Balancing Resources and Demand: A Bi-Objective Mixed-Integer

Programming Approach of Healthcare Districts in Chile

Paulette Castillo

, Victor Bucarey

3 a

, Sebasti

an Davila

1,2

and Franco Quezada

1,2 b

University of Santiago of Chile (USACH), Faculty of Engineering, Industrial Engineering Department, Chile

University of Santiago of Chile (USACH), Faculty of Engineering,

Program for the Development of Sustainable Production Systems (PDSPS), Chile

Institute of Engineering Sciences, Universidad de O’Higgins, Rancagua, Chile

Keywords:

Healthcare System, Districting Problem, Mixed-Integer Linear Programming Formulation.

Abstract:

In the search for equitable and efﬁcient health service delivery, geographical partition into operational districts

is a fundamental factor. This research delves into the intricate challenge of combinatorial optimization of

healthcare districts, with an application to the Metropolitan region of Santiago, Chile, where growing popula-

tion pressures exacerbate concerns about the distribution of healthcare resources. By emphasizing continuity

from primary to secondary healthcare levels, we underline the importance of a good district plan, considering

key parameters such as population homogeneity, compactness, and alignment between capacity and demand.

By applying a mixed integer linear programming model with a bi-objective function, our ﬁndings indicate sub-

stantial scope for improving resource allocation, potentially cutting overages by up to 38.77% at the primary

healthcare level and up to 15% at the secondary healthcare level.

1 INTRODUCTION

One of the foundations of healthcare service delivery

is the partitioning of geographical regions into opera-

tional units or districts. This partitioning aims to opti-

mize resource utilization and ensure equitable access

to healthcare services across diverse demographic and

socio-economic strata. With roots in various disci-

plines, including city planning, administrative juris-

diction allocation, and, notably, healthcare manage-

ment, the concept of districting—also referred to as

re-districting or territory design—emerges as a chal-

lenging combinatorial optimization problem (Validi

and Buchanan, 2022).

Mathematical modeling of the healthcare district-

ing problem entails formulating an objective function

that drives towards creating districts that uphold ge-

ographic cohesion while synchronizing several key

performance indicators, including but not limited to

resource allocation, population balancing, and de-

mand ﬂux. Particularly in settings like Latin Amer-

ica, the diversity and disparity inherent in the pop-

ulation and regional geography fuel the complexity

https://orcid.org/0000-0002-3043-8404

https://orcid.org/0000-0001-5763-5582

of this optimization problem, resulting in a challeng-

ing environment for applying districting models and

strategies (Lin and Kao, 2008).

In this work, we focus on primary and secondary

healthcare planning, where the implications of dis-

tricting are prominently displayed. The pathway

through healthcare often commences with a primary

care encounter, typically with a General Practitioner

(GP), and is subsequently integrated with secondary

care depending on the patient’s needs. A meticulously

crafted districting plan becomes critical to orches-

trating a hierarchical network that seamlessly guides

patients along this journey, from primary care cen-

ters to advanced care centers, while balancing the

load across the entire healthcare infrastructure (R

ıos-

Mercado and L

opez-P

erez, 2013).

In-depth planning of healthcare systems involves

a cascade of intertwined decisions, spanning the loca-

tion of healthcare points, demarcation of service ar-

eas, capacity conﬁguration, resource allocation, and

staff scheduling, to name a few. The design of health-

care service regions, or in other words, the districting

of healthcare services, surfaces as a cardinal dimen-

sion in this multifaceted planning landscape. Effec-

tive districting can pave the way towards minimizing

costs, enhancing capacity utilization, elevating patient

Castillo, P., Bucarey, V., Davila, S. and Quezada, F.

Balancing Resources and Demand: A Bi-Objective Mixed-Integer Programming Approach of Healthcare Districts in Chile.

DOI: 10.5220/0012410100003639

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 13th International Conference on Operations Research and Enterprise Systems (ICORES 2024), pages 341-349

ISBN: 978-989-758-681-1; ISSN: 2184-4372

341

satisfaction, and, importantly, embedding equity into

the accessibility of healthcare services across the so-

ciety (Cort

es et al., 2018; R

ıos-Mercado, 2020). In

general, several criteria are considered in the literature

for planning of healthcare systems, including Popu-

lation Homogeneity, which seeks to minimize devi-

ations in population size, medical expenditure, or the

proportion of elderly individuals in each district. Pop-

ulation Homogeneity or balancing is one of the most

common objectives in healthcare districts as well as

Compactness, which aims to minimize patients’ travel

time inside a district and Contiguity that is deﬁned as

the ability to move between territorial units within a

district without leaving the district. Other criteria in-

clude, for example, Capacity-Demand Match, which

imposes penalties for unmet demand or tries to mini-

mize excess of demand over available capacity; or Va-

riety of Medical Procedures, where each district must

ensure a number of basic procedures.

This research spotlight focuses on unraveling the

healthcare districting problem, speciﬁcally in the con-

text of the City of Santiago, Chile. In recent years,

this city has experienced a surge in population den-

sity, which has concurrently escalated the burden on

healthcare centers. This increased load is notably ev-

ident in the lengthening waiting lists for surgical and

medical consultations (del Gobierno de Chile, 2023).

One hypothesis suggests that a suboptimal distribu-

tion of resources across healthcare centers may be a

latent root cause. By re-evaluating and potentially

re-engineering the allocation of communes within the

City of Santiago relative to the current distribution,

a more balanced and potentially more efﬁcient dis-

tribution of resources, aligned with the demands of

the growing population, might be achieved. Thus, ad-

dressing the Healthcare Districting problem emerges

as a potential strategy to design districts that harmo-

nize population and resource distributions, aiming for

a seamless and equitable healthcare experience for all.

The article is organized as follows. In Section 2,

a review of the relevant literature for the problem of

Primary and Secondary Healthcare Districting is pre-

sented. The problem description and a mathematical

formulation to solve it is introduced in Section 3. Sec-

tion 4.1 showcases the case study along with its re-

spective computational results and discussion. Con-

clusions and further research work are discussed in

Section 5.

2 RELATED WORKS

Different approaches and techniques have been ap-

plied to the healthcare districting problem to meet de-

sirable attributes of the geographical partition, such

as population balance and resource distribution. Two

pioneer works in the area are the one of Ghiggi et

al. (1975) and Pezzella et al. (1981). The for-

mer laid down several key assumptions for districting

the healthcare system, such as the idea that regions

are composed of indissoluble communities with cen-

tralized populations, geographical connectivity of the

districts, and balance of district populations. Their

overarching goal was to devise a hybrid method for

districting, ensuring each district’s self-sufﬁciency

concerning health systems while satisfying planners

and citizens alike. Pezzella et al. (1981) presents a

two-step methodology for partitioning a given geo-

graphical area, ensuring an optimal allocation of the

available healthcare services. The ﬁrst step consists

of determining an optimal partition by taking into ac-

count the demand and capacity of each territorial unit.

Then, in the second step, the ﬁrst partition is aggre-

gated into new partitions by considering the lower and

upper bounds for the population in each district.

In addition, a crucial aspect that multiple studies

have highlighted is the consideration of multiple ob-

jective functions. Datta et al. (2013) emphasizes geo-

graphical compactness, alignment with existing local

authorities, and size and population homogeneity. On

a similar note, Steiner et al. (2015) looked at optimiz-

ing population homogeneity, medical procedure vari-

ety per district, and intradistrict distances in a study

in Brazil. Meanwhile, Yanık et al. (2020) introduced

the innovative concept of gradual assignment. Here,

demands could be split among neighboring districts,

enriching their multi-period, multi-criteria model.

In terms of approaches employed to solve the

healthcare districting problem, both exact and heuris-

tic methods have been observed. However, recogniz-

ing the NP-Hard nature and computational intractabil-

ity of the problem, several researchers have turned to

heuristic and metaheuristic algorithms. Notably, Gray

Wolf Optimizer and Antlion Optimizer algorithms

were implemented in Farughi et al. (2019). These ap-

proaches ﬁnd resonance in the works of Farughi et al.

(2020) that also used the Gray Wolf Optimizer heuris-

tic and compared its performance with an improved

genetic algorithm. Additionally, genetic algorithms to

solve single- and multi-objective healthcare district-

ing problems have also been investigated in Datta et

al. (2013) and Steiner et al. (2015).

The complexity of the problem and the introduc-

tion of newer dimensions like stochastic demand have

led to innovative solutions. A robust optimization

approach is investigated in Darmian et al. (2022),

whereas Fan and Xie (2022) adopted a two-stage dis-

tributionally robust optimization approach consider-

ICORES 2024 - 13th International Conference on Operations Research and Enterprise Systems

342

ing the unpredictability of demand, especially rele-

vant in the backdrop of the pandemic and vaccination

drives.

The application of these methodologies and ap-

proaches is best understood through real-world case

studies. Van Minh et al. (2014) assessed the capa-

bilities of a primary healthcare system in a district,

encompassing both physical and human resources,

in a rural city in central Vietnam. Steiner et al.

(2015) shows the application of a multi-objective op-

timization approach for dealing with a real-world

spatial problem of aggregating the municipalities of

the Parana State in Brazil into some microregions.

Farughi et al. (2019) and Farughi et al. (2020) demon-

strated their model in the South Khorasan Province

of Iran, offering insights into practical implications.

Yanık et al. (2020) applied its multi-criteria and

multi-period formulation with the concept of grad-

ual assignment to Istanbul, Turkey. Lastly, among

the recent studies, Darmian et al. (2022) not only in-

troduced a model considering contiguity and popula-

tion balance but also factored in demand uncertainty,

showcasing its practicality for a real-world case study

of Iran’s healthcare system.

This work is closely related to previous research

on districting, as it considers several relevant fea-

tures of the healthcare districting problem simultane-

ously. However, most studies focus on maximizing

population homogeneity and/or compactness, treating

other features such as capacity-demand match as con-

straints or service levels to be satisﬁed. In this work,

we aim to minimize the largest excess of population

across different healthcare levels over the available

resources in the forming districts, while considering

population homogeneity, compactness, and contigu-

ity as constraints that must be met. Table 1 provides a

summary of pertinent information from selected stud-

ies, including this work.

3 PROBLEM DESCRIPTION AND

MATHEMATICAL

FORMULATION

In this section, we provide a formal deﬁnition of

the problem and mathematical formulation that con-

sider several attributes discussed in the literature, such

as population balance, compactness, contiguity, and

capacity-demand match.

3.1 Problem Deﬁnition and Notation

We ﬁrst formally describe the healthcare districting

problem faced in this article.

The districting problems involve grouping a set of

territorial units to form a district. This can be done for

various purposes, such as administrative or political

reasons. Territorial units represent individual parts of

the territory that structure spatial organization. The

main objective is to establish the assignment of ter-

ritorial units to districts. These units have a certain

allocation of resources. The objective of the district-

ing problem in this article is to establish a territorial

partition, which minimizes the maximum excess of

demand over capacity among all districts for primary

and secondary healthcare services individually.

To achieve this objective, we formulate this prob-

lem as a mixed-integer program based on the span-

ning tree formulation. This formulation allows us to

impose the connectivity of territorial units into a dis-

trict explicitly. Note that a spanning tree of an undi-

rected graph is a subgraph in which any two vertices

are connected by exactly one path and includes all of

the vertices of the graph. Let V be the set of ver-

tices, which represent the territorial units, deﬁned as

V = {1, 2,3,.. ., |V |}. Let E be the set of edges, im-

plying that if there exists an edge between vertex i and

j, then {i, j} ∈ E.

The model requires an auxiliary graph G(V

′

where the set of nodes corresponds to the territorial

units, with an additional node {0}, given by V

′

{0} ∪ V . The set of edges represents border relation-

ships, and the 0 node is adjacent to each territorial

unit. Thus, E

′

= E ∪ {{0,i} | ∀i ∈ V }. Moreover, we

deﬁne a set A of directed edges derived from E, along

with the directed edges emerging from node 0. This is

given by A = {(i, j),( j,i) | {i, j} ∈ E} ∪ {(0,i) | i ∈

V }. The population for each territorial unit is denoted

as p

for all j ∈ V . Similarly, the resources for each

territorial unit j are denoted by r

(resp. r

) and rep-

resent the number of people served by the primary

(resp. secondary) medical supplies within that unit.

Meanwhile, the distance between a pair of territorial

units (i, j) is represented by d

i j

Additionally, we restrict the number of territorial

units in each district to fall within an interval (s

preventing too many or too few territorial units in

each district. In addition, a maximum distance L

max

imposed between any two territorial units that belong

to the same district. This condition, know as compact-

ness, seeks to avoid generating districts with a terri-

tory too large. Thus, the model aims to ﬁnd K disjoint

territory subsets that minimize the largest excess of

population over the resources capacity among the K

Balancing Resources and Demand: A Bi-Objective Mixed-Integer Programming Approach of Healthcare Districts in Chile

343

Table 1: Selected publications on the healthcare districting problem in primary and secondary healthcare services.

Attributes

Article

Population

Homogeneity

Compactness Contiguity Accessibility

Capacity-demand

Match

Variety of Medical

Procedures Offered

Case

study

1 (Ghiggi et al., 1975) x x x

Province of

Imperia Italy

2 (Pezzella et al., 1981) x x x x

Province of

Cosenza Italy

3 (Datta et al., 2013) x x x

East

England

4 (Steiner et al., 2015) x x x

Parana

State, Brazil

5 (Farughi et al., 2019) x

South Khorasan

Province, Iran

6 (Farughi et al., 2020) x x x x

South Khorasan

Province, Iran

7 (Yanık et al., 2020) x x x x x

Istanbul,

Turkey

8 (Fan and Xie, 2022) x x

9 (Darmian et al., 2022) x x x x

Iran

10 This work x x x x

Metropolitan Region

of Santiago, Chile

districts.

It is worth noting that, within the general con-

ditions of the healthcare districting problem for par-

titioning the geographical area, each district d ∈

K = {1, .. ., K} forms a subgraph G

. Together, they

should cover the entire set V . These conditions are

expressed as follows:

̸=

0 d ∈ K (1)

∩ V

′

≡

0 d, d

′

∈ K , d ̸≡ d

′

(2)

∪ V

∪ ... ∪ V

≡ V (3)

3.2 Spanning Tree-Based MIP Model

In this section, we present the Mixed Integer Pro-

gramming (MIP) formulation of the healthcare dis-

tricting problem.

Firstly, we deﬁne the variables involved in the

model, described as follows:

• Let x

i j

∈ {0,1} be a binary variable where x

i j

= 1

if edge {i, j} ∈ E

′

is part of the spanning tree of

G that determines the solution, and x

i j

= 0 other-

wise.

• Let y

i jk

∈ {0, 1} be a binary vector such that y

i jk

1 if arc (i, j) ∈ A is used to deﬁne the connectivity

between the node 0 and territory k, where k ∈ V ,

and y

i jk

= 0 otherwise.

• Let η

and η

be the continuous variables that

quantify the maximum excess of the population

primary and secondary service capacity, respec-

tively, over all districts.

The mathematical formulation is presented as fol-

lows:

min βη

+ (1 − β)η

(4)

subject to:

∑

i∈V

= K (5)

∑

(i, j)∈E

′

i j

= |V

′

| − 1 (6)

∑

(0,i)∈A

0ik

= 1 ∀k ∈ V

(7)

∑

(i, j)∈A

i jk

−

∑

( j,i)∈A

jik

= 0 ∀ j,k ∈ V : i ̸= j

(8)

∑

( j,k)∈A

jkk

= 1 ∀k ∈ V

(9)

i j

≥ y

i jk

+ y

jik

∀(i, j) ∈ E

′

,∀k ∈ V

(10)

∑

k∈V

− r

0 jk

≤ η

∀ j ∈ V

(11)

∑

k∈V

− r

0 jk

≤ η

∀ j ∈ V

(12)

0 j

−

∑

k∈V

0 jk

≤ α

P ∀ j ∈ V

(13)

−

0 j

∑

k∈V

0 jk

≤ α

P ∀ j ∈ V

(14)

0 j

≤

∑

k∈V

0 jk

≤ s

0 j

∀ j ∈ V

(15)

ICORES 2024 - 13th International Conference on Operations Research and Enterprise Systems

344

0 jk

+ y

0 ji

≤ L

max

∀ j,i,k ∈ V : i ̸= k

(16)

i j

∈ {0,1} ∀{i, j} ∈ E

′

(17)

i jk

∈ {0,1} ∀(i, j) ∈ A,∀ k ∈ V

(18)

The objective function (4) aims to minimize the

maximum gap between resources and population

among all districts. Note that the parameters β al-

low us to adjust the objective function to focus on

minimizing the excess at the primary healthcare level

(β = 1), at secondary healthcare level (β = 0) or at

both levels at the same time (β ∈]1, 0[). Constraint

(5) establishes the number of districts to be formed.

It is done by limiting the number of edges that can

be connected to node 0. Constraints (6) ensure that

the number of edges active in a solution deﬁnes a

spanning tree in the auxiliary graph. Constraints (7)-

(10) entails that a path must exist between each node

k ∈ V and the node 0. Constraints (11)-(12) quan-

tify the excess of the population over the available re-

sources in each district for the primary and secondary

healthcare services. Constraints (13)-(14) entail that

the population assigned to each district can only de-

viate an α value from the mean population. Con-

straints (15) ensure that each district has a number of

territorial units between the discrete interval (s

Constraints (16) impose that the distance between

any two territorial units belonging to the same district

must be at most L

max

. Finally, the domain of the deci-

sion variables is represented in Constraints (17)-(18).

Figure 1 illustrates the method for deriving a solu-

tion to the healthcare districting problem via the span-

ning tree approach. Observing the ﬁgure from left to

right, the initial step involves converting a given in-

stance of the problem into a preliminary graph. This

graph represents the territorial units as nodes and their

adjacency as edges; that is, an edge is drawn between

any two neighboring territorial units to signify their

connection. Subsequently, in the central ﬁgure, an

auxiliary graph is constructed by linking each node

to an additional, artiﬁcial node. The constraints de-

lineated by (7) to (10) are then applied to identify a

path from each node k ∈ V to the artiﬁcial node 0.

This step ensures that each territorial unit is allocated

to a district. Moreover, the number of active edges

connected to node 0 is restricted by the predetermined

number of districts to be established. Consequently, a

node k that does not serve as a district center must be

connected to node 0 via a path that traverses through a

sequence of neighboring nodes, which are also inter-

connected. This results in a cohesive cluster of nodes,

thereby fulﬁlling the contiguity requirement for the

districts.

In the next section, we apply the model (4)-(18) to

the Metropolitan region of Santiago of Chile and seek

to analyze how well the current geographical parti-

tion regarding the present population and available re-

sources at each territorial unit.

4 COMPUTATIONAL RESULTS

In this section, we assess the solutions provided by the

formulation (4)-(18) at solving a real instance based

on the Metropolitan region of Santiago, Chile.

4.1 Case Study

The Metropolitan region of Santiago, Chile, as per

the census conducted in 2017, has a total population

of 7,112,808 residents spanning an area of 15,403

. This equates to a density of 461.7 inhabitants

per square kilometer. Demographically, the region

is divided into 52 municipalities, which are grouped

into six provinces: Chacabuco, Cordillera, Maipo,

Melipilla, Santiago, and Talagante. However, when

it comes to health administration, the structure is

different. As stipulated by the Ministry of Health,

the region is segmented into six distinct health-

care services, which are: North Metropolitan Health

Service (North); Western Metropolitan Health Ser-

vice (West);Central metropolitan health service (Cen-

tral); Eastern Metropolitan Health Service (East);

South Metropolitan Health Service (South); South-

East Metropolitan Health Service (South-East). Var-

ious types of healthcare services are located within

these health services. Consequently, different lev-

els are established, leading to the following organi-

zational structure for the establishments:

• Primary Healthcare Centers: They are charac-

terized by being the ﬁrst point of contact with the

patient. Thus, they aim to provide high-quality

initial outpatient care to identify the ailment. The

facilities that offer these services include:

• Secondary Healthcare Centers: This pertains to

emergency hospital care and inpatient hospitaliza-

tion, which includes complex surgical procedures.

Hospitals responsible for providing this service

have high, medium, and low levels of complexity.

Based on the deﬁnition above, this research focuses

on proposing new districts that take into account

the capacities of both primary and secondary health-

care levels. The primary level, which includes Fam-

ily Healthcare Centers (abbreviated as CESFAM in

Balancing Resources and Demand: A Bi-Objective Mixed-Integer Programming Approach of Healthcare Districts in Chile

345

Figure 1: A solution of the healthcare districting problem represented as a spanning tree.

Spanish), plays a vital role in alleviating congestion

in secondary service levels, such as hospitals.

We gathered the information for this case study

from sources within the Ministry of Health and na-

tionally certiﬁed surveys, such as the Census con-

ducted in 2017 to determine the population in each

municipality. The Metropolitan Region of Santiago

currently has 22 secondary healthcare centers and

171 primary healthcare centers. The Ministry of

Health mandates that each primary (resp., secondary)

care center should be assigned at most 30,000 (resp.,

100,000) people, representing its capacity to provide

healthcare services. However, this number is only an

approximation of the actual capacity of each center

and does not consider the heterogeneity of the cen-

ters and their efﬁciency in providing services. To

address this issue, we assume that the capacity of

each center follows a discrete uniform distribution

U((1−ε)

d, (1+ε)

d), where

d represents the assigned

capacity and ε = 0.1 (resp., ε = 0.2) for primary

(resp., secondary) care centers. The lower and up-

per bounds for the number of territorial units in each

district, i.e., (s

), are set to 4 and 15, respectively.

We select each parameter in the formulation by

considering the current geographical partition and its

characteristics. This ensures that the current solution

Figure 2: Current districts of the health care services in the

Metropolitan region of Santiago.

is indeed a feasible solution to formulation (4)-(18).

This consideration is crucial as the primary aim of

this research is to ascertain whether, under the pre-

vailing conditions and regulations of the Metropolitan

region’s healthcare system, there exists an improved

geographical partition that can diminish the popula-

tion excess over the current system’s capacity. By do-

ing so, this could partially alleviate the stress on the

existing system and reduce both the waiting list and

wait times.

Figure 2 displays the current districts stipulated by

the Ministry of Health, and Table 2 shows the com-

position of population and resources of each district

as well as their excess of population over their avail-

able capacity in the primary and secondary level. The

MAPD line stands for Median Absolute Percentage

Deviation, which is calculated as (

∑

− x|/x)/n,

where x

represents the ith data and x denotes the av-

erage value of the n data set.

Table 2: Current composition of healthcare services in the

Metropolitan region of Santiago.

District Population

Secondary

Capacity

Primary

Capacity

Secondary

Excess

Primary

Excess

Central 1,153,995 206,227 550,238 947,768 603,757

East 1,216,177 303,286 695,714 912,891 520,463

South 1,151,126 519,389 1,051,794 631,737 99,332

West 1,202,137 590,346 1,117,597 611,791 84,540

South East 1,356,538 363,570 974,520 992,968 382,018

North 961,721 224,287 766,596 737,434 195,125

MAPD 18.05% 39.03% 35.97% 23.23% 92.15%

4.2 Experimental Setup

The mixed-integer linear programming formulation

(4)-(18) was implemented in Python 3.11 and solved

using GUROBI 10.0.1 with the default settings. All

tests were run on the computing infrastructure of the

Universidad de Santiago de Chile, which consists of

a Lenovo ThinkStation machine with 12th genera-

tion Intel Core i7-12700. We set the machine to use

four 3.60GHz cores and 31GB RAM to solve each in-

stance. We impose a time limit of 43,200 seconds (12

hours) to solve the same instance with three different

ICORES 2024 - 13th International Conference on Operations Research and Enterprise Systems

346

values of the parameter β ∈ {0,0.5,1}.

4.3 Results

Tables 3-5 and Figure 4-6 display the results obtained

for each value of β ∈ {0.0,0.5, 1.0}. Recall that β

determines which healthcare services are a priority

to minimize the overall population surplus concern-

ing its medical offer. When β = 0.0, the emphasis

is solely on the primary healthcare level (β = 1.0).

When β = 1.0, the focus is on the secondary level, Fi-

nally, with β = 0.5, there is a balanced consideration

of both the total exceeding of both the primary and

secondary healthcare level.

Table 3 (β = 1) shows a reduction of 25.07% in

the largest excess of population over resources at the

primary healthcare level, decreasing from 603,757 to

452,258 people. In addition to this reduction, Ta-

ble 3 also indicates a decrease in the largest abso-

lute deviation as a percentage among districts regard-

ing the excess of population over resources at the pri-

mary healthcare level, from 92.15% to 83.13%. These

positive results are also partially observed at the sec-

ondary healthcare level, where the largest excess of

population over resources is reduced from 992,968 to

928,651 people (6.48%). However, the largest mean

absolute deviations as a percentage at the secondary

healthcare level increases from 23.23% to 32.31%,

leading to a reduction in the homogeneity of resource

distribution among districts.

In Table 5 (β = 0), we observe that the largest ex-

cess of the population over resources is reduced from

992,968 to 856,352 people (13.76%), and the largest

mean absolute deviation as a percentage among dis-

tricts is reduced from 23.23% to 13.38%. Nonethe-

less, these improvements come with a slight increase

in the largest excess of population over resources at

the primary healthcare level, rising from 603,753 to

612,519 people (1.45%), and an increase in the largest

mean absolute deviation as a percentage of 2.79%

(from 92.15% to 94.94%). This results in a deteri-

oration of the homogeneity in the distribution of re-

sources at the primary healthcare level.

Table 3: Composition of healthcare services in the

Metropolitan region of Santiago with β = 1.

District Population

Secondary

Capacity

Primary

Capacity

Secondary

Excess

Primary

Excess

Central 1,341,551 486,207 889,183 855,344 452,368

East 953,995 109,457 509,841 844,538 444,154

South 1,402,653 611,261 1,147,692 791,392 254,961

West 1,011,354 412,323 958,347 599,031 53,007

South-East 1,179,203 363,570 822,923 815,633 356,280

North 1,152,938 224,287 828,473 928,651 324,465

MAPD 19.52% 70.24% 40.68% 32.31% 83.13%

Table 4: Composition of healthcare services in the

metropolitan region of Santiago with β = 0.5.

District Population

Secondary

Capacity

Primary

Capacity

Secondary

Excess

Primary

Excess

Central 1,281,187 492,453 950,799 788,734 330,388

East 1,066,277 303,286 696,645 762,991 369,632

South 1,199,911 395,434 967,622 804,477 232,289

West 1,239,516 427,337 983,002 812,179 256,514

South-East 1,094,376 269,316 735,223 825,060 359,153

North 1,160,427 319,279 823,168 841,148 337,259

MAPD 9.17% 33.9% 18.94% 5.31% 26.07%

Table 5: Composition of healthcare services in the

metropolitan region of Santiago with β = 0.

District Population

Secondary

Capacity

Primary

Capacity

Secondary

Excess

Primary

Excess

Central 1,254,276 397,924 641,757 856,352 612,519

East 1,130,793 303,286 665,257 827,507 465,536

South 1,354,982 519,389 1,233,226 835,593 121,756

West 1,202,137 504,195 1,117,597 697,942 84,540

South-East 1,152,682 363,570 793,088 789,112 359,594

North 946,824 118,741 705,534 828,083 241,290

MAPD 19.3% 67.72% 43.50% 13.38% 94.94%

Results in Table 4 (β = 0.5) equate the importance

of the primary and secondary healthcare level in the

objective function. This results in the decrease of both

maximum absolute deviations as a percentaje. For

the case of the secondary healthcare level, this goes

from 23.23% to 5.31%, while at the primary health-

care level, it decreases from 92.15% to 26.07%. On

the other hand, considering the largest excess of popu-

lation over available resources compared to the initial

case, these decrease from 992,968 to 841,148, result-

ing in a reduction of 15.28% for secondary healthcare

level; meanwhile, the excess for primary healthcare

level goes from 603,757 to 369,632, decreasing by

38.77%.

Table 6 and Figure 3 summarize the results for

each value of β. In the case that the decision maker

is more interested in minimizing the largest excess in

Figure 3: Capacity-demand match at primary and secondary

levels according to different values of β.

Balancing Resources and Demand: A Bi-Objective Mixed-Integer Programming Approach of Healthcare Districts in Chile

347

Figure 4: Obtained districts with β = 1.

Figure 5: Obtained districts with β =

0.5.

Figure 6: Obtained districts with β = 0.

Table 6: Summary of results for each value of β.

β η

Gap

1 928,651 452,368 992,968 603,757 6,48% 25,07%

0.5 841,148 369,632 992,968 603,757 15.29% 38.78%

0 856,352 612,519 992,968 603,757 13,76% -1,45%

the primary healthcare level (β = 1), the results sug-

gest that it is possible to achieve an improvement of

up to 25%, i.e., reduce up to 25% the largest excess of

the population over the available resources. However,

this will have an impact on the secondary healthcare

level, which might increase their largest mean abso-

lute deviation as a percentaje by 10%. In the opposite

case, if the decision maker is more interested in mini-

mizing the largest excess in the secondary healthcare

level (β = 0), the results suggest that it is possible to

achieve an improvement of almost up to 14%, how-

ever, as it is expected, an increase in the largest excess

of the primary healthcare level of 1,45% is reported.

In that case, both healthcare levels are considered si-

multaneously in the objective function, i.e., β = 0.5,

the results suggest an improvement in both levels, pri-

mary and secondary healthcare levels. The former

might reduce the largest excess up to 15%, while the

latter might be reduced up to almost 39%. These re-

sults suggest that these decisions must not be taken

independently, and considering both healthcare lev-

els at the same time might provide a more efﬁcient

geographical partition of the Metropolitan region of

Santiago in terms of balance between population and

available resources. It is worth noticing that the dif-

ference between the improvements in the secondary

(resp. primary) healthcare level using β = 0.5 and

β = 0 (resp. β = 1) might be explained due to the for-

mulation (4)-(18) could not ﬁnd an optimal solution

within the imposed time limit. More speciﬁcally, the

optimality gaps reported after the 12 hours of compu-

tation were 58.5% for β = 1, 44.7% for β = 0.5 and

44.4% for β = 0.

5 CONCLUSIONS

The geographical partitioning of healthcare services

is a fundamental pillar for ensuring efﬁcient and eq-

uitable healthcare delivery. This research focuses on

the City of Santiago, Chile, and sheds light on the

complex issue of healthcare districting, a combina-

torial optimization problem that, when addressed ef-

ﬁciently, can bring transformative improvements to

healthcare service delivery.

Our study illustrates the intricate balancing act

required between primary and secondary healthcare

levels. Through mathematical modeling, we have

demonstrated that a conscientiously designed district-

ing plan can play a pivotal role in harmonizing the

healthcare journey for patients, from their initial en-

counters with primary care to more advanced care

needs. The complex interplay between population ho-

mogeneity, compactness, and capacity-demand match

emerges as the cornerstone of this optimization prob-

lem.

Relying on a variety of methodologies, rang-

ing from exact solutions to heuristic approaches, re-

searchers have made commendable progress in ad-

dressing the healthcare districting problem. In our

study, we consider a mixed-integer linear program-

ming formulation with a bi-objective function assess-

ing equity at two layers of the health service system

(namely, primary and secondary). We represent and

assess three scenarios and explore the implications

of prioritizing primary, secondary, or both healthcare

levels. Our ﬁndings underscore that considering both

levels of healthcare simultaneously can lead to a more

balanced geographic partitioning in terms of aligning

population demand with available resources.

Results indicate signiﬁcant potential for improve-

ments. Speciﬁcally, for a balanced approach, the

largest excesses of population over available re-

sources could be reduced by up to 15.29% and

38.78% for secondary and primary healthcare levels,

respectively. While optimizing for a particular level

ICORES 2024 - 13th International Conference on Operations Research and Enterprise Systems

348

offers speciﬁc beneﬁts, it can have consequential ef-

fects on the other level. Thus, decision-makers are

tasked with judiciously evaluating trade-offs.

The intricate nature of this problem, coupled with

the computational challenges encountered during op-

timization, implies that more efﬁcient algorithms or

hybrid methodologies could further improve the solu-

tion quality found in this work.

A logical extension of the current study would

be to consider a range of medical specialties, which

would offer a more comprehensive view of health-

care needs across the city. Alongside this, there is

a clear need to reﬁne and enrich the data concerning

actual demand and the healthcare system’s capacity.

Enhancing data collection and analysis can provide

more accurate insights and lead to better-informed

decisions about resource allocation and distribution.

One strong underlying assumption of this work is that

primary healthcare resources are homogeneously dis-

tributed across districts; however, given the geograph-

ical location of primary centers and district sizes, it is

unrealistic to assume that people can attend any center

within their district. This issue should be addressed

by considering a more realistic model, such as a two-

level districting problem or an integrated approach

that might consider decisions on the location and ca-

pacity of new primary centers. Furthermore, an area

that warrants particular focus is determining the opti-

mal number of districts required to meet healthcare

demands. Pursuing this additional research would

provide a critical perspective for long-term planning

and expansion decisions.

Regarding the modeling approach, there is an ev-

ident requirement to embrace a more dynamic ap-

proach that factors in demand shifts over time and ac-

knowledges the inherent variability in healthcare ca-

pacity. Such dynamic models would render a more

realistic representation of healthcare system needs

and behaviors. To further strengthen this analysis, it

would be valuable to delve into advanced optimiza-

tion techniques such as robust optimization and inte-

grated approaches. These methodologies could gener-

ate more resilient solutions in the face of uncertainties

and help identify optimal locations for future medical

centers.

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