Mammography Unit Location: Reconciling Maximum Coverage and

Budgetary Constraints

Rudivan Paix

ao Barbosa

1 a

, Marcone Jamilson Freitas Souza

1 b

and Gilberto de Miranda Junior

2 c

Programa de P

os-graduac¸

ao em Ci

encia da Computac¸

ao, Universidade de Federal Ouro Preto, Ouro Preto, Brazil

Programac¸

ao de P

os-graduac¸

ao em Engenharia de Produc¸

ao, Universidade Federal de Ouro Preto,

Campus Jo

ao Monlevade, Brazil

Keywords:

Location-Allocation, Mammography Units, Bi-Objective Optimization.

Abstract:

This work addresses the Bi-objective Mammography Unit Location-Allocation Problem. This problem con-

sists in allocating mammography units satisfying two objectives and respecting the constraints of device ca-

pacity for screenings and the maximum travel distance for the service. The ﬁrst objective function maximizes

the coverage of exams performed by the allocated mammography devices, while the second function mini-

mizes the total amount of equipment used. We introduce a mixed-integer linear programming bi-objective

model to represent the problem and apply the Weighted Sum and Epsilon-constraint methods to solve it. The

Epsilon-constraint method was able to generate better Pareto fronts. The instances used for testing come from

real data from two Brazilian states obtained from the Brazilian Health Ministry.

1 BREAST CANCER IN

AMERICAS: HEALTH CARE

BUDGETS vs. HUMAN LIVES

AT STAKE

There were 4 million cancer cases in the Americas in

2020, and this disease was responsible for 1.4 million

deaths in these continents (PAHO, 2020). Also, ac-

cording to this report, breast cancer is the second lead-

ing cause of death among women. Almost 500,000

new breast cancer cases and more than 100,000 deaths

from breast cancer were registered in the Americas.

In Brazil, the situation is no different. In 2020,

there were an estimated 66,280 breast cancer cases,

representing an adjusted incidence rate of 43.73 cases

per 100,000 women (INCA, 2019), with 11.84 deaths

per 100,000 women (INCA, 2022).

When breast cancer is diagnosed in the early

stages, 95% of women affected survive (Witten and

Parker, 2018). On the other hand, mammography

screening is the primary way to detect early-stage

breast cancer (Xavier et al., 2016). Given this, the

Brazilian Ministry of Health recommends that women

https://orcid.org/0000-0002-4997-6261

https://orcid.org/0000-0002-7141-357X

https://orcid.org/0000-0001-5552-0079

aged 50-69 should have the screening biannually

(Brasil, 2017). This institution also recommends that

20% of the female population aged 40-49 undergo

yearly screening.

Although Brazilian public and private health ser-

vices have a sufﬁcient number of mammography

units, screening is not accessible to all women (Mi-

randa and Patrocinio, 2018; Amaral et al., 2017). A

limiting factor to accessing mammography screen-

ing is the determination of the Brazilian Ministry of

Health, which deﬁnes 60 km as the maximum dis-

tance a woman should travel (Brasil, 2017). In (Ama-

ral et al., 2017), the authors showed that some cities

have an oversupply of screenings, while others are

not served by any equipment within a 60 km radius.

A similar result is corroborated in (Rodrigues et al.,

2019), whose authors identiﬁed an unequal distribu-

tion of mammography devices in Brazil, with a sur-

plus of equipment in 17 states and a deﬁcit in the

others (9 states and the Federal District). Although

other factors contribute to discouraging or even mak-

ing screening infeasible, these studies show that the

distance women should travel to undergo mammog-

raphy screenings plays a key role in access to it.

Many studies have proposed mathematical pro-

gramming formulations and heuristics to propose the

best location and allocation for mammography units

(Corr

ea et al., 2018; de Campos et al., 2020; Souza

Barbosa, R., Souza, M. and Miranda Junior, G.

Mammography Unit Location: Reconciling Maximum Coverage and Budgetary Constraints.

DOI: 10.5220/0011852200003467

In Proceedings of the 25th International Conference on Enterprise Information Systems (ICEIS 2023) - Volume 1, pages 187-194

ISBN: 978-989-758-648-4; ISSN: 2184-4992

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

187

et al., 2019; Souza et al., 2020; de Assis et al.,

2022). These studies aim to maximize the coverage

for screenings and present a tool to health managers to

support them in deciding the best location for mam-

mography units and their allocations (assignments),

i.e., the set of locations that each mammography unit

must serve. Although other studies address generic

facility location-allocation problems for health care,

no study was found in the literature to maximize cov-

erage and minimize the number of mammography

units needed for screenings. Reducing the number

of mammography units needed is essential, given the

high cost of these devices and the need for better

management of public health costs. In order to ﬁll

this gap, we introduce the Bi-objective Mammogra-

phy Unit Location-Allocation Problem (BOMULAP)

in this article.

The main contributions of this work are the fol-

lowing:

i) Introducing of the BOMULAP aiming at maxi-

mizing the coverage for screenings and minimiz-

ing the infrastructure costs of health care;

ii) The implementation and comparison of two

well-known solution methods of multi-objective

optimization, the Weighted Sum Method and the

Epsilon-green Constraint Method, to solve the

BOMULAP;

iii) Application of the methodology to realistic cases

extracted from Brazilian public health service

databases.

The remainder of this article is organized as fol-

lows. Section 2 presents a literature review of the

main articles about the Mammography Unit Location

and Mobile Mammography Unit Routing problems

considering the Brazilian reality. In Section 3.1, we

introduce the problem under investigation. In Sec-

tion 3.2, the bi-objective mathematical programming

model is presented. Section 3.3 presents brief descrip-

tions of the proposed solution methods for solving

BOMULAP. In Section 4, the computational results

of the proposed methods are presented and discussed.

Finally, Section 5 presents the conclusions and future

work.

2 SURVEYING THE HISTORY

THUS FAR

In (Shariff et al., 2012), the authors tackle a health

care location-allocation problem from Malaysia.

They introduce a mathematical programming formu-

lation for the problem and then develop a Genetic Al-

gorithm (GA) to obtain good upper bounds in a rea-

sonable time. The analysis of the obtained results es-

tablishes the superior computational performance of

the Genetic Algorithm, which is able to ﬁnd high-

quality solutions for all the instances on the test-bed.

On the other hand, the CPLEX commercial solver was

unable to reach good quality solutions for networks

displaying more than 809 nodes.

(Beheshtifar and Alimoahmmadi, 2015) handle a

multi-objective location-allocation problem for clinic

facilities. The authors propose four objective func-

tions for their mathematical program, based on the

well-known p-median problem, bringing the under-

lying assumption of full demand coverage. The ﬁrst

objective function minimizes the sum of demand-

weighted distance costs. The second objective func-

tion minimizes the standard deviation of the distance

between the patients and the health care clinics. The

third objective function maximizes the suitability of

the selected locations for the new facilities. Finally,

the fourth objective function minimizes the costs of

acquiring land space and installing new health care

clinic facilities. The Non-dominated Sorting Genetic

Algorithm II (NGSA II) is used to yield the Pareto-

efﬁcient front. The analysis of the obtained results

shows that it is possible to reduce the average distance

to provide access to health care services at the expense

of larger land acquisition costs, generally speaking.

In (Corr

ea et al., 2018), the authors present four

formulations for the MULAP. The models are based

on the classical p-median problem. The second dif-

fers from the ﬁrst in allowing violation of the maxi-

mum distance that a mammography device can be as-

signed to serve a locality. In the last two models, the

number of women traveling to be served is consid-

ered, and the maximum distance constraint in the last

one is relaxed. The authors tested the formulations

using the cities of 12 health regions that are up to 100

km away from the city of Ouro Preto. No feasible so-

lutions were found in the established processing time

in the formulations that impose maximum distance.

One of the reasons for this is that several cities in the

state of Minas Gerais are more than 60 km away from

those that host mammography devices.

In (Souza et al., 2019), the authors did a case study

of the MULAP considering the State of Rond

onia,

Brazil. They present two mathematical program-

ming models based on the maximum coverage. The

ﬁrst model considers that a mammography device can

only serve a city if it fully meets its demand. The

second model relaxes the constraint of fully meeting

the demand. They showed that the second formula-

tion better uses the devices’ screening capacity. The

authors suggest using mobile mammography devices

ICEIS 2023 - 25th International Conference on Enterprise Information Systems

188

in cities not served by ﬁxed devices.

In (S

a et al., 2019), the authors did a case study

of the MULAP to the State of Esp

ırito Santo, Brazil,

using the binary mathematical programming model

of Souza et al. (2019). They treated two scenarios.

In the ﬁrst, the location of mammography devices al-

ready installed is not changed, while in the second

scenario, there is permission for free relocation of

these mammography devices. They analyzed the ac-

quisition of new equipment in these two scenarios and

showed how many new mammography devices would

be needed to cover all the demands for mammography

screenings in the state.

In (Souza et al., 2020), the authors did a case

study of MULAP in the state of Minas Gerais, Brazil,

which has the largest number of cities in Brazil. They

used the binary mathematical programming formu-

lation of (Souza et al., 2019). In addition, as MU-

LAP is NP-hard (Church and ReVelle, 1974), they

developed a Variable Neighborhood Search-based al-

gorithm (Hansen et al., 2017) to handle large in-

stances of MULAP, such as the one in this state of

the Brazilian federation. They showed that the pro-

posed VNS algorithm ﬁnds good-quality solutions in

reduced computational time.

A case study of MULAP in Minas Gerais and

Rond

onia states is also done in (de Campos et al.,

2020). They used the relaxed formulation of (Souza

et al., 2019), which allows the demand of a city to be

partially covered by a host city, and also developed

a Simulated Annealing-based algorithm (Kirkpatrick

et al., 1983). The authors analyzed two scenarios to

determine the minimum amount of devices needed to

maximize the coverage for mammography screenings

in the Minas Gerais state, considering 6758 screen-

ings per year as the productivity of mammography

equipment. In the ﬁrst scenario, the relocation of ex-

isting equipment is allowed, while in the second, this

relocation is not allowed. They showed that in the ﬁrst

scenario, it is possible to cover 99.97% of the demand

in Minas Gerais without purchasing any new equip-

ment, while in the second scenario, it would be nec-

essary to purchase 77 devices to achieve this coverage

rate.

(Rosa et al., 2020a) included a new constraint

in the mathematical programming model of (Souza

et al., 2019) to treat MULAP. In this model, women

can only be served by cities within the same health

micro-region in which they reside. The authors also

analyzed the gradual acquisition of new devices to

maximize coverage by screenings. They showed that

with cities clustered into health micro-regions, the

number of equipment needed to maximize coverage

is higher than without this requirement. This result

shows that the existing clustering of cities may need

to be revised.

(Rosa et al., 2020b) introduced the Mammogra-

phy Mobile Unit Routing Problem (MMURP). The

problem consists of maximizing the demand met and

reducing the distance traveled by mobile mammog-

raphy units in Minas Gerais state. They proposed a

hierarchical constructive heuristic algorithm, wherein

a solution is considered better in the ﬁrst level when

it meets a greater number of screenings. If two solu-

tions have the same number of screenings, the chosen

solution is the one with the shorter distance traveled.

The authors showed that in the region studied, which

involves 444 cities, it would be possible to provide

almost 360 thousand more mammography screenings

than is currently provided in the state.

(Rosa et al., 2021) approached the MMURP

through the Iterated Greedy algorithm (St

utzle and

Ruiz, 2018) and named it Smart IG. The Randomized

Variable Neighborhood Descent (RVND) procedure

is executed to reﬁne the solution. Of the 853 cities in

Minas Gerais state, 579 were analyzed. The results

showed that the developed algorithm found solutions

that fully meet the demand of the studied region, and

these results are superior to those obtained through

the constructive algorithm by its previous work (Rosa

et al., 2020b).

In (de Assis et al., 2022), the authors treated the

MULAP allowing the partial fulﬁllment of the de-

mands for screenings of the cities. They proposed an

algorithm based on the General Variable Neighbor-

hood Search (GVNS) (Hansen et al., 2017). The ini-

tial solution is built by a procedure based on the con-

struction phase of the GRASP metaheuristic (Festa

and Resende, 2018). They also introduced a new

representation for the problem solution, in which it

is possible to individualize the screenings performed

by each mammography device. They compared the

results of the proposed algorithm with those of the

Simulated Annealing by (de Campos et al., 2020) and

showed that GVNS obtained better solutions in some

instances.

In (de Freitas Almeida et al., 2022), the authors

deal with the location-allocation of Magnetic Reso-

nance Imaging (MRI) machines. The main objec-

tive is to maximize the coverage for MRI exams con-

sidering the new equipment acquisitions and equity

in regional service supply in Brazil. They propose

three mathematical programming formulations for the

problem. The ﬁrst one aims to maximize the demand

covering; the second minimizes the cost of acquiring

new MRI machines. The third and ﬁnal one aims to

minimize the traveled distance by the patients. Sev-

eral distinct scenarios are investigated for different

Mammography Unit Location: Reconciling Maximum Coverage and Budgetary Constraints

189

numbers of new MRI units acquired. The analysis of

the obtained results shows that for full demand cov-

erage, a total of 812 new MRI machines would be

required, where 753 of those new MRI units would

be allocated to cities with no MRI machines avail-

able. The authors also consider addressing stochastic

demand components and decomposition methods as

solution strategies for future work.

3 MATERIALS AND METHODS

3.1 Problem Statement

The Bi-objective Mammography Unit Location-

Allocation Problem discussed here, denoted by BO-

MULAP, has the following characteristics:

• There is a set N of cities to be covered by mam-

mography screenings;

• There is a set of p mammography units to be allo-

cated to the set N;

• Each mammography unit has a capacity to per-

form Γ screenings annually;

• Each city j has a demand δ for mammography

screenings;

• Only cities with hospital infrastructure are candi-

dates to host mammography units;

• Each city that hosts a mammography machine can

only meet the demand of cities that are no more

than R km away from it;

• A city cannot be served by more than one host

city;

• A host city can only serve another city if it is able

to meet all of its demand.

The objectives are to maximize the coverage of

mammography screenings and minimize the number

of mammography units needed.

3.2 BOMULAP ILP Fomulation

Through equations (1) to (8), we introduce the bi-

objective integer linear programming formulation that

deﬁnes the BOMULAP. This formulation transforms

into an objective function, the constraint used in

(Souza et al., 2020) that limits the number of mam-

mography units.

Initially, we introduce the parameters and deci-

sion variables of the formulation.

Model Parameters:

i j

: Distance from city i to city j;

: Annual demand for mammography screen-

ings in city j;

Γ : Annual mammography screening capacity

of a mammography unit;

p : Maximum number of mammography units

allowed;

R : Maximum distance for service;

: Parameter that assumes the value of 1 if the

city i has hospital infrastructure to host a

mammography device and 0 otherwise;

i j

: {(i, j) ∈ N × N | [(d

i j

∧ d

) ≤ R] ∧ ξ

= 1}

is the adjacency set, ﬁltered by the maxi-

mal service distance R and the infrastruc-

ture availability ξ

Decision Variables:

i j

: 1 if women of city j are served by an

equipment located in city i and 0 otherwise,

∀(i, j) ∈ S

i j

;

: number of mammography units located in

city i, ∀i ∈ N | ξ

= 1.

Problem Formulation:

max f

(x) =

∑

(i, j)∈ S

i j

· x

i j

(1)

min f

(y) =

∑

i∈N | ξ

(2)

s.t.:

∑

(i, j)∈ S

i j

· x

i j

≤ Γ · y

, ∀i ∈ N | ξ

= 1 (3)

∑

(i, j)∈ S

i j

≤ 1, ∀ j ∈ N (4)

i j

≤ x

, ∀(i, j) ∈ S

i j

(5)

≤ p · x

, ∀i ∈ N | ξ

= 1 (6)

i j

∈ {0, 1}, ∀(i, j) ∈ S

i j

(7)

∈ Z

. ∀i ∈ N | ξ

= 1 (8)

The objective functions (1) and (2) aim at max-

imizing the total demand for mammography screen-

ings and minimizing the total number of mammogra-

phy units, respectively. Inequalities (3) are standard

bin-packing constraints ensuring that the capacity of

each mammography unit for annual screenings must

be uphold. Constraints (4) indicate that each city j

needs to be served by some mammography machine

installed in city i if the pair (i, j) is adjacent (service-

ICEIS 2023 - 25th International Conference on Enterprise Information Systems

190

able) or not to be served at all. Constraints (5) force a

mammography unit installed in city i to handle the lo-

cal demand at least, also working as a strong version

of the well-known ﬁxed-charge constraints, strength-

ening the formulation. Inequalities (6) tie together

the mammography service availability to the alloca-

tion of mammography units in a given city i. Finally,

constraints (7) and (8) specify the decision variables’

feasible domains. Please, recall that despite the dis-

tance matrix d

i j

is not directly used in the formula-

tion, this parameter is required to implement the adja-

cency set S

i j

, and therefore its deﬁnition is needed for

completeness.

3.3 Two Multi-Objective Competing

Philosophies

In this Section, the exact methods of multi-objective

optimization proposed for the solution of the BOMU-

LAP are presented.

3.3.1 The Epsilon-Constraint Method

The Epsilon-constraint method (Ritzel et al., 1994)

transforms a multi-objective problem into a mono-

objective problem. One objective is chosen to be opti-

mized, and the others are transformed into additional

inequality constraints in the model.

In the model discussed in this article, the objec-

tive function f

(x) described by Eq. (1) was chosen to

be maximized while the f

(y), described by Eq. (2),

was added as a constraint in the model,

∑

i∈N

≤ p.

Therefore, the formulation to be solved for every se-

lected value of p becomes:

max f

(x) =

∑

(i, j)∈S

i j

· x

i j

(9)

Subject to (3)-(8) and:

∑

i∈N | ξ

≤ p (10)

where p is then prescribed from 1 to 100 in order to

cover the full demand for mammography screenings

and build the desired Pareto front.

3.3.2 The Weighted Sum Method

The Weighted Sum Method (WSM) (Zadeh, 1963)

consists of assigning weights to the objective func-

tions of a multi-objective problem, thus transforming

it into a single-objective problem. The sum of all

weights must be equal to 1.

Hence, for this approach, the objective functions

(1) and (2) are linearly combined with the aid of a

weight λ ∈ [0, 1], resulting:

max λ (

∑

j∈N

) f

(x) − (1 − λ)

(y)

(11)

Function (11) is then maximized after the proper pre-

scribing of λ, in order to yield the Pareto efﬁcient

front.

4 COMPUTATIONAL

EXPERIENCE: WHY ALL THIS

EXTRA BURDEN PAYS OFF

4.1 A Realistic Test-Bed Based on

Real-World Problems

In order to test the two solution methods, we used the

instances related to the states of Esp

ırito Santo (ES)

and Rond

onia (RO) available in (S

a et al., 2019) and

(Souza et al., 2019), respectively. Table 1 presents

the main characteristics of these instances. In this

table, the columns State, nC, p, δ, R, and Γ repre-

sent, respectively, the State of the Brazilian federa-

tion, its number of cities, the number of mammog-

raphy units existing in this State, the existing demand

for mammography screenings, the maximum distance

allowed between an equipment host city and the cities

it serves, and the annual capacity of mammography

screenings for each device.

Table 1: Instance Characteristics.

State nC p δ R Γ

ES 78 30 262732 60 5069

RO 52 8 120636 60 5069

For the test environment, AMPL software with Cplex

20.1.0.0 was used to run the two solution methods

proposed in the previous section. These methods

were tested on a computer equipped with 1 Intel(R)

Core(TM) i7-10750H CPU @ 2.60GHz (12 threads,

fully utilized), 16 GB RAM and Windows 11 Home

system.

4.2 Analyzing the Obtained Results

For the generation of the Pareto front by the Weighted

Sum Method, we ran the model 100 times, adding

∆

= 0.01 to the value of λ at each run, starting it with

Mammography Unit Location: Reconciling Maximum Coverage and Budgetary Constraints

191

the value λ = 0. In turn, to generate the Pareto front

by the Epsilon-constraint method, we also ran the

model 100 times for the ES and RO instances, adding

in one unit the value of p at each execution and start-

ing p with the value 1. We performed the executions

until the gap of 1%. The data of the instances of the

ES and RO states were obtained through the Brazil-

ian government’s website (DATASUS, 2021) and the

Google Maps API, considering travel by car.

Figure 1(a) and Figure 2(a) show the Pareto

fronts of BOMULAP generated by the Weighted Sum

Method in the ES and RO instances, respectively. In

these ﬁgures, the horizontal axes represent the num-

ber of allocated mammography units, and the vertical

axes represent the total demands met. In turn, Fig-

ure 1(b) and Figure 2(b) illustrate the Pareto fronts of

BOMULAP by the Epsilon-constraint method.

(a) Weighted Sum Method.

(b) Epsilon-constraint method.

Figure 1: Pareto fronts for the ES instance.

It is observed that after a given number of installed

mammography devices, increasing them is no longer

worthwhile. In fact, even though there is a demand to

be met, the infrastructure and maximum distance con-

straints prevent further improving the maximal cover-

age, despite the eventual availability of mammogra-

phy units. An alternative to overcome this situation is

integrating the MULAP with the MMURP.

Regarding the two competing multi-objective

philosophies, the Weighted Sum scheme is particu-

larly vulnerable to our way of handling the trade-offs

(a) Weighted Sum Method.

(b) Epsilon-constraint Method.

Figure 2: Pareto fronts for the RO instance.

between maximum coverage and infrastructure de-

ployment since this scheme prefers not to spend any

budget unless the concerns of unattended demands

are relatively higher. On the other hand, the Epsilon-

Constraint scheme can always yield some coverage,

even though a small number of mammography units

is allowed. Therefore, for the features of our speciﬁc

application, the Epsilon-Constraint technique is cer-

tainly preferable when compared to the devised al-

ternative since it is capable of deﬁning the efﬁcient

front with superior resolution, favoring an enhanced

decision-making.

5 INSIGHTS AND UPCOMING

WORK

This paper presents the Weighted Sum and Epsilon-

constraint methods for solving BOMULAP. In all in-

stances, the Epsilon-constraint method provides bet-

ter Pareto fronts than the Weighted Sum method.

The results presented in this paper can assist

health managers in their decision-making, such as de-

ciding where to relocate existing mammography de-

vices and/or purchase new ones.

Future work intends to apply other exact methods,

like the Parallel Partitioning Method (Lemesre et al.,

2007), and solve instances from other Brazilian states.

ICEIS 2023 - 25th International Conference on Enterprise Information Systems

192

Since BOMULAP is NP-hard, developing heuristic

methods, such as Non-dominated Sorting Genetic Al-

gorithm II (NSGA-II) (Deb et al., 2002), to deal with

large instances of the problem is another suggestion.

ACKNOWLEDGMENTS

The authors are grateful for the support provided

by the Universidade Federal de Ouro Preto, the

Coordenac¸

ao de Aperfeic¸oamento de Pessoal de

ıvel Superior - Brazil (CAPES) - Finance Code 001,

CNPq (grants 428817/2018-1, 303266/2019-8, and

307853/2021-7), and FAPEMIG (grant PPM CEX

676/17).

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