully meet the demand of the locations where they are

installed. Finally, the cells x

, x

and x

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

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

and lst

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

, 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

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

zero

,s)

2: s

← s;

3: T ← T

;

4: while (T > T

zero

) do

5: IterT ← 0;

6: while (IterT < SAmax) do

7: IterT + +;

8: Choose s

∈ N (s) at random

9: ∆ = f (s) − f (s

);

10: if ∆ < 0 then

11: s ← s

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

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

(line 24) found during the search.

The initial temperature (T

) 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

) 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

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

onia

State, Brazil, and the last six to the database of Minas

Gerais State, Brazil. In Rond

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

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

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

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

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

encia e Tec-

nologia de Minas Gerais (IFMG), and Universidade

Federal de Ouro Preto (UFOP) for supporting this re-

search.

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