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UNIVERSIDADE FEDERAL DO RIO DE JANEIRO
INSTITUTO COPPEAD DE ADMINISTRAÇÃO
OLAVO ALVES DIOGO
SOLVING DISTRICTING, SCHEDULING AND ROUTING PROBLEMS IN PRIMARY HEALTH CARE: a study applied to Family Clinics in Brazil
Rio de Janeiro
2019
Olavo Alves Diogo
SOLVING DISTRICTING, SCHEDULING AND ROUTING PROBLEMS IN PRIMARY HEALTH CARE:
a study applied to Family Clinics in Brazil
A thesis presented to the Instituto COPPEAD de
Administração, Universidade Federal do Rio de
Janeiro, as part of the mandatory requirements for
the degree of Doctor of Sciences in Business
Administration (D.Sc.)
Advisor: Eduardo Raupp de Vargas
Rio de Janeiro
2019
Olavo Alves Diogo
SOLVING DISTRICTING, SCHEDULING AND ROUTING PROBLEMS IN
PRIMARY HEALTH CARE: a study applied to Family Clinics in Brazil
A thesis presented to the Instituto COPPEAD de Administração, Universidade Federal
do Rio de Janeiro, as part of the mandatory requirements for the degree of Doctor of
Sciences in Business Administration (D.Sc.).
Approved by:
_____________________________________ (President)
Prof. Eduardo Raupp de Vargas, D.Sc. - Advisor
(COPPEAD/UFRJ)
_____________________________________
Prof. Peter Fernandes Wanke, D.Sc.
(COPPEAD/UFRJ)
_____________________________________
Prof. Claudia Affonso Silva Araújo, D.Sc.
(COPPEAD/UFRJ)
_____________________________________
Prof. Virgílio José Martins Ferreira Filho, D.Sc.
(COPPE/UFRJ)
_____________________________________
Prof. Henrique Ewbank de Miranda Vieira, D.Sc.
(FACENS)
Rio de Janeiro
2019
To my three beloved women,
my wife Alice and my daughters Olivia and Aline,
the powerpuff girls.
ACKNOWLEDGEMENTS
Thank God for enabling me to get here.
To my wife Alice and my daughters, Olivia and Aline, for whom I decided to enrol in
this doctorate program despite my age. Thank you very much for your love, for the
daily encouragement and for giving me all the support and conditions necessary to
develop this thesis.
To my parents Paulo Elbio Vidal Diogo and Suely Alves Diogo, in memoriam, who
always encouraged me to study and taught me that knowledge is the only good that
cannot be taken away. Wherever you are, you must be proud right now.
To my advisor Eduardo Raupp de Vargas, who always encouraged me and never let
me down. Thank you for your trust, guidance, advisory, mentoring and friendship.
To professor Peter Wanke, who inspired me, presented me great ideas, and sailed
with me through the universe of journals.
To professors Kleber Figueiredo and Claudia Araújo for all the personal
encouragement and support of the CESS (Centre for Studies in Health Services
Management).
To the friends Thiago Saquetto, Renan Henrique de Oliveira, Edson Muylaert and
Claudio Nunes, the untouchables, for the companionship and support in this long
journey.
To Ticiane Lombardi, who from the inscription in the doctorate program supported me
and gave me the incentive to overcome each obstacle along the way.
To the CNPq (National Council for Scientific and Technological Development) for the
financial support, contributing to make this research possible.
ABSTRACT
DIOGO, Olavo Alves. Solving Districting, Scheduling and Routing Problems in
Primary Health Care: a study applied to Family Clinics in Brazil, 2019. 171f. Tese
(Doutorado em Administração) - Instituto COPPEAD de Administração, Universidade
Federal do Rio de Janeiro, Rio de Janeiro, 2019.
Primary health care in Brazil is provided by basic health units that proactively serve an
adjacent assigned territory. These health units are spread throughout the country, and
in the city of Rio de Janeiro are called Family Clinics. Care workers from these clinics,
known as Community Health Agents (CHAs), are responsible for daily visitation to
households in the territory to initially register the patients of the families in each
household, and then monitor the health condition of these relatives, as well as promote
health and prevent disease. Depending on the level of risk and vulnerability of these
families, visits should be more or less frequent, thus setting certain priorities. Currently,
the procedures for determining the service areas for each CHA, scheduling visits, and
routing of these care workers are done manually. In this manner, coverage is not
achieved with monthly visitation of all families, respecting their priorities, in a
satisfactory way. When this happens is characterised a Territory Alignment Problem.
This thesis intends to present computerised models for solving that problem, including
the processes of districting, scheduling and routing CHAs from a Family Clinic over a
service territory. At a first moment, through a longitudinal bibliometric study, this work
investigates mathematical models to solve the Territory Alignment Problem. The
methodology used encompasses three areas for analysis: social network of authors,
longitudinal co-word analysis, and mapping change analysis. To highlight the
significant changes over time of keywords networks, an alluvial diagram is used to
show the significance clusterings through the subperiods studied. At this point, the
work reports on the most relevant authors on the subject and the most widely used
mathematical models applied to solve the problem. In a second step, the work
investigates methods to solve home health care districting problems. The techniques
most utilised to solve the districting issue are identified. One of these techniques is
applied to analyse its suitability for the (real) case in question. At last, the work presents
a computerised model for the problem of scheduling and routing CHAs. A solution
based on Period Vehicle Routing Problem with Service Priority (PVRP-SP) is
suggested. An algorithm was developed in R code to implement the solution method,
and a classical heuristic for Capacitated VRP (CVRP) was used as routing subroutine.
As contributions, this work shows for the first time the concatenated use of the three
bibliometric analysis techniques and innovates with feasible modifications in the
mapping change analysis applied to small networks. For the districting process, the
proposed algorithm in R code requires few steps for the initial solution, and the local
search mechanism (tabuSearch) contributes to a short computing time. For the
scheduling and routing problems, results found in tests with known benchmark
instances, as well as in a real-life case, demonstrated the practical applicability of the
computer model, with values close to optimal and computing times of a few seconds.
Keywords: districting; scheduling; routing; primary health care; Tabu search
RESUMO
DIOGO, Olavo Alves. Solving Districting, Scheduling and Routing Problems in
Primary Health Care: a study applied to Family Clinics in Brazil, 2019. 171f. Tese
(Doutorado em Administração) - Instituto COPPEAD de Administração, Universidade
Federal do Rio de Janeiro, Rio de Janeiro, 2019.
Os cuidados primários de saúde no Brasil são providos por unidades básicas de saúde
que atendem proativamente a um território adscrito. Essas unidades no Rio de Janeiro
são chamadas de Clínicas da Família. Seus agentes comunitários de saúde (ACSs)
são responsáveis pela visitação aos domicílios no território para inicialmente cadastrar
os membros de cada família e, posteriormente, monitorar seus estados de saúde, bem
como promover a saúde e prevenir doenças. Dependendo do nível de risco e
vulnerabilidade dessas famílias, as visitas devem ser mais ou menos frequentes,
estabelecendo assim prioridades. Atualmente, os procedimentos para determinar as
áreas de serviço para cada ACS, o agendamento de visitas e o roteamento desses
profissionais são feitos de forma manual. Assim sendo, a cobertura não é alcançada
de forma satisfatória, com a visitação mensal de todas as famílias, respeitando suas
prioridades. Quando isso acontece, é caracterizado um Problema de Alinhamento de
Território. Esta tese pretende apresentar modelos automatizados para a solução
desse problema, incluindo os processos de distritamento, agendamento e roteirização
de ACSs de uma Clínica da Família em seu território. Inicialmente, por meio de um
estudo bibliométrico longitudinal, este trabalho investiga modelos matemáticos para
resolver o Problema de Alinhamento de Território. A metodologia utilizada engloba
três áreas de análise: rede social de autores, análise longitudinal de palavras-chave e
mudança de mapeamento. Para destacar as mudanças significativas ao longo do
tempo das redes de palavras-chave, um diagrama aluvial é usado para mostrar os
agrupamentos de significância através dos subperíodos estudados. Neste ponto, o
trabalho relata os autores mais relevantes sobre o assunto e os modelos matemáticos
mais utilizados para resolver o problema. Em uma segunda etapa, o trabalho investiga
métodos para resolver problemas de distritamento. As técnicas mais utilizadas para
resolver a questão são identificadas. Uma dessas técnicas é aplicada para analisar
sua adequação a um caso real. Por fim, o trabalho apresenta um modelo automatizado
para o problema de agendamento e roteirização de ACSs. Sugere-se uma solução
baseada no Problema de Roteirização de Veículos com Prioridade de Serviço (PVRP-
SP). Um algoritmo foi desenvolvido em R para implementar o método de solução, e
uma heurística clássica para VRP Capacitado (CVRP) foi usada como sub-rotina de
roteirização. Como contribuições, este trabalho mostra pela primeira vez o uso
concatenado das três técnicas de análise bibliométrica e inova com modificações
factíveis na análise de mudança de mapeamento aplicada a pequenas redes. Para o
processo de distritamento, o algoritmo proposto em R requer poucas etapas para a
solução inicial, e o mecanismo de busca local (tabuSearch) contribui para um tempo
de computação curto. Para os problemas de agendamento e roteirização, resultados
encontrados em testes com instâncias conhecidas, bem como em um caso real,
demonstraram a aplicabilidade prática do modelo computacional, com valores
próximos do ótimo e tempos de computação de poucos segundos.
Palavras-chave: distritamento; agendamento; roteirização; cuidados primários de
saúde; busca Tabu
LIST OF FIGURES
Figure 1.1. Cycles of development before creation of the SUS ................................ 20
Figure 2.1. The steps of analysis methodology ........................................................ 46
Figure 2.2. Social network of authors, 1963-2016 ................................................... 52
Figure 2.3. Longitudinal view - overlapping and evolution maps. ............................ 55
Figure 2.4. Cluster elements, 2000-2003. ............................................................... 58
Figure 2.5. Cluster elements, 2004-2007. ............................................................... 58
Figure 2.6a. Cluster elements, 2008-2010. ............................................................... 61
Figure 2.6b. Cluster elements, 2008-2010. ............................................................... 61
Figure 2.7a. Cluster elements, 2011-2016. ............................................................... 62
Figure 2.7b. Cluster elements, 2011-2016. ............................................................... 63
Figure 2.7c. Cluster elements, 2011-2016. ............................................................... 63
Figure 2.8. Alluvial diagram - evolution of significance clusterings. ......................... 67
Figure 3.1. Research model. ................................................................................... 87
Figure 3.2. Social network of authors, 1963-2017 ................................................... 90
Figure 3.3. Results from literature review and SNA. ................................................ 97
Figure 3.4. Territory partitioned into 10 basic units. ............................................... 101
Figure 3.5. Tabu search results of the algorithm proposed with 40 iterations. ...... 106
Figure 3.6. Real instance: CLSC territory partitioned into 36 basic units ............... 107
Figure 3.7. Real instance: Assis Valente Family Clinic territory partitioned into 36
basic units ................................................................................................ 108
Figure 4.1. Hierarchy of human resource planning process. ................................. 122
Figure 4.2. Quantity of home health care articles. ................................................. 127
Figure 4.3. Routes for the first week. ..................................................................... 145
LIST OF TABLES
Table 1.1. Indices, parameters and variables .......................................................... 23
Table 1.2. Indices, parameters and variables .......................................................... 31
Table 2.1. Productivity of authors, 1963-2016 ......................................................... 53
Table 2.2. Selected parameters for longitudinal analysis........…………….…………54
Table 2.3a. Quantitative and impact measures for the themes of period 2000-2016 64
Table 2.3b. Quantitative and impact measures for the themes of period 2000-2016 65
Table 2.4. Themes and mathematical models ......................................................... 72
Table 3.1. Productivity of authors, 1963-2017 ......................................................... 91
Table 3.2. Home care articles, 1963-2017............................................................... 93
Table 3.3. Home care districting articles, 1963-2017 .............................................. 95
Table 3.4. Workload vi in each basic unit i ............................................................ 102
Table 3.5. Travel time dij between basic units i and j ............................................ 102
Table 3.6a. Comparison of algorithms for initial solution ......................................... 104
Table 3.6b. Comparison of algorithms for initial solution ......................................... 105
Table 4.1. Risk indicators and risk score ............................................................... 121
Table 4.2. Classification of family risk ................................................................... 121
Table 4.3. Classification scheme based on constraints ......................................... 135
Table 4.4. Indices, parameters and variables ........................................................ 137
Table 4.5. Comparison on benchmark instances .................................................. 142
Table 4.6. Visit frequency and schedule ................................................................ 143
Table 4.7. Planned routes and schedule ............................................................... 144
LIST OF ABBREVIATIONS B&P Branch-and-Price Method
BIHCRSP Bi-objective Home Care Routing and Scheduling Problem
CHA Community Health Agent
ConVRP Consistent Vehicle Routing Problem
CRH Caregivers Routing Heuristic
CVRP Capacitated Vehicle Routing Problem
DSS Decision Support System
ESTPMA Earliest Start Time Priority with Minimum Distance Assignment
FCT Family Clinic Team
FHT Family Health Team
FLT Facility Location Problem
GA Genetic Algorithm
GIS Geographic Information System
GRASP Greedy Randomised Adaptive Search Procedure
HC Home Care
HHC Home Health Care
HHCRSP Home Health Care Routing and Scheduling Problem
HSA Harmony Search Algorithm
IDEF0 Integrated Definition for Function Modelling
ILP Integer Linear Programming
ILS Iterated Local Search
LRP Location Routing Problem
MILP Mixed-Integer Linear Programming
MIP Mixed-Integer Programming
MOWSD Meals-On-Wheels Service Districting
MTSPTW Multiple Travelling Salesman Problem with Time Windows
M-VRP Multi-period Vehicle Routing Problem
NRP Nurse Rostering Problem
PSO Particle Swarm Optimisation
PVRP Period Vehicle Routing Problem
PVRP-SC Period Vehicle Routing Problem with Service Choice
PVRP-SP Period Vehicle Routing Problem with Service Priority
RVND Random Variable Neighbourhood Descent
SHHCRSP Stochastic Home Health Care Routing and Scheduling Problem
SNA Social Network Analysis
TS Tabu Search
TSP Travelling Salesman Problem
VOS Visualisation of Similarities
VRP Vehicle Routing Problem
VRPPD Vehicle Routing Problem with Pickup and Delivery
VRPTW Vehicle Routing Problem with Time Windows
VRPTWSyn Vehicle Routing Problem with Time Windows and Synchronisation
WSRP Workforce Scheduling and Routing Problem
SUMMARY
1 INTRODUCTION ................................................................................................ 17
1.1 PRIMARY HEALTH CARE IN BRAZIL ........................................................... 19
1.2 METHODOLOGY ............................................................................................. 22
1.2.1 Longitudinal Bibliometric Analysis ............................................................. 23
1.2.2 Districting Process ..................................................................................... 23
1.2.2.1 Initial Solution ......................................................................................... 25
1.2.2.2 Optimising the Initial Solution ................................................................. 26
1.2.3 Scheduling and Routing Processes ........................................................... 30
1.3 FINDINGS ....................................................................................................... 35
1.4 CONTRIBUTIONS ............................................................................................ 35
1.5 ORIGINALITY AND VALUE .............................................................................. 37
1.6 REFERENCES ................................................................................................ 38
2 1ST PAPER: THE TERRITORY ALIGNMENT PROBLEM: A LONGITUDINAL
BIBLIOMETRIC ANALYSIS APPLIED TO HOME CARE SERVICES ....................... 41
2.1 INTRODUCTION .............................................................................................. 42
2.2 MATERIALS AND METHODS ........................................................................... 44
2.2.1 Social Network Analysis of Authors ........................................................... 46
2.2.2 Longitudinal Keyword Analysis .................................................................. 47
2.2.3 Mapping Change Analysis ......................................................................... 47
2.2.3.1 Cluster the Keyword Networks ............................................................... 48
2.2.3.2 Generate and Cluster the Bootstrap Replicate Networks ....................... 48
2.2.3.3 Identify Significant Assignments ............................................................. 49
2.2.3.4 Construct Alluvial Diagram ..................................................................... 49
2.3 THE FIELD OF RESEARCH ON THE TERRITORY ALIGNMENT PROBLEM ...... 50
2.3.1 Social Network Analysis of Authors from 1963 to 2016 ............................. 51
2.3.2 Longitudinal Co-word Analysis .................................................................. 53
2.3.2.1 Subperiod 2000-2003 ............................................................................. 57
2.3.2.2 Subperiod 2004-2007 ............................................................................. 57
2.3.2.3 Subperiod 2008-2010 ............................................................................. 59
2.3.2.4 Subperiod 2011-2016 ............................................................................. 59
2.3.3 Real-world and Bootstrap-world Networks ................................................ 66
2.3.4 Alluvial Diagram......................................................................................... 66
2.4 DISCUSSION .................................................................................................. 68
2.5 CONCLUSION AND FUTURE WORK ............................................................... 70
2.6 REFERENCES ................................................................................................ 73
3 2ND PAPER: THE HOME CARE DISTRICTING PROBLEM: AN APPLICATION
TO FAMILY CLINICS ............................................................................................... 80
3.1 INTRODUCTION .............................................................................................. 81
3.2 THE FAMILY CLINICS AND THE TERRITORY ALIGNMENT PROBLEM ............ 83
3.2.1 The Territory Alignment Problem of Family Clinics .................................... 83
3.3 LITERATURE REVIEW AND SOCIAL NETWORK ANALYSIS ............................ 86
3.3.1 Social Network Analysis of Authors - 1963 to 2017 ................................... 88
3.3.2 Results from the Literature Review and Social Network Analysis ............. 92
3.4 A POSSIBLE SOLUTION ................................................................................. 97
3.4.1 Algorithms Implemented ............................................................................ 99
3.4.2 Comparison of Results ............................................................................ 100
3.5 CONCLUSIONS ............................................................................................. 109
3.6 REFERENCES .............................................................................................. 110
4 3RD PAPER: SCHEDULING AND ROUTING PROBLEM WITH SERVICE
PRIORITY IN PRIMARY HEALTH CARE: A SOLUTION FOR FAMILY CLINICS IN
BRAZIL ................................................................................................................... 115
4.1 INTRODUCTION ............................................................................................ 116
4.2 BACKGROUND ............................................................................................. 117
4.2.1 The Family Risk Scale of Coelho-Savassi ............................................... 120
4.2.2 Human Resource Planning Process ........................................................ 121
4.2.3 The Scheduling and Routing Problem Applied to Family Clinics ............. 123
4.3 RELATED LITERATURE ................................................................................ 126
4.3.1 Home Health Care Articles ...................................................................... 126
4.3.2 Home Health Care Scheduling and Routing Problem Articles ................. 128
4.4 MATHEMATICAL MODEL APPLIED TO FAMILY CLINICS ............................... 135
4.4.1 Clarke and Wright Algorithm .................................................................... 139
4.4.2 CW_VRP Algorithm ................................................................................. 141
4.5 COMPUTATIONAL RESULTS ........................................................................ 141
4.5.1 Comparison on Benchmark Instances ..................................................... 141
4.5.2 Application to the Real Case of a Family Clinic ....................................... 142
4.6 DISCUSSION ................................................................................................ 146
4.7 CONCLUSION AND FUTURE WORK ............................................................. 147
4.8 REFERENCES .............................................................................................. 148
5 CONCLUSION ................................................................................................. 154
5.1 FINAL CONSIDERATIONS .......................................................................... 155
5.2 RESULTING WORKS .................................................................................... 156
5.2.1 Full Paper Presented In Conference ....................................................... 157
5.2.2 Abstracts Approved for Presentation In Conferences .............................. 157
5.3 REFERENCES .............................................................................................. 157
APPENDIX A – LONGITUDINAL BIBLIOMETRIC ANALYSIS ...................................... 159
17
1 INTRODUCTION
This work was motivated by the belief in the Brazilian health care model, represented
by the integrated networks of SUS (Unified Health Care System) at its various levels.
In particular, we believe in the importance of primary care for the success of the system
as a whole.
Analysing the performance of family clinics in Rio de Janeiro in relation to the quality
of services provided to the population, and confronting what is recommended in the
PNAB (National Policy on Primary Care) for the management of these basic health
units (BRASIL, 2012), we envisage the possibility of contributing to human resource
management processes in a specific field, the process of territorialisation and mapping
of the service area of the family health team (FHT) – the Territory Alignment Problem.
The location of the Family Clinic and the designated territory is chosen by the municipal
government, which also divides the territory into districts, a process based only on the
number of residents in the region. Each district is then subdivided into service basic
units (micro areas) and assigned to a community health agent (CHA). As it is not known
which homes will require more or less visits, since the level of risk and vulnerability of
each family is not known beforehand, this process generates unbalanced districts,
where an agent will have more visits than others.
The PNAB determines that the work of the CHAs should be organised by the head
nurse of each family health team. However, it does not specify this process in a detailed
way regarding the management of home visits, but must take into account the risk and
vulnerability of families. The only guidelines are about the number of hours worked for
each CHA and that each household should be visited monthly. In this way, each clinic
uses a different procedure for organising the work of the CHAs, stipulating specific
goals for home visits. In general, these procedures do not use automated tools, the
work scheduling of CHAs is performed manually by spreadsheets and there is no
design of the routing for home visits.
More specifically, the present work intends to assist in the process of distributing CHAs
in their micro areas in a balanced manner in terms of workload, and thus to ensure that
all residents of the territory attached to the clinic are visited on a monthly basis by these
18
agents, according to the needs determined by the risk and vulnerability level of these
families. In other words, we are talking about solving in an automated way the territory
alignment problem, which, in the case of Family Clinics, involves the processes of
districting, scheduling of visits and routing of community health agents.
In an exploratory first phase, we investigate the most used mathematical methods to
solve the various issues of the territory alignment problem, as well as the techniques
for their solutions. By means of three concatenated techniques of bibliometric analysis
(social network of authors, longitudinal analysis, and mapping change) it was possible
to identify the most relevant authors, their research groups, and the most used
modelling approaches with their respective solution methods. This work resulted in the
first article produced (Chapter 2).
The exploratory phase identified the most referenced work on districting in home care.
Confronted with three other models, the work of Blais, Lapierre and Laporte (2003)
was more adequate to the case of Family Clinics. This model was then developed in
an algorithm in R code. The results of this development were presented in the second
article produced (Chapter 3).
Considering a hierarchy in human resource management processes, the districting
process would be at a strategic level. Once the micro areas are established for each
CHA (tactical level), the operational level processes of scheduling visits and routing of
health workers are facilitated. These latter processes are discussed in Chapter 4 (the
third article submitted), where a scheduling and routing model herein called PVRP-SP
(Period Vehicle Routing Problem with Service Priority) is proposed and implemented
in an algorithm developed in R code.
This Chapter 1 initially presents a background on the primary health care development
in Brazil in Section 1.1. The objective of this section is to contextualise the territory
alignment problem in the development of primary health care.
The methodology studied and applied throughout the research is presented in Section
1.2. The idea here is to show in a concise way all the methodology presented in each
of the three articles developed (Chapter 2 to 4), but with some details that were not
covered there. Reading this chapter will make the future reading of each article easier
19
and faster. This section is divided into three parts: longitudinal bibliometric analysis
(Subsection 1.2.1); districting process (Subsection 1.2.2); and scheduling and routing
processes (Subsection 1.2.3). The content of Subsection 1.2.1 is considered a by-
product of the work developed. The intention of describing in detail this methodology
was to leave a legacy for the new researchers, who can make use of these techniques
in future works. The reader can skip this subsection in a first reading and move on to
Sections 1.2.2 and 1.2.3, where the mathematical models and their solution methods
are detailed, respectively, for the districting, and the scheduling and routing processes.
Section 1.3 discusses the findings of the research work and its applications. Section
1.4 points out the contributions this work can bring to both academia and the health
sector in Brazil. And Section 1.5 deals with the relevance and originality of this work.
At last, Chapter 5 presents the conclusions, final considerations and also resulting
works approved for conference presentations.
1.1 PRIMARY HEALTH CARE IN BRAZIL
The historic Alma-Ata Declaration of 1978 established the principles that guided
primary health care in the universal sphere:
Primary health care is essential health care based on practical, scientifically sound and socially acceptable methods and technologies made universally accessible to individuals and families in the community through their full participation and at a cost that the community and country can afford to maintain at every stage of their development in the spirit of self-reliance and self-determination. It forms an integral part both of the country's health system, of which it is the central function and main focus, and of the overall social and economic development of the community. It is the first level of contact of individuals, the family and community with the national health system bringing health care as close as possible to where people live and work, and constitutes the first element of a continuing health care process. (WHO, 1978)
In Brazil, the experiences with primary health care presented several cycles of
development (PAIM, 2003; MENDES, 2012). As can be seen in Figure 1.1, there were
five cycles before the creation of the SUS (Unified Health System) in 1988, then
regulated in 1990: (i) creation of Health Centres by the University of São Paulo (USP)
in the 1920s, which used health education, health promotion and prevention of
20
diseases, in a segmented form of medical care; (ii) creation of the Special Public Health
Service (SESP) and implementation of primary health care units in the 1940s, with
preventive and curative actions restricted to infectious diseases; (iii) State Health
Secretariats in the mid-1960s with actions to prevent infectious diseases, with a special
focus on the maternal and child group; (iv) academic institutions and health
secretariats in the late 1970s, inspired by the Alma-Ata International Conference, carry
out pilots of community medicine; and (v) the creation of Integrated Health Actions
(AIS) in the 1980s, through agreements between states and municipalities funded by
INAMPS (National Institute of Medical Assistance and Social Security) and replaced in
1987 by SUDS (Unified and Decentralised Health System). A sixth cycle emerges with
the creation of SUS and a reorientation of the care model, which becomes universal,
equitable and integral (PAIM, 2003).
Figure 1.1. Cycles of development before creation of the SUS
1920 1940
USP Health Centres
SESPSpecial Public Health Service
1960
State Health Secretariats
1970
Academic institutions and health secretariats
AIS Integrated
Health Actions
1980 1990
Cycles
Years
1
2
3
4
5
21
Created in the early 1990s, the PACS (Community Agents Program) extended the
health actions aimed at the mother and child group to rural populations and the urban
periphery. In 1993, it expanded its objectives with educational actions in the
communities. Starting in 1994, the first teams of the Family Health Program (PSF) were
formed, incorporating and expanding the action of community health agents (CHAs).
The seventh cycle of development of primary care in Brazil began (PAIM, 2003).
In 1997, the PSF is presented as the new assistance model in the SUS for primary
health care through the Basic Operational Norm (BRASIL, 1997).
In 2006, the Pact for Health was launched, involving the three spheres of government
and encompassing three dimensions: the Pact for Life, the Pact in Defence of the SUS
and the Management Pact. The Pact for Health focused on the need to respect local
specificities in the organisation and development of Family Health as a priority strategy
for reorganisation and strengthening of Primary Care (GIL; MAEDA, 2013). Also in
2006, the National Policy on Primary Care (PNAB) was regulated, revising the
guidelines and norms of the organisation of Primary Care, for the PFS and for the
PACS.
In 2012, the PNAB was reviewed and published (BRASIL, 2012), where the PSF
ceases to be a program and is now called the Family Health Strategy (ESF).
The state of Rio de Janeiro, and more specifically its capital, took a long time to
implement the PNAB, which was published in 2006. The first Family Clinic in the city
(Padre Velloso Social Inclusion Pole) was inaugurated on February 3, 2009 in the
Botafogo neighbourhood and in that year only two other basic health units were
inaugurated. The year 2010 was much more profitable and 21 Family Clinics were
launched. The strong pace continued in 2011 with 29 new units. In 2012, more than
20 Family Clinics were opened. Currently, there are 109 Family Clinics in operation in
Rio de Janeiro, within 67 of the city’s neighbourhoods. This represents 67.25% of the
total service area to be covered, according to data from the Rio de Janeiro city
government (RIO DE JANEIRO, 2017).
The routine of a Family Clinic is mainly dependent on the daily visits made by the CHA
to the homes of the residents of the designated territory. In practice, the current
22
planning for coverage of service territories of Family Clinics does not consider certain
criteria that could favour the good service delivery, since the Family Health Teams
(FHT) are distributed only taking into account the number of households to be served
(BRASIL, 2012). Criteria such as workload of community agents (number of home
visits) and contiguity and compactness of service areas (to reduce travel time and
avoid crossing routes) are not met in the current planning of the teams.
There is therefore a problem of territory alignment, as the territory needs to be divided
into subareas (in a process called districting), each covered by an FHT so that each
Community Health Agent (CHA) belonging to an FHT has comparable workloads.
Furthermore, since there is currently no systematic (computer-based or otherwise)
application to plan and try to optimise visits in a logical way, the solutions obtained are
hardly optimal. Field surveys conducted at Family Clinic units in 2012 (ARAÚJO, 2012;
DIOGO; ARAÚJO, 2013) and 2015 (SILVEIRA, 2015) showed that the CHA visited
monthly, respectively, only 56.9 and 52.0% of the households in the assigned territory.
Aiming at improving the aforementioned situation, we set out to develop mathematical
models to enhance the process of household visits of CHAs. Such models include the
processes of districting the territory as well as the scheduling of visits, and routing of
the CHAs or health teams when applicable.
1.2 METHODOLOGY
In this section, the methodology used throughout this study is presented, in important
details that were not shown in the papers submitted to journals (Chapters 2 to 4) in the
effort to make them concise and lean within the imposed limits of pages. We take care,
whenever possible, not to make the information redundant in relation to the papers,
which are complementary to the one presented here.
1.2.1 LONGITUDINAL BIBLIOMETRIC ANALYSIS
The methodology used for building the Social Network of Authors is depicted in Section
A.1 of Appendix A.
23
In Section A.2 of Appendix A, an approach to carry out the analysis of the evolution of
a specific research field is shown, being in the present case the mathematical models
to solve the territory alignment problem.
Section A.3 of Appendix A describes the methodology for mapping change analysis.
1.2.2 DISTRICTING PROCESS
In this subsection a methodology based on Blais, Lapierre, and Laporte (2003) is
presented as the solution for the districting problem. The indices, parameters, and
decision variables for the modelling approach are presented in Table 1.1 below.
Table 1.1. Indices, parameters and variables
Indices Description
𝑖, 𝑗
𝑚
𝑘
Index of basic units, 𝑖, 𝑗 ∈ {1,2, … , 𝑛}
Number of districts per territory
Index of districts, 𝑘 ∈ {1,2, , … 𝑚}
𝑛𝑘 Number of basic units of district 𝐷𝑘
Parameters Description
𝛼, 𝛽 Control parameter in the interval [0, 1]
𝑣𝑖 Number of visits made to basic unit 𝑖
𝑑𝑖𝑗
𝜆
�̅�
Distance (or travel time) between the centers of the basic units 𝑖 and 𝑗
Relation between the total travel time 𝑇𝑘 and the total workload 𝑊𝑘
The average workload
Variables Description
𝑊𝑘
𝑇𝑘
Total workload of district 𝐷𝑘
Total travel time in district 𝐷𝑘
𝑉𝑘 Total visit time in district 𝐷𝑘
The model considers five constraints (indivisibility of basic units, respect for
neighbourhood boundaries, connectivity, mobility and workload balance), with the first
three being normal restrictions and the other two in a weighted bi-objective function:
24
𝑓(𝑠) = ∝ 𝑓1(𝑠) + (1−∝)𝑓2(𝑠) (1.1)
where 𝑓1(𝑠) and 𝑓2(𝑠) evaluate the degree of mobility and the workload balance of
solution 𝑠, respectively, and α is a control variable in the interval [0, 1]. The degree of
mobility of the solution 𝑠 is evaluated as
𝑓1(𝑠) = ∑ (∑ 𝑣𝑖𝑣𝑗𝑑𝑖𝑗𝑖,𝑗∈𝐷𝑘,𝑖<𝑗 )𝑚𝑘=1 / [(𝑛𝑘(𝑛𝑘 − 1)/2)(∑ 𝑣𝑖𝑖∈𝐷𝑘 )
2] (1.2)
where 𝑚 is the number of districts, 𝑖 and 𝑗 are the basic units of district 𝐷𝑘, 𝑑𝑖𝑗 is the
distance (or travel time) between the centres of the basic units 𝑖 and 𝑗 either using
public transportation or walking, 𝑛𝑘 is the number of basic units of 𝐷𝑘, and 𝑣𝑖 is the
number of visits made to unit 𝑖.
In Equation 1.2, the numerator calculates for each 𝑘 the total distance travelled within
the district 𝐷𝑘 each analysed period of time. The denominator is a scale factor, where
(𝑛𝑘(𝑛𝑘 − 1)/2) represents the quantity of pairs (𝑖, 𝑗) in the numerator. The lower the
value of 𝑓1(𝑠) the greater the degree of mobility within the district or territory.
The function of a balanced workload is given by
𝑓2(𝑠) = (∑ 𝑚𝑎𝑥{𝑊𝑘𝑚𝑘=1 − (1 + 𝛽)�̅�, (1 − 𝛽)�̅� − 𝑊𝑘 , 0}) / �̅� (1.3)
where 𝑊𝑘 is the workload in district 𝑘.
In order to calculate 𝑓2(𝑠), a piecewise linear function is used, which considers that the
workload has higher penalty if it ends up being outside the interval [(1 − 𝛽) �̅�, (1 +
𝛽) �̅�], where �̅� is the average workload, and 0 ≤ 𝛽 ≤ 1.
The workload 𝑊𝑘 is the sum of 𝑉𝑘 (total visit time) and 𝑇𝑘 (total travel time) in district 𝑘
in a given period (e.g. one year). Indeed, the figures 𝑊𝑘, 𝑉𝑘, and 𝑇𝑘 are dependent on
the districting solution and in turn total travel time ∑ 𝑇𝑘𝑚𝑘=1 and the total visit time ∑ 𝑉𝑘
𝑚𝑘=1
in the territory are also dependent on the solution. In other words, depending on the
solution, 𝑇𝑘 can be decreased by an optimum grouping of customers and optimum
planning of community worker visits. 𝑉𝑘 is also dependent on the solution because less
time spent on the trip permits more time in the visit.
25
The model uses a parameter 𝜆 for the relation between the total travel time 𝑇𝑘 and the
total workload 𝑊𝑘 , historically considered as 18 per cent (CLSC, 2000; BLAIS, 2001).
Blais, Lapierre, and Laporte (2003) chose as a solution the metaheuristic Tabu search
developed by Bozkaya, Erkut, and Laporte (2003) for political districting, but with a
different objective function. From an initial solution, the Tabu search of Bozkaya
iteratively goes from one solution to another in its neighbourhood by doing two kinds
of movements: it either moves one basic unit from its current district to an adjacent
district (transferring), or it swaps two basic units on the border of two different adjacent
districts (swapping).
1.2.2.1 INICIAL SOLUTION
The initial solution is an iterative one built using seed basic units, which are units
randomly chosen to belong to each of the districts. By using these seeds, the districts
are one at a time built by adding to each step a basic unit adjacent to district 𝑘 that has
a lower workload. Note that the decision variables, that is, the basic units 𝑖 that will
compose each district 𝑘, are not explicit in the objective function (Equations 1.1 to 1.3).
This way it is not possible to use any commercial solver tool, hence the need to use
the algorithm of our own.
Accurately, considering 𝑆(𝑘) as the set of basic units in district 𝑘 and 𝑆′(𝑘) the set of
basic units not designated adjacent to district 𝑘, the basic unit 𝑖∗ is included in district
𝑘∗ if and only if 𝑘∗ and 𝑖∗ satisfy
𝑚𝑖𝑛𝑘 𝑚𝑖𝑛𝑖 ∈ 𝑆´(𝑘) {𝑔 (𝑖, 𝑘)} (1.4)
and
𝑔 (𝑖, 𝑘) = ∑ 𝑣ℎ𝑣𝑗𝑑ℎ𝑗ℎ,𝑗∈ 𝑆(𝑘)∪{𝑖} (1.5)
We developed our own algorithm logic and the application by using R code, from the
formulas presented in (1.1) to (1.5). The algorithm pseudocode is depicted below.
26
Initial Solution Algorithm
1: generate matrix n x n 𝑔𝑛(𝑖, 𝑗) 𝑖𝑛: 𝑣𝑖 , 𝑑𝑖𝑗 𝑜𝑢𝑡: 𝑚𝑎𝑡𝑟𝑖𝑥 𝑔𝑛 = 𝑔(𝑖, 𝑗) , 𝑓𝑜𝑟 𝑖, 𝑗 =
1, . . . , 𝑛 2: choose seeds 𝑖𝑛: 𝑖, 𝑗 𝑜𝑢𝑡: 𝑠𝑒𝑒𝑑(𝑘), 𝑓𝑜𝑟 𝑘 = 1, … , 𝑚 3: generate matrix n x m 𝑔𝑠𝑒𝑒𝑑(𝑖, 𝑘) = 𝑔𝑛 (𝑖, 𝑘𝑠), 𝑓𝑜𝑟 𝑖 = 1, … , 𝑛; 𝑘𝑠 = 𝑠𝑒𝑒𝑑(𝑘) 4: generate matrix n x m 𝑔(𝑖, 𝑘) from 𝑔𝑠𝑒𝑒𝑑(𝑖, 𝑘) zeroing 𝑔𝑠𝑒𝑒𝑑 (𝑠𝑒𝑒𝑑(𝑘), 𝑘) 5: add a basic unit 𝑠𝑘ℎ for each district 𝑘 𝑤ℎ𝑖𝑐ℎ 𝑔(𝑖, 𝑘) = min(𝑖) , 𝑓𝑜𝑟 𝑖 = 1, … , 𝑛 6: compute 𝑓(𝑠𝑘), 𝑤ℎ𝑒𝑟𝑒 𝑠𝑘 = 𝑐(𝑠𝑒𝑒𝑑(𝑘), 𝑠𝑘1, … , 𝑠𝑘ℎ, … ), 𝑓𝑜𝑟 ℎ = 1, … , (𝑛𝑘 − 1 )
𝑖𝑛: 𝑚, 𝑣𝑖, 𝑔, 𝑠𝑘, 𝛼, 𝛽, 𝜆 𝑜𝑢𝑡: 𝑓(𝑠𝑘) 7: generate a new matrix n x m 𝑔(𝑖, 𝑘) 𝑧𝑒𝑟𝑜𝑖𝑛𝑔 𝑔(𝑠𝑘, 𝑘) 8: repeat the steps (5) to (7) respecting workload balance until 𝑘 = 𝑚, then stop 9: compute 𝑓(𝑠) = ∑ 𝑓(𝑠𝑘)𝑘
1.2.2.2 OPTIMISING THE INITIAL SOLUTION
In order to optimise the initial solution, the model uses as a local search mechanism
the metaheuristic Tabu search. Our implementation uses the tabuSearch package
from R library (DOMIJAN, 2012), which is more complete and developed than the one
used by Bozkaya, Erkut, and Laporte (2003), the latter being restricted to two types of
movement (transferring and swapping).
Tabu search (TS) is based on the idea of imposing restrictions to prevent a stochastic
search from falling into infinite loops and other undesirable behaviour. Tabu search
algorithm is divided into three parts: preliminary search, intensification, and
diversification (FOUSKAKIS; DRAPER, 2002).
Preliminary search, the most important part of the algorithm, works as follows. From a
specified initial configuration, TS examines all neighbours and identifies the one with
the highest value of the objective function. Moving to this configuration might not lead
to a better solution, but TS moves there anyway; this enables the algorithm to continue
the search without becoming blocked by the absence of improving moves, and to
escape from local optima. If there are no improving moves (indicating a kind of local
optimum), TS chooses one that least degrades the objective function. In order to avoid
returning to the local optimum just visited, the reverse move now must be forbidden.
This is done by storing this move, or more precisely a characterization of this move, in
a data structure - the tabu list - often managed like a circular list (empty at the beginning
27
and with a first-in-first-out mechanism), so that the latest forbidden move replaces the
oldest one. This list contains a number of elements defining forbidden (tabu) moves,
the tabu list size. The tabu list as described may forbid certain relevant or interesting
moves, as exemplified by those that lead to a better solution than the best one found
so far. In view of this, an aspiration criterion is introduced to allow tabu moves to be
chosen anyway if they are judged to be sufficiently interesting. The aspiration criterion
is simply a comparison between the value of the tabu move and the aspiration value,
which is usually the highest value found so far.
The next stage is intensification, which begins at the best solution found so far and
clears the tabu list. The algorithm then proceeds as in the preliminary search phase. If
a better solution is found, intensification is restarted. The user can specify a maximum
number of restarts; after that number the algorithm goes to the next step. If the current
intensification phase does not find a better solution after a specified number of
iterations, the algorithm also goes to the next stage. Intensification provides a simple
way to focus the search around the current best solution.
The final stage, diversification, again starts by clearing the tabu list, and sets the s
most frequent moves of the run so far to be tabu, where s is the tabu list size. Then a
random state is chosen and the algorithm proceeds to the preliminary search phase
for a specified number of iterations. Diversification provides a simple way to explore
regions that have been little visited to date. After the end of the third stage, the best
solution (or 𝑘 best solutions) found so far may be reported, or the entire algorithm may
be repeated (always storing the 𝑘 best solutions so far) a specified number of times.
The TS algorithm pseudocode is depicted below.
Tabu search Algorithm
1: Begin; 2: Randomly choose a configuration 𝑖𝑠𝑡𝑎𝑟𝑡, set 𝑖 ∶= 𝑖𝑠𝑡𝑎𝑟𝑡 , and evaluate the objective
function 𝑓(𝑖); set the aspiration value 𝛼 ∶= 𝑙𝑜, a small number; determine 𝑙 ∶=𝐿𝑖𝑠𝑡𝑙𝑒𝑛𝑔𝑡ℎ, the length of the tabu list; set 𝑀𝑜𝑣𝑒 ∶= 0 and 𝑖𝑚𝑎𝑥 ∶= 𝑖𝑠𝑡𝑎𝑟𝑡;
3: Repeat: 4: Preliminary Search
28
5: Add 𝑖 to the tabu list at position 𝑙; set 𝑙 ∶= 𝑙 − 1. If 𝑙 = 0 then set 𝑙 = 𝐿𝑖𝑠𝑡𝑙𝑒𝑛𝑔𝑡ℎ; set 𝑀𝑜𝑣𝑒 ∶= 𝑀𝑜𝑣𝑒 + 1, 𝑖𝑛𝑏ℎ𝑑 ≔ 𝑖, and 𝑐𝑛𝑏ℎ𝑑 ≔ 𝑙𝑜𝑤, a small number;
6: For each neighbour 𝑗 of 𝑖 do: If 𝑓(𝑗) > 𝛼 do: If 𝑓(𝑗) ≥ 𝑐𝑛𝑏ℎ𝑑 then set 𝑖𝑛𝑏ℎ𝑑 ≔ 𝑗 and 𝑐𝑛𝑏ℎ𝑑 ≔ 𝑓(𝑗); If 𝑓(𝑗) ≤ 𝛼 do; If 𝑗 is in the tabu list go to the next neighbour; Else if 𝑗 is non-tabu and 𝑓(𝑗) ≥ 𝑐𝑛𝑏ℎ𝑑 then set 𝑖𝑛𝑏ℎ𝑑 ≔ 𝑗 and 𝑐𝑛𝑏ℎ𝑑 ∶=
𝑓(𝑗); 7: Set 𝛼 ∶= min (𝛼, 𝑐𝑛𝑏ℎ𝑑) and 𝑖 ∶= 𝑖𝑛𝑏ℎ𝑑; 8: If 𝑓(𝑖) ≥ 𝑓(𝑖𝑚𝑎𝑥) then 𝑖𝑚𝑎𝑥 ≔ 𝑖; 9: If 𝑀𝑜𝑣𝑒 ≠ 𝑚𝑎𝑥𝑚𝑜𝑣𝑒𝑠 go back to Preliminary Search; 10: Else go to Intensification; 11: Intensification 12: Repeat: 13: Set 𝑖 ∶= 𝑖𝑚𝑎𝑥 and clear the tabu list; 14: Repeat: 15: Do the Preliminary Search; Until a better solution than 𝑖𝑚𝑎𝑥 is found. If no improvements after
𝑛𝑖𝑛𝑡 iterations go to Diversification; 16: Until 𝑛𝑖𝑚𝑝𝑟 replications;
17: Diversification 18: Clear the tabu list and set the 𝑙 most frequent moves to be tabu; 19: Randomly choose a configuration 𝑖; 20: Evaluate 𝑓(𝑖); 21: Repeat: 22: Do the Preliminary Search; 23: Until 𝑛𝑑𝑖𝑣 repetitions have occurred; 24: Until the whole algorithm has been repeated 𝑟𝑒𝑝 times; 25: 𝑖𝑚𝑎𝑥 is the approximation to the optimal solution; 26: End.
The tabuSearch package from R library is a Tabu search algorithm for optimising
binary strings. It takes a user defined objective function, in the present case 𝑓(𝑠), and
reports the best binary configuration found throughout the search i.e. the one with the
highest objective function value. The results can be plotted and summarised using
plot.tabu and summary.tabu functions.
This way, we must take some precautions before calling the local search engine
directly. The binary vector (sbin) to be delivered to the Tabu search metaheuristic must
have a length 𝑛 𝑥 𝑚, consisting of 𝑚 blocks of 𝑛 positions each. In each block, the
order of the vector will indicate the identification of the basic units that make up each
district. For example, in the case of 36 basic units to be distributed in 6 districts, we
will have a binary vector of length 216. Each district will be represented by a block of
29
36 positions. If a basic unit belongs to a particular district, the order of the block, from
1 to 36, should contain the value 1 according to its identification.
A special function (calcfsbin) transforms the input of the function that calculates 𝑓(𝑠)
in order to always treat a binary vector (sbin) from the Tabu search metaheuristic. In
fact, the function calcfsbin delivers 1/𝑓(𝑠), since metaheuristic searches for the highest
value of the objective function and, in our case, we try to minimise this function. The
calcfsbin function pseudocode is shown below.
Calcfsbin function (sbin, n, m, ...)
1: 𝐹𝑜𝑟 𝑖 = 1 𝑡𝑜 𝑛, 𝑘 = 1 𝑡𝑜 𝑚, 𝑑𝑜: 𝑠𝑏𝑘[𝑖] = 𝑠𝑏𝑖𝑛[𝑖 + (𝑘 − 1) ∗ 𝑛]
2: 𝐼𝑓 (𝑟𝑜𝑤𝑆𝑢𝑚𝑠(𝑠𝑏𝑘) < 3) 𝑡ℎ𝑒𝑛 𝑓(𝑠) = 0; 𝑟𝑒𝑡𝑢𝑟𝑛 𝑓(𝑠) 3: 𝐹𝑜𝑟 𝑖 = 1 𝑡𝑜 𝑛, 𝑑𝑜:
𝑐ℎ𝑒𝑐𝑘𝑠𝑢𝑚[𝑖] = ∑ 𝑠𝑏𝑘[𝑖]𝑘 4: 𝐼𝑓 𝑎𝑛𝑦 𝑐ℎ𝑒𝑐𝑘𝑠𝑢𝑚[𝑖] = 0 𝑜𝑟 > 1 𝑡ℎ𝑒𝑛 𝑓(𝑠) = 0; 𝑟𝑒𝑡𝑢𝑟𝑛 𝑓(𝑠) 5: 𝐹𝑜𝑟 𝑖 = 1 𝑡𝑜 𝑛, 𝑘 = 1 𝑡𝑜 𝑚, 𝑑𝑜:
𝐼𝑓 𝑠𝑏𝑘[𝑖] > 0 𝑡ℎ𝑒𝑛 𝑠𝑖𝑛𝑡𝑘[𝑖] = 𝑖 𝑒𝑙𝑠𝑒 𝑠𝑖𝑛𝑡𝑘[i] = 0 6: 𝐹𝑜𝑟 𝑖 = 1 𝑡𝑜 𝑛, 𝑘 = 1 𝑡𝑜 𝑚, 𝑑𝑜: : 𝑠𝑘 = 𝑤ℎ𝑖𝑐ℎ 𝑠𝑖𝑛𝑡𝑘[𝑖] > 0 7: compute 𝑓(𝑠𝑘), 𝑖𝑛: 𝑚, 𝑣𝑖 , 𝑔, 𝑠𝑘, 𝛼, 𝛽, 𝜆 𝑜𝑢𝑡: 𝑓(𝑠𝑘) 8: compute 𝑓(𝑠) = ∑ 𝑓(𝑠𝑘)𝑘 9: compute 𝑓(𝑠) = 1/𝑓(𝑠); 𝑟𝑒𝑡𝑢𝑟𝑛 𝑓(𝑠)
The call of the tabuSearch function should be as follows:
tabuSearch (size=216, iters = 600, objFunc = calcfsbin, config = sbin, neigh = 216,
listSize = 9, nRestarts = 10, repeatAll = 1, verbose = TRUE)
where
size The length of the binary configuration, equal to 𝑛 𝑥 𝑚. For instance, 216
is the size for 36 basic units and 6 districts (typical for a Family Clinic).
iters The number of iterations in the preliminary search of the algorithm.
30
objFunc A user supplied method that evaluates the objective function for a given
binary string. The objective function is required to take as an argument a
vector of zeros and ones.
config A starting configuration (the Initial Solution).
neigh A number of neighbour configurations to check at each iteration. The
default is all, which is the length of the string. If neigh < size, the
neighbours are chosen at random.
listSize Tabu list size.
nRestarts The maximum number of restarts in the intensification stage of the
algorithm.
repeatAll The number of times to repeat the search.
verbose If TRUE, the name of the current stage of the algorithm is printed e.g.
preliminary stage, intensification stage, diversification stage.
1.2.3 SCHEDULING AND ROUTING PROCESSES
In this subsection a methodology is presented as the solution for the home health care
scheduling and routing problem related to the Family Clinics. The problem can be seen
as a variant of the period vehicle routing problem with service choice (PVRP-SC)
(FRANCIS; SMILOWITZ; TZUR, 2006), which we are calling here as PVRP with
service priority (PVRP-SP), where there is an upper limit for the total time of each route
and the set of nodes has cohorts according to patient’s priorities. These priorities
correspond to the Coellho-Savassi scale of vulnerability and risk (COELHO; SAVASSI,
2004). The indices, parameters, and decision variables for the modelling approach are
presented in Table 1.2 below.
Table 1.2. Indices, parameters and variables
Indices Description
𝑖, 𝑗
𝑙, 𝑚
Index of nodes (family homes), 𝑖, 𝑗 ∈ {2,3, … , 𝑁}, 1is the origin node (Family Clinic)
Index of CS risk scale, 𝑙, 𝑚 ∈ {0,1,2,3}
31
𝑠 Index of schedules, 𝑠 ∈ {1,2,3, … |𝑆|}
𝑑 Index of days in the period, 𝑑 ∈ {1,2,3, … , |𝐷|}
Parameters Description
𝑉 The set of nodes (family homes)
𝐴
S
𝐷
𝑇
The set of arcs (𝑖, 𝑗) between each pair of nodes 𝑖, 𝑗 ∈ 𝑉 ∪ {1}
The set of service schedules
The set of days
The upper limit for total time of each route
𝑡𝑖𝑗
𝑟𝑖
𝑉𝑙
𝑉𝑠
Travel time from 𝑖 to 𝑗, for 𝑖, 𝑗 ∈ 𝑉 and 𝑖 ≠ 𝑗
Duration of visit at family home 𝑖, for 𝑖 ∈ 𝑉 (assuming that 𝑟1 = 0)
The set of nodes with CS risk scale equal to 𝑙. 𝑉 = 𝑉0 ∪ 𝑉1 ∪ 𝑉2 ∪ 𝑉3 and 𝑉𝑙 ∩ 𝑉𝑚 = 0; 𝑙, 𝑚 ∈ {0,1,2,3}
The set of nodes in schedule 𝑠 ∈ 𝑆. 𝑉𝑠 is a subset of 𝑉
Variables Description
𝑥𝑖𝑗𝑑
𝑦𝑖𝑗𝑠
1 𝑖𝑓 𝑎𝑟𝑐(𝑖, 𝑗) 𝑏𝑒𝑙𝑜𝑛𝑔𝑠 𝑡𝑜 𝑡ℎ𝑒 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑜𝑛 𝑑𝑎𝑦 𝑑 ∈ 𝐷 𝑓𝑜𝑟 𝑖, 𝑗 ∈ 𝑉𝑠 𝑎𝑛𝑑 𝑖 ≠ 𝑗, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 0
Accumulated time flow on schedule 𝑠 ∈ 𝑆 after travelling through arc (𝑖, 𝑗) and before starting the service at 𝑗, for 𝑖, 𝑗 ∈ 𝑉𝑠 and 𝑖 ≠ 𝑗.
As a starting point for the formulation of the problem, we consider in general the
directed graph
𝐺 = (𝑉 ∪ {1}, 𝐴) (1.6)
where 𝑉 is the set of nodes (family homes); 1 is the origin node (Family Clinic); and
𝐴 is the set of arcs (𝑖, 𝑗) between each pair of nodes 𝑖, 𝑗 ∈ 𝑉 ∪ {1} . The parameter
𝑡𝑖𝑗 describes the travel time from 𝑖 to 𝑗, for each arc (𝑖, 𝑗); and 𝑟𝑖 is the duration of visit
at family home 𝑖, for each node 𝑖 ∈ 𝑉. All the routes start and end at node 1, and a
route is determined by a series of arcs linked together. The total time of a route is the
summation of the travel times of the arcs that compose it (𝑡𝑖𝑗) together with the times
of visitation of each node (𝑟𝑖), having as an upper limit 𝑇.
Daily routes will be constructed, attending visits to the patients' homes according to
their priorities. For each schedule 𝑠, the model considers variables and parameters
related to the arcs (𝑖, 𝑗) with 𝑖, 𝑗 ∈ 𝑉𝑠, 𝑠 ∈ 𝑆. The set of nodes 𝑉𝑠 is, in fact, a subset of
𝑉, composed of the combination of sets 𝑉𝑙 or a partition of these, for 𝑙 = {0,1,2,3}, 𝑉𝑙
32
being the set of nodes with corresponding Coelho-Savassi (CS) risk scale 𝑙, which
varies from 0 to 3 (𝑅0 to 𝑅3).
For calculating the travelling times, it is possible to consider the problem in a symmetric
sense. However, we will use the asymmetric sense in the mathematical formulation in
order to consider priorities among the family homes in an easier manner. In this case,
each arc (𝑖, 𝑗) can be substituted by the pairs (𝑖, 𝑗) and (𝑗, 𝑖).
The mathematical formulation for the problem becomes:
(PVRP-SP) 𝑚𝑖𝑛 𝑧 = ∑ ∑ 𝑦𝑖1𝑠
𝑖∈𝑉𝑠 𝑑∈𝐷 (1.7)
subject to:
∑ 𝑥𝑖𝑗𝑑
𝑖∈𝑉𝑠∪{1}
= 1, 𝑗 ∈ 𝑉𝑠, 𝑑 ∈ 𝐷 (1.8)
∑ 𝑥𝑖𝑗𝑑
𝑗∈𝑉𝑠∪{1}
= 1, 𝑖 ∈ 𝑉𝑠, 𝑑 ∈ 𝐷 (1.9)
∑ 𝑦𝑗𝑖𝑠
𝑖∈𝑉𝑠∪{1}
− ∑ 𝑦𝑖𝑗𝑠
𝑖∈𝑉𝑠∪{1}
− ∑ 𝑡𝑗𝑖
𝑖∈𝑉𝑠∪{1}
𝑥𝑗𝑖𝑑 = 𝑟𝑗 , 𝑗 ∈ 𝑉𝑠, 𝑑 ∈ 𝐷 (1.10)
(𝑡𝑖𝑗 + 𝑟𝑖)𝑥𝑖𝑗𝑑 ≤ 𝑦𝑖𝑗
𝑠 ≤ (𝑇 − 𝑟𝑗)𝑥𝑖𝑗𝑑 , 𝑖, 𝑗 ∈ 𝑉𝑠 ∪ {1}, 𝑑 ∈ 𝐷 (1.11)
𝑥𝑖𝑗𝑑 ∈ {0,1}, 𝑦𝑖𝑗
𝑠 ≥ 0, 𝑖, 𝑗 ∈ 𝑉𝑠 ∪ {1}, 𝑑 ∈ 𝐷 (1.12)
The objective function (1.7) minimises the accumulated time flow on each schedule
𝑠 ∈ 𝑆. Constraints (1.8) and (1.9) ensure that to each node (family home) arrives one
33
and only one agent (route) and that just one agent (route) leaves from each node
(family home). Constraint (1.10) ensures the added time of each route. The Equations
1.8 to 1.10 prevent the formation of sub-paths among the nodes in 𝑉𝑠. Inequality (1.11)
relates 𝑥 to 𝑦 and ensures that the total time on any route does not surpass the upper
limit 𝑇.
We developed our own algorithm logic and the application by using R code for the
solution of the PVRP-SP problem. Specifically, to solve the routing problem with the
solution formulated in the previous section, we select the Naveen Kaveti (2017)
algorithm developed in R code and called CW_VRP, which implements the Clarke and
Wright (1964) heuristic in both parallel and sequential versions to find greedy routes.
The author strongly recommends using the parallel version in case of building more
than one route. In our implementation, this algorithm in its parallel version is triggered
as a subroutine for each schedule 𝒔 determined by the PVRP-SP model. The algorithm
pseudocode is depicted below.
PVRP-SP Algorithm
1: plan schedule 𝑠 for each week 𝑖𝑛: 𝑛𝑜𝑑𝑒𝑠 𝑉, 𝑛𝑜𝑑𝑒𝑠 𝑤𝑖𝑡ℎ 𝐶𝑆 𝑟𝑖𝑠𝑘 𝑠𝑐𝑎𝑙𝑒 𝑉𝑙 , 𝑟𝑖 𝑜𝑢𝑡: 𝑉𝑠 𝐹𝑜𝑟 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑤𝑒𝑒𝑘 𝑑𝑜: 𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠0 = 𝑠𝑢𝑏𝑠𝑒𝑡(𝑛𝑜𝑑𝑒𝑠 𝑉, 𝑁𝑜𝑑𝑒 == 1)
𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠1 = 𝑠𝑢𝑏𝑠𝑒𝑡(𝑛𝑜𝑑𝑒𝑠 𝑉, 𝑁𝑜𝑑𝑒 > 1 𝑎𝑛𝑑 𝐶𝑆 ≥ 2)
34
𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠10 = 𝑗𝑜𝑖𝑛(𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠0, 𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠1) 𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠 = 𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠10 𝑑𝑒𝑚𝑎𝑛𝑑0 = 𝑠𝑢𝑏𝑠𝑒𝑡(𝑑𝑒𝑚𝑎𝑛𝑑𝑠 𝑟𝑖, 𝑁𝑜𝑑𝑒 == 1) 𝑑𝑒𝑚𝑎𝑛𝑑1 = 𝑠𝑢𝑏𝑠𝑒𝑡(𝑑𝑒𝑚𝑎𝑛𝑑𝑠 𝑟𝑖, 𝑁𝑜𝑑𝑒 > 1 𝑎𝑛𝑑 𝐶𝑆 ≥ 2) 𝑑𝑒𝑚𝑎𝑛𝑑10 = 𝑗𝑜𝑖𝑛(𝑑𝑒𝑚𝑎𝑛𝑑0, 𝑑𝑒𝑚𝑎𝑛𝑑1)
𝑑𝑒𝑚𝑎𝑛𝑑 = 𝑑𝑒𝑚𝑎𝑛𝑑10; go to (2) 𝐹𝑜𝑟 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑤𝑒𝑒𝑘 𝑑𝑜:
𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠2 = 𝑠𝑢𝑏𝑠𝑒𝑡(𝑛𝑜𝑑𝑒𝑠 𝑉, 𝑁𝑜𝑑𝑒 > 1 𝑎𝑛𝑑 𝐶𝑆 == 1) 𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠20 = 𝑗𝑜𝑖𝑛(𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠0, 𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠2) 𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠 = 𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠20 𝑑𝑒𝑚𝑎𝑛𝑑2 = 𝑠𝑢𝑏𝑠𝑒𝑡(𝑑𝑒𝑚𝑎𝑛𝑑𝑠 𝑟𝑖, 𝑁𝑜𝑑𝑒 > 1 𝑎𝑛𝑑 𝐶𝑆 == 1) 𝑑𝑒𝑚𝑎𝑛𝑑20 = 𝑗𝑜𝑖𝑛(𝑑𝑒𝑚𝑎𝑛𝑑0, 𝑑𝑒𝑚𝑎𝑛𝑑2)
𝑑𝑒𝑚𝑎𝑛𝑑 = 𝑑𝑒𝑚𝑎𝑛𝑑20; go to (2) 𝐹𝑜𝑟 𝑡ℎ𝑒 𝑡ℎ𝑖𝑟𝑑 𝑤𝑒𝑒𝑘 𝑑𝑜: 𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠31 = 𝑠𝑢𝑏𝑠𝑒𝑡(𝑛𝑜𝑑𝑒𝑠 𝑉, 𝑁𝑜𝑑𝑒 > 1 𝑎𝑛𝑑 𝐶𝑆 == 3) 𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠4 = 𝑠𝑢𝑏𝑠𝑒𝑡(𝑛𝑜𝑑𝑒𝑠 𝑉, 𝑁𝑜𝑑𝑒 > 1 𝑎𝑛𝑑 𝐶𝑆 == 0)
q𝑡𝑙 = ((𝑙𝑒𝑛𝑔𝑡ℎ(𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠4) + 𝑙𝑒𝑛𝑔𝑡ℎ(𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠31))/2)/𝑙𝑒𝑛𝑔𝑡ℎ(𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠4) 𝑁𝑜𝑑𝑒. 𝑡ℎ𝑖𝑟𝑑 = 𝑞𝑢𝑎𝑛𝑡𝑖𝑙𝑒(𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠4, 𝑞𝑡𝑙) 𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠41 = 𝑠𝑢𝑏𝑠𝑒𝑡(𝑛𝑜𝑑𝑒𝑠 𝑉, 𝑁𝑜𝑑𝑒 < 𝑁𝑜𝑑𝑒. 𝑡ℎ𝑖𝑟𝑑 𝑎𝑛𝑑 𝐶𝑆 == 0) 𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠30 = 𝑗𝑜𝑖𝑛(𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠0, 𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠31) 𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠30 = 𝑗𝑜𝑖𝑛(𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠30, 𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠41) 𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠 = 𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠30 𝑑𝑒𝑚𝑎𝑛𝑑31 = 𝑠𝑢𝑏𝑠𝑒𝑡(𝑑𝑒𝑚𝑎𝑛𝑑𝑠 𝑟𝑖, 𝑁𝑜𝑑𝑒 > 1 𝑎𝑛𝑑 𝐶𝑆 == 3) 𝑑𝑒𝑚𝑎𝑛𝑑41 = 𝑠𝑢𝑏𝑠𝑒𝑡(𝑑𝑒𝑚𝑎𝑛𝑑𝑠 𝑟𝑖, 𝑁𝑜𝑑𝑒 < 𝑁𝑜𝑑𝑒. 𝑡ℎ𝑖𝑟𝑑 𝑎𝑛𝑑 𝐶𝑆 == 0) 𝑑𝑒𝑚𝑎𝑛𝑑30 = 𝑗𝑜𝑖𝑛(𝑑𝑒𝑚𝑎𝑛𝑑0, 𝑑𝑒𝑚𝑎𝑛𝑑31) 𝑑𝑒𝑚𝑎𝑛𝑑30 = 𝑗𝑜𝑖𝑛(𝑑𝑒𝑚𝑎𝑛𝑑30, 𝑑𝑒𝑚𝑎𝑛𝑑41)
𝑑𝑒𝑚𝑎𝑛𝑑 = 𝑑𝑒𝑚𝑎𝑛𝑑30; go to (2) 𝐹𝑜𝑟 𝑡ℎ𝑒 𝑓𝑜𝑢𝑟𝑡ℎ 𝑤𝑒𝑒𝑘 𝑑𝑜: 𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠42 = 𝑠𝑢𝑏𝑠𝑒𝑡(𝑛𝑜𝑑𝑒𝑠 𝑉, 𝑁𝑜𝑑𝑒 > 𝑁𝑜𝑑𝑒. 𝑡ℎ𝑖𝑟𝑑 𝑎𝑛𝑑 𝐶𝑆 == 0) 𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠40 = 𝑗𝑜𝑖𝑛(𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠0, 𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠42) 𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠 = 𝑛𝑒𝑤𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠40 𝑑𝑒𝑚𝑎𝑛𝑑42 = 𝑠𝑢𝑏𝑠𝑒𝑡(𝑑𝑒𝑚𝑎𝑛𝑑𝑠 𝑟𝑖, 𝑁𝑜𝑑𝑒 > 𝑁𝑜𝑑𝑒. 𝑡ℎ𝑖𝑟𝑑 𝑎𝑛𝑑 𝐶𝑆 == 0) 𝑑𝑒𝑚𝑎𝑛𝑑40 = 𝑗𝑜𝑖𝑛(𝑑𝑒𝑚𝑎𝑛𝑑0, 𝑑𝑒𝑚𝑎𝑛𝑑42)
𝑑𝑒𝑚𝑎𝑛𝑑 = 𝑑𝑒𝑚𝑎𝑛𝑑40; go to (2) 2: compute vehicle capacity 𝑖𝑛: 𝑣𝑖𝑠𝑖𝑡 𝑡𝑖𝑚𝑒𝑠 𝑟𝑖, 𝑠𝑒𝑡 𝐷, 𝑇 𝑜𝑢𝑡: 𝑣𝑒ℎ𝑖𝑐𝑙𝑒 𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑦 3: compute distance matrix 𝑖𝑛: 𝑛𝑜𝑑𝑒𝑠 𝑉𝑠, 𝑐𝑜𝑜𝑟𝑑_𝑥, 𝑐𝑜𝑜𝑟𝑑_𝑦 𝑜𝑢𝑡: 𝑡𝑟𝑎𝑣𝑒𝑙 𝑡𝑖𝑚𝑒𝑠 4: compute the savings 𝑠𝑖𝑗 = 𝑡𝑖1 + 𝑡1𝑗 − 𝑡𝑖𝑗 for 𝑖, 𝑗 = 2, … , 𝑛 and 𝑖 ≠ 𝑗
5: create 𝑛 − 1 vehicle routes (1, 𝑖, 1) for 𝑖 = 2, … , 𝑛 6: order the savings in a nonincreasing way 7: start from the top of the savings list 8: choose a saving 𝑠𝑖𝑗
9: check if there are two routes, one containing arc(1, 𝑗) and the other containing arc(𝑖, 1), that can feasibly be merged
10: if so, combine these two routes by deleting (1, 𝑗) and (𝑖, 1) and introducing (𝑖, 𝑗) 11: repeat the steps (4) to (10) until no further improvement is possible, then stop 12: repeat the steps (1) to (11) for each schedule 𝑠, then stop
1.3 FINDINGS
35
At first, the most relevant authors on the studied subject and the most widely used
mathematical models applied to solve the territory alignment problem were reported
(see Chapter 2 to 4).
In sequence, the work investigated methods to solve home health care districting
problems (see Chapter 3). The techniques most utilised to solve the districting issue
were identified. One of these techniques was chosen and an algorithm was developed
in R code. Its suitability was tested in a fictitious instance and in two real-life cases.
The proposed algorithm requires few steps for the initial solution and the local search
mechanism (tabuSearch) contributes to a short computing time.
At last, the work presented a computerised model for the problem of scheduling and
routing community health agents (see Chapter 4). A solution based on Period Vehicle
Routing Problem with Service Priority (PVRP-SP) is suggested. Another algorithm was
developed in R code to implement the solution method, and a classical heuristic for
Capacitated VRP (CVRP) was used as routing subroutine. Results found in tests with
known benchmark instances, as well as in a real-life case, demonstrated the practical
applicability of the computer model, with values close to optimal and computing times
of a few seconds.
1.4 CONTRIBUTIONS
This work shows for the first time the concatenated use of the three bibliometric
analysis techniques and innovates with feasible modifications in the mapping change
analysis applied to small networks. In fact, it can be considered a by-product of the
work as a whole, but it was fundamental for the research, in the sense of identifying
the academic communities with their main authors among thousands of publications,
as well as the best techniques used to solve the mathematical models, with their
evolutions over time.
The main contribution of this work concerns the human resource planning process,
which seems to represent one of the main issues of primary health care unit managers
in their decision-making processes. In the human resource planning process of home
care organisations there is a hierarchy in the operations management decisions (see
36
Chapter 4), including healthcare planning process and related operations research
problems (MATTA et al., 2014; SAHIN; MATTA, 2015).
The strategic level includes decisions that must be taken in the long term over a period
of one to three years, which in the case of Family Clinics means addressing issues
such as what types of care services will be provided, for which type of patients, based
on the quality of service measured over the coverage area, as well as taking into
account an estimate of overall demand (e.g., annual volumes of patient visits).
Demands for home visits may change within the time frame considered, due to
worsening or improving the health-disease conditions of patients or by increasing or
decreasing the category in the family risk and vulnerability scale. These changes in
demands will force a realignment of the territory, with a new Districting solution
(Partitioning Problem). We address the Districting Problem of territories in home care
operations in Chapter 3.
Decisions at the tactical level are taken over a horizon of six to 12 months considering
the decisions made at the strategic level and addressing their implementation. For
example, the districting process simplifies the resource allocation problem, since
patients are first assigned to a district and then assigned the health teams that will
provide assistance to the families in the district, including the community health agents
(CHAs).
Operational level decisions, with a time horizon in weeks to months, are taken so that
the flow of activities occur within the standards set at the higher hierarchical levels and
can thus be controlled. In the case of home care organisations, the main decisions at
this level refer to the assignment of care workers to patients. Especially in the case of
Family Clinics, the task is to determine the number of CHAs that will make up the
Family Health Team and the designation of the micro area for each of them.
Decisions of detailed operational level affect the planning, coordination and
supervision of day-to-day activities. In home care organisations these decisions
include the scheduling of visits and the routing of health workers across the territory
(see Chapter 4). At this level, the major operational research issues to consider are
the Vehicle Routing Problem (VRP) and the Travelling Salesman Problem (TSP)
(SAHIN; MATTA, 2015).
37
1.5 ORIGINALITY AND VALUE
Basic health units do not have standardised planning for human resource processes,
especially with respect to community health agents (CHAs), since there are no detailed
guidelines in this regard in the PNAB (BRASIL, 2012). The only recommendation in
this sense is for the head nurses of each family health team (FHT) to plan home visits
to be performed by their CHAs. Currently, this planning is done manually, bringing
inefficiency to the process.
In October 2018, Rio's City Hall announced its plan for the restructuring of Primary
Care, with a cut of 239 health teams, of which 184 were family health and 55 were oral
health teams (JUNQUEIRA, 2018). There will be 1,400 fewer jobs. The Family Health
Strategy (FHS) program that currently benefits about 70% of the population will retreat
to the level of early 2016, which was 55%. The secretary general of the Civil House
said that “the restructuring plan for Primary Care took into account the productivity of
the teams and the social development index (IDS) of the region”. “We know the impact
of the deficit in Primary Care. About 50% of people looking for UPAs (Emergency Care
Units) should be in Primary Care (Family Clinics), but are not absorbed, because it is
a care that is not adequate”, said the municipal secretary of Health (JUNQUEIRA,
2018).
This news shown above testifies to the lack of efficiency of the Family Clinics in Rio de
Janeiro. These facts could be avoided with a better management of human resource
processes in these health organisations.
Recently, the Inter-American Development Bank (IDB) said that “policies should focus
on improving the efficiency of health care, investing in interventions that deliver the
best health outcomes, and appropriate implementation of these interventions"
(GIORGI, 2018). In this context, where the increase in health budgets is unlikely,
according to the IDB study, a large part of the countries in the region (Latin America)
could significantly improve their indicators if they achieved an advance of efficiency.
To this end, still according IDB and related in (GIORGI, 2018), governments would
have to "improve institutions and governance; regulate the prices of medicines; and
38
provide full primary treatment". These changes are essential in the face of an aging
population, the growing incidence of chronic diseases, and socioeconomic advances
that translate into greater demand for quality and universal health services.
In this sense, the present study shows a great deal of relevance in contributing to the
increase of efficiency in the human resource planning processes, so that there is a
total coverage of the service area of the Family Clinics, which will have an assistance
service with more resoluteness. This increase in the efficiency of Family Clinics should
decrease hospital urgency and emergency queues, bringing even greater efficiency in
the integrated SUS network system.
1.6 REFERENCES
ARAÚJO, P. B. Qualidade na prestação de serviços das clínicas da família: Caso Assis Valente. Rio de Janeiro: Faculdade CNEC Ilha do Governador, 2012.
BLAIS, M. Le découpage territorial pour les services de soins de santé à domicile. Montréal: Université de Montréal, 2001.
BLAIS, M.; LAPIERRE, S. D.; LAPORTE, G. Solving a home–care districting problem in an urban setting. Journal of the operational research society, 2003. v. 54, n. 11, p. 1141–1147. Disponível em: <http://dx.doi.org/ 10.1057/palgrave.jors.2601625>.
BOZKAYA, B.; ERKUT, E.; LAPORTE, G. A tabu search heuristic and adaptive memory procedure for political districting. European journal of operational research, 2003. v. 144, n. 1, p.12-26.
BRASIL. Ministério da Saúde. Norma operacional básica do sistema único de saúde - NOB-SUS 96. Brasília, DF, 1997.
BRASIL. Ministério da Saúde. Política nacional de atenção básica - PNAB. Brasília, DF, 2012. Disponível em: < http://portal.saúde.gov.br/portal/arquivos/pdf/volume_4_completo.pdf>. Acesso em: 1 set. 2015.
CLARKE, G.; WRIGHT, J. W. Scheduling of vehicles from a central depot to a number of delivery points. Operations research, 1964. v. 12, n. 4, p. 568-581.
CLSC - Centre Local de Santé Communautaire. Côte-des-Neiges. Montreal, CA, 2000.
COELHO, F.; SAVASSI, L. Aplicação da escala de risco familiar como instrumento de priorização das visitas domiciliares. Revista brasileira de medicina de família e comunidade, 2004. v. 1, n. 2, p. 19-26. Disponível em: <http://www.rbmfc.org.br/index.php/rbmfc/issue/view/2/>. Acesso em: 7 nov. 2018.
39
DIOGO, O. A.; ARAÚJO, P. B. Qualidade na prestação de serviços das clínicas da família: caso Assis Valente. TerCi, 2013. v. 2, n. 2, p. 58-78. Disponível em: <http://www.cnecrj.com.br/ojs/index.php/temiminos/article/download/11/11>. Acesso em: 22 jan. 2019.
DOMIJAN, K. tabuSearch: R based tabu search algorithm. R package version 1.1, 2012. Disponível em: < https://CRAN.R-project.org/package=tabuSearch>.
FRANCIS, P.; SMILOWITZ, K.; TZUR, M. The period vehicle routing problem with service choice. Transportation science, 2006. v. 40, n. 4, p. 439-454.
FOUSKAKIS, D; DRAPER, D. Stochastic optimization: a review. International statistical review, 2002. v. 70, n. 3, p. 315–349
GIL, C. R. R.; MAEDA, S. T. Modelos de atenção à saúde no Brasil. In: SOARES, C. B.; CAMPOS, C. M. S. (Org.). Fundamentos de saúde coletiva e o cuidado de enfermagem. Barueri: Manole, 2013, p. 325-347.
GIORGI, J. A ineficiência dos sistemas de saúde da América Latina. Folha de São Paulo, São Paulo, 13 nov. 2018. Disponível em: <http://gehosp.com.br/2018/11/13/ineficiencia-saude/?fbclid=IwAR3NNL9wTVhOSOzrKDoB4C9Fb4bYt9xrx7LO_AegZLwQLin4JmMThzuNKIE>. Acesso em: 27 jan. 2019.
JUNQUEIRA, F. Prefeitura anuncia o corte de 239 equipes de saúde da família e saúde bucal na cidade do Rio. Jornal Extra, Rio de Janeiro, 30 out. 2018. Disponível em: < https://extra.globo.com/noticias/rio/prefeitura-anuncia-corte-de-239-equipes-de-saude-da-familia-saude-bucal-na-cidade-do-rio-23200159.html>. Acesso em: 27 jan. 2019.
KAVETI, N. CW_VRP: implements Clarke-Wright savings algorithm to find greedy routes. R package documentation, 2017. Disponível em: < https://rdrr.io/github/kavetinaveen/HeuristicsVRP/man/CW_VRP.html>. Acesso em: 6 nov. 2018.
MATTA, A. et al. Modelling home care organisations from an operations management perspective. Flexible services and manufacturing journal, 2014. v. 26, n. 3, p. 295-319.
MENDES, E. V. O cuidado das condições crônicas na APS: o imperativo da consolidação da ESF. Brasília, DF: CONASS/OMS/MS, 2012.
PAIM, J. S. Gestão da atenção básica nas cidades. In: NETO, E. R.; BÓGUS, C. M. (Org.). Saúde nos aglomerados urbanos. Brasília, DF: Organização Pan-Americana de Saúde, 2003, p. 183-212.
RIO DE JANEIRO (Cidade). Secretaria Municipal de Saúde. Clínicas da família. Rio de Janeiro, RJ, 2017. Disponível em: <http://www.rio.rj.gov.br/web/sms/clinicas-da-familia#>. Acesso em: 8 fev. 2017.
SAHIN, E.; MATTA, A. A contribution to operations management-related issues and models for home care structures. International journal of logistics - research and applications, 2015. v. 18, n. 4, p. 355-385.
SILVEIRA, R. S. Qualidade na prestação de serviços das clínicas da família: caso Maria Sebastiana de Oliveira. Rio de Janeiro: Faculdade CNEC Ilha do Governador, 2015.
40
WHO – World Health Organisation. Declaration of Alma-Ata. In: INTERNATIONAL CONFERENCE ON PRIMARY HEALTH CARE, 1978, Alma-Ata, USSR. Proceedings of international conference on primary health care. Alma-Ata: WHO, 1978, p. 1-3. Disponível em: <https://www.who.int/publications/almaata_declaration_en.pdf>. Acesso em: 24 jan. 2019.
41
2 1ST PAPER: THE TERRITORY ALIGNMENT PROBLEM: A LONGITUDINAL
BIBLIOMETRIC ANALYSIS APPLIED TO HOME CARE SERVICES
42
THE TERRITORY ALIGNMENT PROBLEM: A LONGITUDINAL BIBLIOMETRIC
ANALYSIS APPLIED TO HOME CARE SERVICES
ABSTRACT
Territory alignment usually involves facility location, districting of work basic areas and
resources assignment, however, for some applications in health services may also
include the problem of routing of health teams. Typical design requirements are
districts that are similar in size or that reduce travel times to service customers within
the work basic areas. Through a longitudinal bibliometric study, this paper investigates
mathematical models to solve the territory alignment problem, seeking to foster
improvements in health services operations, such as home care. The methodology
used encompasses three areas for analysis: social network of authors, longitudinal co-
word analysis, and mapping change analysis. The latter is usually applied in large
networks; however, herein, it was adapted to small and medium networks, and used
Tabu search as a fast local search scheme. The work reports on the most relevant
authors on the subject and the most widely used mathematical models applied to solve
the problem.
Keywords: methodology; networks and graphs; optimisation; multi-objective;
heuristics; health service
2.1 INTRODUCTION
Home health services are implemented as a supplement or replacement to the
hospital-centred health model for a variety of reasons, especially in developed
countries. Many issues have to be considered for delivering such services. Seen as a
system, these services comprise a complex network interconnecting an ecosystem
that includes various actors, from physicians to patients. And for this complex network
to work well, a large number of decisions must be made involving design and
operations (BRICON et al., 2005; LANZARONE; MATTA; SAHIN, 2012; GUTIERREZ;
JULIO VIDAL, 2013). One of the main issues in this complex task is the so-called
43
territory alignment problem, which involves the location of a clinic or hospital, the
allocation of health teams, and the distribution of these teams in basic units of service
(DASKIN; DEAN, 2005). Earlier territory alignment schemes have not included routing
of teams over the basic units; however, recently, the entire task has indeed been
performed with routing solutions. The present work considers the territory alignment
problem with respect to four issues: (1) location of the territory centre (clinic or
hospital); (2) allocation of teams; (3) division of the territory into districts of service
(districting); and (4) the routing of the health teams.
Which mathematical model would be optimal for solving the territory alignment problem
related to health care services? What strategy to choose as the theoretical model?
Exact methods have the advantage of providing the best solutions, but have the
disadvantage of high processing time and no guarantees that their solutions can be
implemented in practice. Heuristics may provide quick solutions that are easy to
implement in practice, but which are often of lower quality (KALCSICS; NICKEL;
SCHRÖDER, 2005). The success of metaheuristics derives from several factors, such
as the general applicability of the approach, ease of implementation, quality of the
solution, and relatively low computational effort. In addition, hybrid metaheuristics
methods have been used to render the resulting procedure more effective than any
single component (KIM; NARA; GEN, 1994; KURODA; KAWADA, 1994; GLOVER;
KELLY; LAGUNA, 1995; PIRLOT, 1996; ROACH; NAGI, 1996).
In order to find good mathematical models to solve the territory alignment problem
applied to health services, we used a methodology that combines three methods of
analysis in a novel manner: social network analysis, longitudinal analysis, and mapping
change analysis. Indeed, with the increasing of academic publication it is almost
impossible to read all the documents related to a research domain. The methodology
proposed here presents a way to find the most relevant academic articles on the
subject under study without having to read the entirety of them.
The analysis of the social network has the intention of finding the most-published
authors in the search field, which is achieved by analysing the cluster structures in
terms of the formation of academic communities (NEWMAN, 2001; PERIANES-
RODRÍGUEZ; OLMEDA-GÓMEZ; MOYA-ANEGÓN, 2010). The analysis of the social
network of authors was also done to identify the first articles related to home health
44
care services and thereafter be able to drill down in the search field. After that, a
longitudinal keyword analysis is performed to understand the concepts treated in each
time period. The fundamental idea here consider that co-occurrence of words has the
power of describing the content of the articles under analysis (CALLON; COURTIAL;
LAVILLE, 1991). In immediate sequence, a graph can be made in which the keywords
are the nodes and the weight of edges between them represents their equivalence
indexes. However, in the graph creation process, choices, such as the minimum
number of co-occurrences, the minimum number of edges per nodes, the maximum
number of nodes in the network, and so on, must be made. The resulting network is
idiosyncratic by nature. For this reason, we used the third technique of mapping
change analysis, which entails a stochastic analysis for each period of time in the
original (real-world) network, first by bootstrapping this network and then generating
more than 1000 bootstrapped networks. Both original and bootstrapped networks
undergo a clustering process which produces a modular description of each network.
At this step, a stochastic process is used in order to determinate the most confidence
modular description. The authors of this method, Rosvall and Bergstrom (2010),
applied the technique to large networks and then used a simulated annealing scheme
(KIRKPATRICK; GELATT; VECCHI, 1983) for this step. In this paper, we applied the
mapping change method to small and medium networks and used the Tabu search
scheme instead of simulated annealing. Finally, to highlight the significant changes
over time of keywords networks, an alluvial diagram is used to show the significance
clusterings through the subperiods studied.
Section 2 presents the methodology used and an explanation regarding the
innovations on the original method of mapping change. Section 3 describes the data
grouping technique and summarises the body of knowledge on the topic. Section 4
explores the significance of the results. Section 5 presents the conclusion and
suggestions for future work.
2.2 MATERIALS AND METHODS
Bibliometric research is one of the most used methods to assess the efficiency of
scientific publications (PRITCHARD, 1969; HOOD; WILSON, 2001; GODIN, 2006;
45
ABRAMO; D’ANGELO; DI COSTA, 2011; WAINER; VIEIRA, 2013; KARLSSON et al.,
2015; MAYR; SCHARNHORST, 2015; DREW et al., 2016). By its turn, social network
analysis (SNA) is one of the tools often used in bibliometric researches (LARIVIÈRE;
GINGRAS; ARCHAMBAULT, 2006; PERIANES-RODRÍGUEZ; OLMEDA-GÓMEZ;
MOYA-ANEGÓN, 2010). SNA allows identifying the formation of academic
communities, to understand their interrelationships and the strength of their
interconnections (NEWMAN, 2001). In the present study we used SNA to know the
most prolific authors about the subject in question and the possible formation of
academic communities that share research on the same theme. In addition, it was
possible to identify the first publication using mathematical methods applied to
healthcare, and from this first publication a citation network was created to evaluate
the evolution of the methods used to solve the problems related to health services.
After identifying the main authors and their works, the keywords used in their articles
are collected and grouped for a longitudinal analysis in order to verify the evolution of
the mathematical methods used for the territory alignment problem as well as its
applications for health services. To do so, the entire analysis period of the citation
network (2000-2016) was subdivided into subperiods and, for each subperiod,
prevailing concepts and themes were detected. However, in this process clusters are
generated from the existing connections between the main groups of keywords. For
the formation of the clusters, some premises are adopted and in this way the resulting
networks present certain idiosyncrasies.
In order to eliminate the idiosyncrasies resulting from the clustering process, mapping
change analysis is used. The step begins with an analysis of possible partitions of the
clusters (modular description 𝑀) in higher affinity subnetworks. Then starts a
bootstrapping process where more than 1000 clusters configurations are generated
considering a Poisson distribution for the formation of links between the nodes of each
network. Next, a search algorithm is used to identify the most repeating partitions with
95% confidence. The algorithm is followed in its final phase by a local search scheme
in order to optimise the process.
With the configurations of the most confidence modular descriptions for each
subperiod, it is possible to draw up an alluvial diagram that shows the evolution over
the time of the most relevant themes (cluster nodes).
46
Figure 2.1 shows a schematic diagram of the steps that constitute the methodology
used in this work.
Figure 2.1. The steps of analysis methodology
The following subsections present a little more detail on the implementation of the
methodology steps employed.
2.2.1 SOCIAL NETWORK ANALYSIS OF AUTHORS
The bibliometric mapping has assumed a relevant position in the bibliometric literature
(BÖRNER; CHEN; BOYACK, 2003). However, a greater concern has been given to
the construction of the maps, with the graphical representation thereof receiving
considerably less attention (VAN ECK; WALTMAN, 2010). The present work uses the
software package VOSviewer version 1.6.4 for the construction and representation of
bibliometric maps. The program employs the method of VOS (visualisation of
similarities), which supports only distance-based maps (VAN ECK; WALTMAN, 2007).
2.2.2 LONGITUDINAL KEYWORD ANALYSIS
Alluvial diagram
Significance clusters Highlight the significant changes
Mapping change analysis
Bootstrapping Most confidence modular description
Longitudinal keyword analysis
Concepts and thematics in each subperiod Idiosyncratic networks
Social Network Analysis of authors
Most published authors 1st article in healthcare
47
The use of bibliometric information to construct maps that show the evolution of a
scientific research theme over time is a very popular method today (GARFIELD, 1994).
The present work uses the approach presented by Cobo et al. (2011) to reveal the
evolution of the methods used to solve the various aspects of the territory alignment
problem. A longitudinal keyword analysis is performed to understand the concepts
treated in each time period. The co-occurrence of words describes the content of the
articles under analysis (CALLON; COURTIAL; LAVILLE, 1991). A graph can be made
in which the keywords are the nodes and the weight of edges between them represents
their equivalence indexes. Here, the results are represented by using strategic
diagrams and the conceptual evolution is depicted by thematic areas.
The strategic diagram is a quadrant plot, where the x-axis represents the cluster's
centrality and the y-axis density (CALLON; COURTIAL; LAVILLE, 1991). Themes can
appear in any of the four quadrants, with: (1) motor-themes appearing in the first
quadrant and representing well-developed themes that drive academic research; (2)
in the second quadrant appear themes that are very specialized and therefore not of
interest to the entire academic community; (3) in the third quadrant appear themes with
poor development, which are still emerging or tend to disappear; and (4) in the fourth
quadrant appear important themes that are in development. The approach indicated
above was implemented in a step-by-step process by using SciMAT version 1.1.03
software.
2.2.3 MAPPING CHANGE ANALYSIS
The methodology of Rosvall and Bergstrom (2010) is used. However, as the method
was devised for large networks, some modifications were made to adapt it to small and
medium networks. The Rosvall/Bergstrom method assesses the level of confidence of
the clustering of a network. The final result of the procedure shows the most significant
structural changes in the research field over the subperiods.
The approach focuses on weighted directed networks and assumes that the weight of
edges follows a Poisson distribution. The method consists of four steps: (1) cluster the
keyword networks for each subperiod; (2) generate and cluster the bootstrap replicate
48
networks for each subperiod; (3) identify significance clusters for each subperiod; and
(4) construct an alluvial diagram to enlighten changes between subperiods.
2.2.3.1 CLUSTER THE KEYWORD NETWORKS
The process begins with the partition of the original network G into the modular
description 𝑀. The partition is supported by the map equation as the objective function
(ROSVALL; AXELSSON; BERGSTROM, 2009), whose minimisation over all possible
partitions 𝑀 will produce the best partition.
The map equation can be performed by cluster_infomap function of igraph package in
R library (CSARDI, G.; NEPUSZ, 2006).
2.2.3.2 GENERATE AND CLUSTER THE BOOTSTRAP REPLICATE NETWORKS
The bootstrap method is necessary in order to eliminate the idiosyncrasy introduced
by a computational process used for clustering the original networks (ROSVALL;
BERGSTROM, 2010). The bootstrapping in this case is a parametric process which
resamples each link weight wαβ (node α to node β with weight w) of the original network
G considering a Poisson distribution with mean equal to the original link weight wαβ.
Afterwards, as stated by Rosvall and Bergstrom (2010), the same clustering method
is used to partition the resulting bootstrap network 𝐺𝑏∗ to generate the bootstrap
modular description 𝑀𝑏∗. The process must be repeated to generate more than one
thousand modular descriptions 𝑀∗= {𝑀1∗, 𝑀2
∗, . . . , 𝑀𝐵∗ }, where B > 1000.
In order to generate the bootstrap-world networks, we use the boot package in R
library. Let 𝑎𝑑𝑗𝑚 be the real-world network adjacency matrix representing the links
(edges) between the nodes with respective weights. To convert the adjacency matrix
𝑎𝑑𝑗𝑚 into a graph 𝑔 we use the igraph package in R library (CSARDI, G.; NEPUSZ,
2006). To perform a parametric bootstrap, we first need to define the bootstrap function
(which in our case corresponds to a Poisson distribution) and then call the boot function
(CANTY; RIPLEY, 2016).
49
2.2.3.3 IDENTIFY SIGNIFICANT ASSIGNMENTS
To identify the most significant nodes assigned to a module, a search algorithm is used
for finding the largest subset of nodes that appears in original modular description 𝑀
and that also appears in at least 95% of all bootstrap modular descriptions 𝑀∗. In order
to choose the largest subset, the total PageRank of the cluster is used. The PageRank
of the cluster can be performed by page_rank algorithm of igraph package in R library.
Then, we follow the local search with Tabu search instead of the standard simulated
annealing scheme originally used by Rosvall and Bergstrom (2010). The Tabu search
scheme can be performed by tabuSearch package in R library (DOMIJAN, 2012).
2.2.3.4 CONSTRUCT ALLUVIAL DIAGRAM
To highlight significant changes over time in keyword networks, the results of the most
significant clusters for each subperiod can be summarised in an alluvial diagram. Each
significant cluster for a subperiod (𝐺𝑖) seizes a column in the alluvial diagram and is
connected horizontally to clusters of previous and successive meanings by flow fields.
Furthermore, each block in a column of the diagram represents a cluster node and the
height of the block corresponds to the size of the network element. The elements are
sorted from bottom to top by size (PageRank).
The alluvial diagram uses stream fields to show the changes in cluster structures
(nodes) and in level of significance between two adjacent subperiods. The height of a
stream field at each end represents the total size of the nodes that participate of this
transition. Therefore, by following each stream field from a cluster to an adjacent
column, it is possible to analyse the most significant transitions. The alluvial diagram
can be constructed by alluvial_ts function of alluvial package in R library
(BOJANOWSKI; EDWARDS, 2016).
The following section will be the application of each introduced part.
50
2.3 THE FIELD OF RESEARCH ON THE TERRITORY ALIGNMENT PROBLEM
In this article, the integrative literature review (COOPER, 1984; GANONG, 1987;
BROOME, 2000) was adopted as a method of grouping data and synthesis of
knowledge about the proposed theme in order to answer the following question: What
mathematical programming model would be the most suitable for solving the territory
alignment problem in health care operations?
Articles containing discourse on mathematical programming and heuristics models
were included, and dissertations, theses, editorial notes, books and conference
proceedings were excluded. We opted not to set a specific time period, but rather limit
the scope to studies published until 2016, written in English, Spanish and Portuguese.
A detailed reading was undertaken only after the social network analysis, in order to
obtain a list of articles related to the proposed solution.
JSTOR and Web of Science were the databases selected for the literary search of the
main journals related to Operations Research, Management Science and Decision
Sciences. It is worth pointing out that the concurrent search of the two databases was
done in February 7, 2017.
The keywords and search logic used were: districting problem AND home health care
services AND mathematical modelling AND operations management OR districting
AND community health AND clinic AND multi-criteria optimization OR heuristics AND
optimization AND logistics AND territory design.
As search results yielded 2,106 articles in JSTOR and 443 in the Web of Science
databases, i.e., a total of 2,549 articles. Note that the two databases are also included
in other databases, such as Emerald, for example. The organisation of the articles,
with the necessary actions of export, stocks, imports, classification and archiving, was
carried out with the help of EndNote X7, Mendeley Desktop 1.16.1, JabRef 3.4, and
SciMAT-v1.1.03 tools.
Of the 2,549 items found in the integrative literature review search, those published in
Econometrics and Marketing journals were also excluded. The sample was thus
reduced to 2,316 articles and 3,957 authors.
51
2.3.1 SOCIAL NETWORK ANALYSIS OF AUTHORS FROM 1963 TO 2016
Having defined the sample to be studied, the metadata in the databases (authors,
publication year, journal, abstract, keywords, among other data) were extracted. Figure
2.2 shows the social network of authors in terms of those with a minimum of two
publications.
Considering a network with up to 500 items, 350 connected items and 47 clusters were
found, based on 715 links. The main node of each cluster is represented by the author
with the most published articles, with the number of co-authorships as tiebreaker.
Table 2.1 shows a sample of the number of: (1) articles published by authors (greater
than or equal to 14) in the period 1963-2016; (2) co-authorships; (3) cluster ID; and (4)
number of items (quantity of cluster participants).
Table 2.1 considers overlapping communities. Most of the clustering methods aim at
detecting standard partitions, i. e. partitions in which each vertex is assigned to a single
community. However, in real graphs vertices are often shared between communities,
and the issue of detecting overlapping communities has become quite popular in the
last years (FORTUNATO, 2010). The present work uses software package VOSviewer
version 1.6.4 that implements techniques to detect overlapping communities.
52
Figure 2.2. Social network of authors, 1963-2016
53
Table 2.1. Productivity of authors, 1963-2016
Order Author Documents Co-authorships Cluster ID (Items)
1 Laporte, G. 402 634 9 (11)
2 Cordeau, J. F. 69 145 9 (11)
3 Gendreau, M. 67 128 16 (10)
4 Semet, F. 27 49 26 (6)
5 Ricca, F. 24 50 17 (9)
6 Kalcsics, J. 21 46 11 (10)
7 Eiselt, H. A. 19 27 32 (3)
8 Nickel, S. 19 45 11 (10)
9 Labbe, M. 17 33 13 (10)
10 Bertsimas, D. 15 6 1 (20); 5 (16)
11 Hall, N. G. 15 20 20 (9)
12 Nobert, Y. 15 24 25 (6)
13 Scozzari, A. 15 34 17 (9)
14 Bard, J. F. 14 9 1 (20)
15 Coelho, L. C. 14 23 22(8)
16 Puerto, J. 14 34 17 (9)
Blais, Lapierre, and Laporte (2003) were the first to address a territory alignment
problem related to health care operations. The authors present a solution for the
districting problem of a community clinic in Côte-des-Neiges, a borough in Montreal,
QC, Canada. Considering (BLAIS; LAPIERRE; LAPORTE, 2003) main connections
(references and citations) the period 2000-2016 was then selected in order to perform
an analysis of the thematic evolution. The search indicated a sample of 173 authors
with 99 published articles.
2.3.2 LONGITUDINAL CO-WORD ANALYSIS
A longitudinal co-word analysis was performed on the 99 articles from 2000 to 2016
considering keywords (author keywords, source keywords, and extracted words). The
purpose of longitudinal co-word analysis was to identify the most used mathematical
models (exact solutions, algorithms, heuristics and metaheuristics) that could
contribute to the solution of the territory alignment problem through a thematic
evolution study.
54
The knowledgebase of 386 words (keywords) was manually clustered in 64 groups.
Then, as the first step in the longitudinal analysis, we considered following four
subperiods: (1) 2000 to 2003; (2) 2004 to 2007; (3) 2008 to 2010; and (4) 2011 to 2016.
Each subperiod produces a map. Table 2.2 shows the selected parameters for the
longitudinal analysis (COULTER; MONARCH; KONDA, 1998; HIRSCH, 2005; VAN
RAAN, 2006; ALONSO et al., 2009; COBO et al., 2011).
Table 2.2. Selected parameters for longitudinal analysis
Step Step name Selection
1 Subperiods (1) 2000 – 2003; (2) 2004-2007; (3) 2008-2010; (4) 2011-2016
2 Unit of analysis Words (Author’s words, Source’s words, and Extracted words)
3 Data reduction at least ‘2’ documents per subperiod
4 Kind of matrix co-occurrence
5 Network reduction edges with a value greater or equal to ‘1’ per subperiod
6 Normalisation equivalence index
7 Cluster algorithm simple centres algorithm
8 Document mapper core and secondary mappers
9 Quality measures h-index and sum citations
10 Longitudinal inclusion index
The results can be viewed two ways: via the longitudinal view or via the period view.
In the longitudinal view (see Figure 2.3), the overlapping map (upper) and evolution
map (lower) are depicted. The period view (see next subsection) shows detailed
information for each subperiod, as follows: the respective strategic diagram,
quantitative and qualitative measures, and the network configuration.
55
Figure 2.3. Longitudinal view – overlapping and evolution maps
56
By inspecting the longitudinal view (Figure 2.3), we see from the evolution map (lower)
that heuristics was the dominant thematic in the 2000-2003 subperiod. The solid line
between heuristics in subperiod 2000-2003 and Tabu search in subperiod 2004-2007
indicates that the name of one theme is part of the other theme. In period 2008-2010,
two themes were detected: multi-criteria and health services. The dotted line between
Tabu search and multi-criteria means that the themes share elements other than the
label of the themes. Operations research, heuristics, and multi-criteria were the three
detected thematics in the subperiod 2011-2016. Again, the dotted line between multi-
criteria in the previous subperiod and operations research indicates sharing elements.
The solid lines between the subperiods 2008-2010 and 2011-2016 means that the
labels appear in both subperiods. The thickness of the edges is related to the inclusion
index, and the volume of the nodes indicates the quantity of published documents for
each theme.
The overlapping map in Figure 2.3 (upper) illustrates the evolution of the keywords.
The circles represent subperiods, and the numbers inside are the corresponding
quantities of keywords. Above the arrows is shown the amount of keywords that are
shared (between subperiods), which are discarded (upper outcoming) or are new ones
(upper incoming). Similarity index is shown in parentheses.
The following subsections will discuss the strategic diagrams and the cluster elements
for each studied subperiod.
57
2.3.2.1 SUBPERIOD 2000-2003
In the strategic diagram for subperiod 2000-2003 we see heuristics as a motor-theme
(first quadrant). Figure 2.4 shows the elements of the heuristics cluster in subperiod
2000-2003. We see that the strongest connection occurs between facility location and
graph theory (weight = 0.60) nodes. The second strongest link is between heuristics
and routing problem (weight = 0.33) nodes, and between the heuristics and TSP
(travelling salesman problem, weight = 0.33) nodes. This fact denotes a concern about
the routing problem and the use of heuristics to solve it. Districting and applications
(weight = 0.25) nodes also presents a strong connection. Applications is a word group
composed of 16 items that may indicate that the districting problem was considered
for several applications in subperiod 2000-2003. The sales territory alignment node is
linked to the heuristics and graph theory nodes with the same weight (0.17). Facility
location was the most published theme in the subperiod, with five documents.
2.3.2.2 SUBPERIOD 2004-2007
The strategic diagram for subperiod 2004-2007 shows Tabu search as a motor-theme.
Figure 2.5 illustrates the elements of the Tabu search cluster in subperiod 2004-2007.
The three strongest links are between Tabu search and routing problem nodes,
between heuristics and TSP nodes, and between time windows and VRP (Vehicle
Routing Problem) nodes, which have a weight equal to 0.67. Tabu search node has a
strong link with VRP (weight = 0.44). Districting has edges to applications (weight =
0.12) and to Tabu search and VRP nodes, both with the same weight and equal to
0.08. We observe the growth of the districting theme from the previous subperiod
(2000-2003) to this subperiod (2004-2007); this theme was the most published in this
studied subperiod, with four documents.
58
Figure 2.4. Cluster elements, 2000-2003 Figure 2.5. Cluster elements, 2004-2007
59
2.3.2.3 SUBPERIOD 2008-2010
The strategic diagram for subperiod 2008-2010 shows two clusters: multi-criteria
(central) and health services (motor-theme). The cluster multi-criteria in Figure 2.6a
has as its strongest link that between the districting and simulated annealing nodes,
with weight equal to 1, which means that all documents in each node share the same
keywords. The second strongest link is between the spatial analysis and applications
(weight = 0.4) nodes, which may indicate that the spatial analysis approach was
considered for several applications in subperiod 2008-2010. The third strongest link is
between spatial analysis and clustering (weight = 0.25) nodes, which likely indicates
the objective for using the mentioned method. Facility location, optimisation, heuristics,
and applications were the most published themes, respectively, with nine, six, five, and
five documents.
The cluster health services (Figure 2.6b) presented as the strongest link at the edge
between branch-and-price and set partitioning problem (weight = 1). There are two
edges in the second place: those departing from column generation to set partitioning
problem and branch-and-price (weight = 0.67). Health services was the most published
theme, with four documents.
2.3.2.4 SUBPERIOD 2011-2016
In the strategic diagram for the last studied subperiod (2011-2016), two clusters appear
as motor-themes: operations research and heuristics. Furthermore, the cluster multi-
criteria appears in the lower-left quadrant, representing a declining theme.
The cluster operations research (Figure 2.7a) presents the metaheuristics node with
the strongest link to GRASP (greedy randomised adaptive search procedure) node
(weight = 1). Then, in the same cluster, the time windows node with the second
strongest link in connection with VRP (weight = 0.67) appears. Districting and health
services are the most published themes, with 16 documents.
Figure 2.7b shows another motor-theme, heuristics, which in turn presents clustering
as one of the most important nodes, with the strongest cluster links for mixed-integer
60
programming and heuristic nodes, respectively with weight equal to 0.17, and 0.27.
Heuristics is the most published theme, with five documents.
The third cluster (Figure 2.7c), multi-criteria, represents a declining theme, since it
appeared in the previous subperiod as a central theme and is now located in the lower
left quadrant. The edges connecting multi-criteria node to decision-making and
adaptive large neighbourhood search nodes are the cluster's strongest, as well as the
edge between decision-making and applications, all with a weight equal to 0.07.
Decision-making is the most published themes, with eight documents.
Table2. 3 (a and b) summarises the quantitative and impact measures for the themes
of period 2000-2016.
61
Figure 2.6a. Cluster elements, 2008-2010 Figure 2.6b. Cluster elements, 2008-2010
62
Figure 2.7a. Cluster elements, 2011-2016
63
Figure 2.7b. Cluster elements, 2011-2016 Figure 2.7c. Cluster elements, 2011-2016
64
Table 2.3a. Quantitative and impact measures for the themes of period 2000-2016
Subperiod
Strategic diagram Documents
Cluster nodes Theme (kind)
Density (range)
Centrality (range)
Core (secondary)
Quantity h-index Total
citations
2000-2003 Heuristics (motor) 37.36 (1) 21.92 (1) 7 (6) 6 (5) 116 (171)
1. Districting 2. Routing Problem 3. Heuristics 4. Applications 5. TSP
6. Facility Location 7. Sales Territory
Alignment 8. Graph Theory
2004-2007 Tabu Search (motor)
66.20 (1) 12.50 (1) 6 (7) 5 (4) 144 (159)
1. Districting 2. Tabu Search 3. VRP 4. Applications 5. Routing Problem
6. Facility Location 7. Heuristics 8. Time Windows 9. TSP
2008-2010
Multi-criteria (central)
36.43 (0.5) 28.38 (0.5) 13 (13) 7 (8) 300 (249)
1. Districting 2. Heuristics 3. Simulated
Annealing 4. Multi-criteria 5. GIS 6. Facility Location 7. Metaheuristics
8. Optimisation 9. Financing 10. Sales Territory
Alignment 11. Applications 12. Spatial Analysis 13. Clustering
Health Services (motor)
44.79 (1) 32.89 (1) 7 (5) 4 (4) 74 (99)
14. Scheduling 15. Integer
Programming 16. Planning 17. Health Services
18. Assignment Problem
19. Column Generation
20. Branch-and-price 21. Set Partitioning
Problem
65
Table 2.3b. Quantitative and impact measures for the themes of period 2000-2016
Subperiod
Strategic diagram Documents
Cluster nodes Theme (kind)
Density (range)
Centrality (range)
Core (secondary)
Quantity h-index Total
citations
2011-2016
Operations Research (motor)
88.34 (1) 67.14 (1) 27 (23) 5 (6) 98 (114)
1. Districting 2. Routing Problem 3. Facility Location 4. Health Services 5. Mathematical
Programming 6. Algorithms 7. GRASP 8. Time Windows 9. Metaheuristics 10. Service Operations
11. Assignment Problem
12. Optimization 13. Modelling 14. Operations
Management 15. VRP 16. Operations
Research 17. Engineering 18. Computer Science 19. Scheduling 20. Integer
Programming
Heuristics (motor)
13.4 (0.67) 48.2 (0.67) 5 (16) 4 (5) 36 (86)
21. Sales Territory Alignment
22. Heuristics 23. GIS 24. Clustering
25. Financing 26. TSP 27. Mixed-Integer
Programming 28. Logistics
Multi-criteria (declining)
6.12 (0.33) 27.73 (0.33) 6 (16) 2 (5) 10 (88)
29. Decision-Making 30. Adaptive Large
Neighbourhood Search
31. Multi-criteria
32. Graph Theory 33. Applications
66
2.3.3 REAL-WORLD AND BOOTSTRAP-WORLD NETWORKS
The SciMAT program outcomes (Section 2.3.2) represent the real-world clusters for
each selected subperiod. However, since these cluster assignments are results of a
computing process, the resulting network presents some idiosyncrasies. In order to
increase the level of confidence in the clustering process, bootstrap is an indispensable
step (ROSVALL; BERGSTROM, 2010).
For instance, considering subperiod 2008-2010, the number of resamples is equal to
1008. Then, we can rebuild the output of bootstrapping into 1008 adjacency matrices
and then form 1008 bootstrap-world networks. The next step is the clustering
procedure (for each bootstrapped network) to generate 1008 bootstrap modular
descriptions 𝑀𝑏∗. The modular description 𝑀𝑏
∗ is a vector which shows the modularity
(division in subsets) and the information of PageRank for each node in the cluster.
After that, a local search algorithm based on Tabu search is done to find the largest
significant subsets. In the example of subperiod 2008-2010, the most significant
network corresponded to modular description 𝑀462∗ ; thus, its PageRank information
can be used to build an alluvial diagram.
2.3.4 ALLUVIAL DIAGRAM
Figure 2.8 shows the resulting alluvial diagram that emphasises and condenses the
structural differences between significance clusters over time. Each cluster node in the
network is represented by an equivalent colour block in the alluvial diagram. Changes
in the clustering assignments from one subperiod to the adjacent are represented by
the movements of the ribbons linking the blocks at each subperiod.
The alluvial diagram for the keyword data reveals the significant structural changes
that have occurred in the research field of the territory alignment problem applied to
health services over the past seventeen years.
67
Figure 2.8. Alluvial diagram – evolution of significance clusterings
68
2.4 DISCUSSION
Regarding subperiod 2000-2003, although the cluster is heuristics, the highest position
in the column indicates the most significant element as the routing problem node. The
heuristics node is the second most significant element. These facts may indicate the
use of heuristics for solving routing problems, as previously thought (see section 3.2.1).
Heuristics will persist through the periods, falling in relevance in the second and third
subperiods at lowest level; however will reach the subperiod 2011-2016 with middle
importance.
The routing problem will continue to be a concern in the subperiod 2004-2007, albeit
assuming second position in terms of relevance and will appear again in subperiod
2011-2016 at third position.
Facility location figures in the subperiod 2000-2003 with a relatively high (third)
significance, falls in the next subperiod (2004-2007) to the second lowest position in
the cluster, and undergoes a climbing process until subperiod 2011-2016, where it
reaches a medium importance.
Graph theory had a medium relevance in subperiod 2000-2003 and will appear again
only in subperiod 2011-2016, with a low significance position (third lowest).
Sales territory alignment appears in an intermediate level in the subperiod 2000-2003
and will reach some positions, to perform in the subperiod 2011-2016 in relevant
position.
TSP was present in subperiod 2000-2003 at the third lowest position; it continues in
the next subperiod (2004-2007), does not appear explicitly in subperiod 2008-2010,
and reappears in subperiod 2011-2016 at the thirteenth lowest position. Despite the
low positions, TSP was widely used since subperiod 2000-2003 for solving routing
problems, and is well considered to this day.
Districting appears in the subperiod 2000-2003 with low relative significance (seventh
position), soars to first position with presence highlighted in subperiod 2004-2007, and
continues through the subperiods, albeit with falling significance in subperiod 2008-
2010, it reach subperiod 2011-2016 with relative importance.
69
Applications represent a group of keywords encompassing other applications than
home care services. This cluster element comes up in the first subperiod (2000-2003)
at the lowest position, oscillates up and down through the other subperiods up to the
last one (2011-2016) where arrives at a low level. It means that the mathematical
models were applied to many industries over the time.
Tabu search places as a strong element (second) in the subperiod 2004-2007 and is
associated to districting, the most relevant node in the cluster. The first article
mentioning Tabu search within the research field appeared in 2003 (BLAIS;
LAPIERRE; LAPORTE, 2003), and the subject grew in importance in the next
subperiod (2004-2007).
VRP was first treated in this research in a secondary but well cited document
(BLAKELEY et al., 2003) in the subperiod 2000-2003. It appears with a moderate
presence in the next subperiod (2004-2007), where routing problem was a relevant
issue. VRP will reach some positions, to perform in the subperiod 2011-2016 in an
intermediate level.
Time windows appears with a moderate element (third lowest position) in the cluster
related to subperiod 2004-2007, but with a strong relation to VRP (see section 3.2.2).
This is evidence that vehicle routing problem with time windows (VRPTW) was subject
of interest and well discussed in that subperiod. Time windows will grow up in
importance and reach subperiod 2011-2016 at tenth highest position.
Although many new elements come up in the cluster related to subperiod 2008-2010,
some do not persist in the next subperiod (2011-2016). This category includes set
partitioning problem, branch-and-price, planning, column generation, simulated
annealing and spatial analysis. Scheduling remains in the next subperiod at almost the
same level. Financing will deep fall to the fourth lowest position. Assignment problem
will fall some positions reaching 2011-2016 at the eleventh lowest level. Clustering will
decrease a little in the next subperiod. Metaheuristics appears in an intermediate
position in this subperiod but will soar to fourth position in subperiod 2011-2016. GIS
(geographic information system) will go down to the lowest position in the next
subperiod. Integer programming remains at the same low position in 2011-2016.
70
Health services was a motor-theme in the subperiod 2008-2010 (see section 3.2.3)
and highlighted other themes, such as districting and simulated annealing. Health
services appears as a moderate node in this subperiod and will soar to eighth position
in the next subperiod.
Optimization and Multi-criteria will departure from subperiod 2008-2010 at a low
position and reach the next subperiod in an intermediate level.
Finally, in the subperiod 2011-2016 modelling, operations research, routing problem,
and metaheuristics are the four most relevant themes. These four elements—with
great synergy—are followed in significance by GRASP, computer science, districting,
and health services. Apparently, the concern still pertain to the search for optimised
solutions to solve logistics problems such as districting and staff routing related to
health care operations. For instance, Lanzarone, Matta, and Sahin (2012) stated that:
(1) Lahrichi et al. (2006) had reviewed the proposed solution of Blais, Lapierre, and
Laporte (2003) and solved a possible district overload problem; and (2) once districting
process is done, the staff assignment can be carried out. This fact highlights the
concern with the assignment problem, since the districting problem having been
solved.
We must remember that the territory alignment problem may be divided into four
issues: (1) location of the territory centre; (2) allocation of teams in basic units; (3)
districting; and (4) routing the teams in each basic unit. Considering that issues (1) and
(2) generally are covered by the Facility Location theme, we can focus our attention on
three themes: Facility Location, Districting, and Routing Problem. Therefore, and on
the basis of the above discussion, we have summarised these themes for each
subperiod with the most cited documents and the mathematical model used in the
Table 2.4.
2.5 CONCLUSION AND FUTURE WORK
Considering the four issues of the territory alignment problem (location, allocation,
districting, and routing), we conclude that although all of them already have good
solutions for home health care applications, districting and routing remain a concern
for optimisation.
71
For issues related to location and allocation, Kalcsics et al (2005), for example,
presented a methodology introducing a heuristic based on geometric ideas. This
heuristic is quick compared to the location-allocation (loc-alloc) approach with the
heuristic AllocMinDist (KALCSICS et al., 2002b), and the resulting territories are
balanced compared to location-allocation with TRANSP (exact solution) and split
resolution with heuristic AssignMAX (SCHRÖDER, 2001).
Regarding the issue of districting, according to (LANZARONE; MATTA; SAHIN, 2012),
Lahrichi et al. (2006) presented a good solution in their review of the procedure of
Blais, Lapierre, and Laporte (2003). The approach of these authors, using the
metaheuristic Tabu search applied to the problem of home care districts, may be used
in any instance involving a community health clinic. The implementation of this
technique proved to be quite feasible; therefore, its application is indicated in real
cases of health clinics, in conjunction with appropriate surveys of operational data on
their territory, basic units and team sizes, in order to ascertain suitability. In addition,
with the improvement suggested by Lahrichi et al. (2006) the districting problem seems
to have a good solution for home care applications.
Once the districting phase has been accomplished with the proper partitioning of the
territory into districts, a routing phase of the visitation teams should be implemented.
At this stage should be further considered (1) the differentiation of patients; (2) whether
the patient needs to be visited more frequently; and (3) whether the visiting time should
be longer or shorter. Therefore, heuristics with time windows (CORDEAU; LAPORTE;
MERCIER, 2004; PRIVÉ et al., 2006; YANIK; BOZKAYA; DEKERVENOAEL, 2014) or
another similar scheme should be added to the model.
After the analysis of social networks of authors, longitudinal analysis of co-words, and
observation of the evolution of themes over time, the theoretical gap vis-à-vis a future
model can be seen in terms of three aspects: (1) the choice of the best heuristics for
solving routing problems related to home care services; (2) the best approach to
scheduling/scaling routing of health teams visiting patients at home; and (3) optimal
integration with a graphical interface, represented by a geographic information system.
72
Table 2.4. Themes and mathematical models
Themes Highlighted in subperiods
Main citations – times cited Mathematical models
Facility Location
2000-2003
1. Kalcsics, Nickel, and Puerto (2003) – 18 2. Kalcsics et al. (2002a) – 36 3. Francis, Lowe, and Tamir (2000) – 40 4. Zoltners, Lorimer, and Sally (2000) – 45
1. algorithm, finite dominating sets 2. same as (1) above 3. location model, demand point aggregation 4. algorithm, sales territory alignment
2008-2010
1. Bozkaya, Yanik, and Balcisoy (2010) – 13 2. Kalcsics et al. (2010) – 13 3. Hinojosa et al. (2008) – 59 4. Hu, Ding, and Shao (2009) – 25 5. Mu and Wang (2008) – 65
1. hybrid heuristic, Genetic algorithm, location-routing, GIS 2. MIP, discrete location 3. integer programming, Lagrangian approach 4. Immune co-evolutionary algorithm, VRP, GIS 5. modified scale-space clustering, GIS
2011-2016 1. Carello and Lanzarone (2014) – 12 2. Al-Nory et al. (2014) – 15 3. Lanzarone, Matta, and Sahin (2012) – 21
1. cardinality-constrained assignment model, home care 2. DSS, Graph theory, location 3. mathematical programming, stochastic patient demand, home care
Districting
2000-2003 1. Muyldermans, Cattrysse, and Oudsheusden (2003) – 17 2. Blais, Lapierre, and Laporte (2003) – 43
1. heuristic, capacitated arc routing 2. algorithm, Tabu search, home care
2004-2007 1. Caro et al. (2004) – 35 2. Haugland, Ho, and Laporte (2007) – 40
1. integer programming, GIS 2. stochastic programming, VRP, Tabu search
2008-2010 1. Fernandez et al. (2010) – 9 2. Rios-Mercado and Fernandez (2009) – 43 3. Skiera and Albers (2008) – 22
1. heuristic, GRASP, geographically dispersed territory design 2. Reactive GRASP, territory design 3. algorithm, sales territory alignment
2011-2016
1. Bozkaya et al. (2011) – 9 2. Benzarti, Sahin, and Dallery (2013) – 14 3. Duque, Anselin, and Rey (2012) – 22 4. Tong and Murray (2012) – 24
1. algorithm, Tabu search, GIS 2. mixed-integer programming,, home care 3. heuristic, MIP, clustering 4. spatial optimization, GIS
Routing Problem
2000-2003
1. Blais and Laporte (2003) – 16 2. Clossey, Laporte, and Soriano (2001) – 16 3. Laporte and Palekar (2002) – 13 4. Blakeley et al. (2003) – 27
1. generalised TSP, graph transformation 2. heuristic, TSP, turn penalties 3. clustered TSP 4. periodic VRP, automated route-scheduling, GIS
2004-2007
1. Cordeau, Laporte, and Mercier (2004) – 31 2. Privé et al. (2006) – 31 3. Gendreau, Laporte, and Semet (2006) – 57
1. site-dependent VRP, time windows, Tabu search 2. heuristics, heterogeneous VRP, time windows 3. integer programming, dynamic coverage, relocation
2011-2016
1. Yanik, Bozkaya, and Dekervenoae (2014) – 5 2. Cappanera and Scutella (2015) – 5 3. Lei, Wang, and Laporte (2016) – 2
1. CVRPMPDTW, time windows, Genetic algorithm, Hybrid heuristic 2. integer linear programming (ILP), Skill VRP, home care 3. enhanced multi-objective evolutionary algorithm (MOEA)
73
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3 2ND PAPER: THE HOME CARE DISTRICTING PROBLEM: AN APPLICATION
TO FAMILY CLINICS
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THE HOME CARE DISTRICTING PROBLEM: AN APPLICATION TO FAMILY
CLINICS
ABSTRACT
There are five major issues associated with the provision of home health care services:
facility location, districting, assignment, scheduling, and routing problems. This article
investigates methods to solve home health care districting problems in order to
improve the management of Family Clinic operations. Family Clinic is a community
health care centre which comprises a network of clinics staffed by a group of general
practitioners and nurses providing primary health care services in a certain area of
Brazil. The most relevant authors on this subject and the techniques most utilised to
solve the districting issue are identified through a Social Network Analysis of the
authors. Finally, one of these techniques is applied to analyse its suitability for the
(real) case in question, and paths are suggested to resolve territory alignment
problems in the implementation of Family Health strategies.
Keywords: metaheuristics; districting; health care; territory alignment; family clinics;
algorithm
3.1 INTRODUCTION
Primary health care is intended mainly for promoting health and preventing disease,
as well as monitoring chronic non-transmittable diseases, controlling endemic
diseases and zoonoses by combating disease vectors, and health surveillance. The
public Unified Health System in Brazil (SUS) has organised a primary health care
network through the so-called Family Health Strategy, which involves the three
spheres of government (federal, state and municipal). The responsibility of
implementing this strategy, and with it, effectively creating the necessary operational
infrastructure was delegated to the municipalities. In the city of Rio de Janeiro, primary
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health care units are referred to as Family Clinics (RIO DE JANEIRO, 2015c) and they
operate in a specific region, providing care for up to 24,000 people.
In practice, the current districting method for serving Family Clinic regions does not
consider service quality. Family Clinic (district) Teams (FCT) are defined by only taking
into account the number of households to be served (RIO DE JANEIRO, 2015b, 2015c,
2017a, 2017b; FAUSTO; FONSECA, 2013; BRASIL, 2012; ANVISA, 2007). This
number was arbitrarily defined as 1,000 households per FCT, assuming an average of
four people per household. Criteria such as adjacency of service regions, workload
(given the difficulty of access to some remotely located or “informal” households), and
household compactness (a measure of density) are not currently considered. There is
therefore a problem of territory alignment, as the territory needs to be divided into
subareas (in a process called districting), each covered by an FCT so that each
Community Health Agent (CHA) belonging to an FCT has comparable workloads.
Furthermore, since there is currently no systematic (computer-based or otherwise)
application to plan and try to optimise visits in a logical way, the solutions obtained are
hardly optimal. Aiming at improving the aforementioned situation, we set out to develop
mathematical models to enhance the process of household visits of CHAs. Such
models include the processes of districting the territory as well as the routing of the
CHAs or health teams when applicable. This paper proposes a solution for the former
(the districting issue) of the stated overall problem.
We investigate, by systematically analysing research social networks, the most
appropriate methods to be adopted for solving territory alignment problems that could
eventually be applied to the Family Clinics case. The sequence of the article is divided
as follows: first, we present the characterisation of the territory alignment problem.
Then, the relevant literature is critically analysed based on a social network of authors.
Academic communities that have utilised similar objectives and/or techniques to solve
problems similar to Family Clinics are also identified. In the fourth section, we develop
an algorithm based on one of the identified techniques, which was successfully applied
at a local community health centre in Quebec, Canada, with the aim of testing the
feasibility of its application to the case at hand. The final section presents our
concluding thoughts and some ideas for future research.
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3.2 THE FAMILY CLINICS AND THE TERRITORY ALIGNMENT PROBLEM
The routine of a family clinic is dependent on the daily visits made by the CHAs to
households within a designated territory. During these visits, health workers ask for
patient vitals (typically those with diabetes and high blood pressure), collect possible
exam requests and help schedule future medical appointments.
Each clinic typically has six teams, each of which includes one physician (usually a
general practitioner), one registered nurse, one nurse assistant, six CHAs, and one
sanitary surveillance agent. Each FCT is responsible in average for serving 3,450
people, but this number may reach 4,000 people (RIO DE JANEIRO, 2015c, 2017b).
Currently, there are 109 Family Clinics in operation in the city of Rio de Janeiro, within
67 of the city’s neighbourhoods. This represents 67.25% of the total service area to be
covered, according to data from the Rio City government (RIO DE JANEIRO, 2017b).
3.2.1 THE TERRITORY ALIGNMENT PROBLEM OF FAMILY CLINICS
There are five main problems related to the provision of home health services (SAHIN;
VIDAL; BENZARTI, 2013). The first is optimising health care facility location. The
second one comprises partitioning each territory into appropriate districts. After
partitioning the territory, the different resources must be assigned to regions in a more
equitable way. After that, scheduling and routing the field teams can then be defined.
Companies that work with sales representatives or service providers usually define
their operations by dividing up the geographic territory where they intend to work within
a region. These companies’ sales or service teams must be appropriately dimensioned
and distributed throughout the territory so that each member of the team has a
comparable workload. The task of optimising the team’s allocation requires that the
territory be divided into regions (also called “districts”) and that each region be made
up of basic units (sub-regions) to be covered by a member of the sales or service team.
In management science research, this is known as alignment or realignment of sales
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or service territories (HESS; SAMUELS, 1971; ZOLTNERS; SINHA, 1983, 2005;
ZOLTNERS; LORIMER, 2000). More recently, this has been called “territory design”
in both the operations management and operations research literature (RÍOS-
MERCADO; FERNANDEZ, 2009; FERNANDEZ et al., 2010; SALAZAR-AGUILLAR;
RÍOS-MERCADO; GONZÁLEZ-VELARDE, 2012; LÓPEZ-PÉREZ ; RÍOS-
MERCADO, 2013).
We believe that adopting the districting approach used in service and sales
organisations into home health care (HHC) services would allow for the improvement
of patients’ care quality, as well as the improvement of caregivers’ work conditions.
With patients grouped in one district, the responsiveness of health professionals would
increase, and in succession would lead to more satisfied patients. Moreover, this
procedure allows for the decrease of unproductive travel time and, therefore, the
increase of productive time (time spent on patient care) besides improving the
efficiency of the care delivery process.
An additional relevant impact of considering the districting procedure is the
improvement it may cause in working conditions for health professionals. The
districting procedure strives for the equilibrium of workload between districts, and this
would result in a more levelled workload among the various teams. Furthermore, since
each district will be under the responsibility of a single team, a more intimate
relationship will develop between caregivers and patients, with consequent increase in
the quality and resolution of care (BENZARTI; SAHIN; DALLERY, 2013).
Various mathematical programming and heuristics models can be used to determine
the size of service teams and their allocation per territory. Some examples are the
seminal paper from Hess and Samuels (1971) and Zoltners and Sinha (1983). For
solving territory design problems, Kalcsics, Nickel, and Schröder (2005) propose some
"building blocks" of a basic model:
Basic units: A territory is comprised of a set 𝑉 of basic units, which are
geographic objects in the plan: points (addresses), lines (sections of streets), or
geographical areas (ZIP codes etc.), which are usually given as polygons.
Several attributes called measurements of activity are associated with each
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basic unit. Instances are workload for services or number of visits to customers
within a region, or number of inhabitants.
Territory centres: In general, one centre is associated with each territory.
Generally the territory’s centre coincides with the centre of one of the basic units
that is part of the territory. Therefore, 𝑉𝑐 ⊂ 𝑉 is the set of territory centres. These
centres can be pre-determined and fixed or subject to planning.
Number of territories: It is given a priori and is expressed by 𝑝.
Single designation of basic units: It is mandatory that each basic unit be included
in one and only one territory. For this reason, the territories define a partition of
the set 𝑉 of basic units. If 𝐵𝑖 ⊆ 𝑉 denotes the i-th territory, then
𝐵1 ∪ ⋯ ∪ 𝐵𝑝 = 𝑉 (3.1)
and
𝐵𝑖 ∩ 𝐵𝑗 = ∅ , 𝑖 ≠ 𝑗 (3.2)
Balancing: The activity in a territory should be balanced. The total measurement
of activity of the basic units contained in them is given by:
𝑤 (𝐵𝑖) = ∑ 𝑤𝑣𝑣∈𝐵𝑖 (3.3)
which is the size of 𝐵𝑖.
Adjacency: Explicit information of the neighbourhood for basic units is
necessary in order to obtain adjacent districts.
Compactness: The model considers two methods for the attribute of
compactness. First, minimising the weighted total distance ∑ ∑ 𝑤𝑣𝑣∈𝐵𝑖𝑑𝑖𝑣
𝑝𝑖=1
(Euclidean, Euclidean square, or based on network) from the centres of the
districts to the basic units. Second, a geometric approach derives a measure of
compactness based on convex hulls.
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Objective: Partitioning set 𝑉 of basic units into a number 𝑝 of territories that meet
the specific criteria of planning such as balancing, compactness, and adjacency.
3.3 LITERATURE REVIEW AND SOCIAL NETWORK ANALYSIS
Social Network Analysis (SNA) has become one of the contemporary popular research
methods often used in bibliometric studies (GOMES; KLIEMANN NETO, 2015;
POZZEBON; DELGADO, 2015). By using SNA, researchers examine the structure of
researcher communities, attempt to describe the structures of researchers’ networks,
and model the existing connections to then identify the relations between communities
(NEWMAN, 2001).
As seen in Figure 3.1, the methodology used to structure and enable the analysis of
social networks is an integrative review of the literature (COOPER, 1984; GANONG,
1987; BROOME, 2000) with the following research question: Which mathematical
programming models would be best suited to solve the territory alignment problem in
health care operations?
Figure 3.1. Research model
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Articles were included about mathematical programming and heuristics models, but
dissertations, theses, and editorial notes were excluded as well as books and annals
from congresses. No date limits were established for searching the databases and the
resulting articles had publication dates from 1963 to 2017. Detailed reading would only
be performed at the end of the social network analysis.
JSTOR and Web of Science were the databases selected for carrying out the literary
search, because they are main journals related to Operations Research, Management
Science, and Decision Sciences. An initial search was performed from August 18 to
September 3, 2015 in both databases. An update of the search in the databases was
performed on February 7, 2017, and one last revision was made on September 29,
2017.
The keywords and search logic used were: (districting problem AND home health care
services AND mathematical modelling AND operations management) OR (districting
AND community health clinic AND multi-criteria optimization) OR (heuristics AND
optimization AND logistics AND territory design).
In the first search, 2,106 articles were found in the JSTOR database and 427 in Web
of Science, totalling 2,533 articles; both databases also search other databases such
as Emerald, for example. Sixteen articles were added in the first update, totalling
2,549. Eleven articles were added in the most recent update, totalling 2,560. The
organisation of the articles with the necessary actions of export, import, classification,
and filing was performed with the aid of the tools EndNote X7 and SciMAT v1.1.03.
Once the sample was defined, the metadata available in the databases were extracted
such as authors, title, year of publication, abstract, keywords, citations, references,
among others.
3.3.1 SOCIAL NETWORK ANALYSIS OF AUTHORS – 1963 TO 2017
For the social network analysis of authors, of the 2,560 articles found in the literature
search, those published in journals of Econometrics and Marketing were excluded
because, although they used heuristic and exact methods of linear programming, they
88
were not aligned with our research purpose. Our resulting sample was then 2,327
articles with 4,020 authors. The aim of the exercise was to identify the most relevant
authors that could contribute to solving the territory alignment problem. Figure 3.2
shows the social network of authors with their most relevant nodes from 1963 to 2017,
which was obtained when we considered publications with two or more co-authors. We
used the VOSviewer software version 1.6.4 for the construction and representation of
the bibliometric maps.
Considering a network with up to 500 items, 350 connected items and 47 clusters were
found, based on 715 links. The main node of each cluster is represented by the author
with the highest number of published articles, with the number of co-authors as
tiebreaker. Table 3.1 shows a sample of: the number of articles published by authors
(greater than or equal to 10) in the period 1963-2017; co-authorship; cluster
identification; and number of items per cluster. Table 3.1 considers overlapping
communities. Most of the clustering methods aim at detecting standard partitions, i. e.
partitions in which each vertex is assigned to a single community. However, in real
graphs vertices are often shared between communities, and the issue of detecting
overlapping communities has become quite popular in the last years (FORTUNATO,
2010). The present work uses software package VOSviewer version 1.6.4 that
implements techniques to detect overlapping communities.
Based on Figure 3.2 and Table 3.1, the most important node of the social network of
authors refers to Gilbert Laporte with 402 articles published. Other important authors
with a strong connection with Laporte in relevant publications on the topic were Marko
Blais, Burcin Bozkaya, Frederica Ricca, Bruno Simeone, and Jörg Kalcsics.
89
90
Figure 3.2. Social network of authors, 1963-2017
91
Table 3.1. Productivity of authors, 1963-2017
Order Author Documents Co-authorships Cluster ID (Items)
1 Laporte, G. 402 634 9 (11)
2 Cordeau, J. F. 69 145 9 (11)
3 Gendreau, M. 67 128 16 (10)
4 Semet, F. 27 49 26 (6)
5 Ricca, F. 24 50 17 (9)
6 Kalcsics, J. 21 46 11 (10)
7 Eiselt, H. A. 19 27 32 (3)
8 Nickel, S. 19 45 11 (10)
9 Labbe, M. 17 33 13 (10)
10 Bertsimas, D. 15 6 1 (20); 5 (16)
11 Hall, N. G. 15 20 20 (9)
12 Nobert, Y. 15 24 25 (6)
13 Scozzari, A. 15 34 17 (9)
14 Bard, J. F. 14 9 1 (20)
15 Coelho, L. C. 14 23 22(8)
16 Puerto, J. 14 34 17 (9)
17 Boctor, F. F. 13 28 31 (3)
18 Crainic, T. G. 13 13 10 (11)
19 Federgruen, A. 13 12 15 (10); 37 (2)
20 Renaud, J. 13 28 31 (3)
21 Bektas, T. 12 23 24 (6)
22 Powell, W. B. 12 2 5 (16)
23 Simeone, B. 12 29 17 (9)
24 Vigo, D. 12 21 23 (6)
25 Desarbo, W. F. 11 13 2 (20)
26 Fisher, M. L. 11 15 -
27 Ghiani, G. 11 18 18 (9)
28 Mesa, J. A. 11 25 22 (8)
29 Potts, C. N. 11 11 20 (9)
30 Simchi-Levi, D. 11 15 1 (20)
31 Weintraub, A. 11 11 14 (10)
32 Balakrishnan, A. 10 10 10 (11)
33 Hertz, A. 10 17 38 (2)
34 Magnanti, T. L. 10 12 10 (11)
35 Mercure, H. 10 18 25 (6)
36 Sherali, H. D. 10 0 -
92
3.3.2 RESULTS FROM THE LITERATURE REVIEW AND SOCIAL NETWORK
ANALYSIS
Table 3.2 summarises in chronological order the home care articles found in literature
review and social network analysis with the most cited authors and article information,
such as decision type, scope, and modelling approach/solution method.
Table 3.3 shows a cut-out of home care districting articles, where we find more detailed
information: citations, particular input/setting, objective function, decision variable,
constraints, basic districting model, modelling approach, solution method and, case
study.
As seen in Figure 3.3, although there has been an evolution of HHC services in both
developed and developing countries given the relevance of these services, the quantity
of research papers concerning this subject within operations management is still
relatively small (26 articles found). These findings are also reinforced by Sahin and
Matta (2015) and Benzarti, Sahin, and Dallery (2013). Most of the research works on
home care deals with the problem of assigning health professionals to patients or the
problem of team routing. Assessing factors that may generate complexity in managing
operations was another issue focused on by many authors. The districting problem in
home care operations was mainly addressed by four group of authors: Blais, Lapierre,
and Laporte (2003); Benzarti, Sahin, and Dallery (2013); Gutiérrez and Vidal (2015);
and Lin et al. (2017).
93
Table 3.2. Home care articles, 1963-2017
Citation Decision type Scope Modelling Approach / Solution Method
Sachidanand, Miller, and Weaver (1997)
Scheduling and routing
Integrated spatial DSS for scheduling and routing home health care nurses
GIS and Clarke and Wright (1964) heuristic with improvements
Blais, Lapierre, and Laporte (2003)
Districting Districting procedure for a local community health clinic
Multi-criteria approach by using algorithm and metaheuristic Tabu search
Chahed et al. (2009) Scheduling and
routing
Production-delivery (drug supply chain) related to chemotherapy at home
Exact solution with Integer Programming using branch and bound algorithm
Hertz and Lahrichi (2009)
Assignment Patient assignment algorithm for home care services
Mixed-integer programming (MIP) and metaheuristic Tabu search
Lanzarone, Matta, and Sahin (2012)
Assignment Assigning human resources to patients with stochastic demand
Mathematical programming models for different types of home care service providers
Benzarti, Sahin, and Dallery (2013)
Districting Optimising compactness and workload balance criteria.
Mixed-integer programming (MIP) model based on multi-criteria assumptions
Gutiérrez and Vidal (2013)
Logistics Critical review of models and methods used to support logistics decisions
Framework for planning, management decisions, and services processes
Sahin, Vidal, and Benzarti (2013)
Assessment Assessing factors that may generate complexity in managing operations
Framework identifying complexity factors in home care organisations
Fanti and Ukovich (2014)
Modelling
Review about approaches to model and simulate healthcare systems
Discrete event systems models and methods for problems in healthcare management
Gutiérrez et al. (2014) Assessment Logistics management diagnosis for the home care institutions in Colombia
Evaluation of six work axes and the maturity level of service processes
Holm and Angelsen (2014)
Assessment
Describes how nurses and health workers spend their working time in Norway
Descriptive retrospective study of day-schedules and driving routes for staff, using GIS
Matta et al. (2014) Modelling Describes the most relevant processes associated with home care operations
IDEF0 (Integrated Definition for Function Modelling) activity-based model
Aiane, El-Amraoui, and Mesghouni (2015)
Scheduling and routing
Optimising routes and rosters for staffs, satisfying specific constraints
Mixed-integer linear programming (MILP) model using ILOG/CPLEX
Bastos et al. (2015) Routing
Web based application for optimisation of home care professionals visits
Clarke and Wright (1964) heuristic and a variation using a second order heuristic
Cappanera and Scutellà (2015)
Assignment, scheduling, and
routing
Integrated pattern-based approach to optimise home care services
Integer linear programming (ILP) with two balancing functions maxmin and minmax
En-Nahli, Allaoui, and Nouaouri (2015)
Assignment and routing
Effective feasible working plan for each resource on a daily basis
Multi-objective approach based on a mixed-integer linear programming (MILP)
94
Table 3.2. Continued
Citation Decision type Scope Modelling Approach / Solution Method
Gutiérrez and Vidal (2015)
Districting Context of a rapid-growing city, which results in problems for the population
Bi-objective Mixed-integer programming (MIP) solved with lexicographic approach
Rodriguez et al. (2015) Staff
dimensioning
The amount of personnel required to activities with uncertain demands
Two-stage approach based on integer linear stochastic programming
Sahin and Matta (2015) Assessment
Characterisation of home care operations and their decision-making models
Interviews and international review of the literature related to home care operations
Castillo-Salazar, Landa-Silva, and Qu (2016)
Scheduling and routing
Scenarios of workforce scheduling and routing problems (WSRP)
Mathematical programming and benchmark computation times using the Gurobi solver
En-Nahli et al. (2016) Routing
Vehicle Routing Problem (VRP) with time windows and synchronisation
Iterated local search (ILS) with a variant of Random Variable Neighborhood Descent method
Hewitt, Nowak, and Nataraj (2016)
Planning and routing
Examines appropriate planning horizon length and the routing cost of planning
Consistent VRP (ConVRP) with Stochastic Customers solved using algorithms
Lin et al. (2016) Assignment Therapist assignment with time periods selections and weight allocations
MIP model with linear objective function and quadratic constraints
Redjem and Marcon (2016)
Scheduling and routing
Patients receiving multiple caregivers with precedence and coordination constraints
Caregivers Routing Heuristic (CRH) tested using several instances
Yalcindag et al. (2016) Assignment
Assignment of patients to care givers taking into account travel times
Kernel regression technique using the travel times observed from previous periods
Lin et al. (2017) Districting Meals-On-Wheels service districting (MOWSD) for home care in Hong Kong
Integrated mixed-integer programming (MIP) model solved by a greedy heuristic
95
Table 3.3. Home care districting articles, 1963-2017
Citation (Times cited)
Particular input/ setting
Objective function
Decision variable
Constraint Basic districting model
Modelling approach
Solution method
Case study
Blais, Lapierre, and Laporte (2003)
(43)
Travel time / distance between basic units
Workload / number of visits per basic unit
Relation between the total travel time and the total workload
Admissible percentage deviation of the average workload
Minimise total travel distance / time (mobility)
Minimise workload unbalance (equilibrium)
Basic units per district
Indivisibility of basic units
Respect of borough boundaries
Contiguity
The home care districting problem
Multi-criteria optimisation model
Algorithm and metaheuristic Tabu search
Community health clinic (CLSC) of Côte-des-Neiges, Quebec, Montreal, Canada
Benzarti, Sahin, and Dallery (2013)
(14)
Number of patients' profiles
Number of visits required by a patient of each profile
Average duration of a visit relative to each profile
Number of patients living in each basic unit
Distance between basic units
Maximum distance allowed between two basic units in the same district
Average care workload among all districts
Admissible percentage deviation of the average workload
Model 1: balance care workload (visit time)
Model 2: minimise compactness (distance between basic units in the same district)
Basic units per district
Total care workload per district
Maximum deviation of care workload
Accessibility
Model 1: compactness (consider travel time)
Model 2: allowable care workload within each district
The home care districting problem
Mixed-integer programming (MIP)
CPLEX Not applied on a real case
96
Table 3.3. Continued
Citation (Times cited)
Particular input/ setting
Objective function
Decision variable
Constraint Basic districting model
Modelling approach
Solution method
Case study
Gutiérrez and Vidal (2015)
(0)
Sets of basic units assigned to one and only one district
Set of different types of medical activities
Set of different types of patients
Set of type of medical staff
Number of annual visits required in each basic unit for each medical activity, for each type of patient
Service time for each medical activity and each type of patient
Time to travel between basic units
Security levels in each basic unit
Minimise the total distance travelled by the medical staff in each district
Minimise the sum of the total workload deviations from the average workload
Inclusion of each pair of basic units into each district
Assignment of each individual basic unit to a district
Lower and upper deviations of the workload of each district
Each basic unit is assigned to only one district
Visit, travel, and total workloads for each district
Average workload
The home care districting problem in the context of a rapid-growing city
Bi-objective Mixed-integer programming (MIP)
Lexicographic approach
Real data instances from a home care institution which delivers services in the largest cities in Colombia
Lin et al. (2017)
(0)
Sets of basic units visited by walking or by driving
Average number of meal packages to be delivered to each basic unit
Estimated workload of serving at each basic unit (service time plus travel time)
Available travel (driving or walking) mode to visit each basic unit
Walking duration between basic units
Driving duration between basic units
Driving duration between depot and basic units
Minimise the total number of districts created according to operational factors
Basic units per district
Estimated driving duration from depot to district
Compactness
Indivisibility of basic units
Capacity limitation (maximum number of meal packages)
Delivery time period limitation (customers prefer hot meals)
Meals-On-Wheels service districting (MOWSD)
Mixed-integer programming (MIP)
Greedy heuristic method
Salvation Army-Tai Po Integrated Home Care Service Centre (SA-TPIHCS) in Hong Kong
97
Figure 3.3. Results from literature review and SNA
3.4 A POSSIBLE SOLUTION
A community health care clinic belongs to a network of clinics staffed by a group of
general practitioners and nurses providing primary health care services to people in a
certain geographic area.
The Blais, Lapierre, and Laporte (2003) model is the one that best suits the case of
community clinics because it does not differentiate between types of patients or types
of health professionals for their care. The varying patient health conditions can be
considered simply in the scheduling and routing stages of managing the health teams.
The Benzarti, Sahin, and Dallery (2013) model considers different patient profiles, each
requiring a different length of care. Based on the preferences that HHC managers
have, the model favours either the compactness or care workload balance criteria.
98
Gutiérrez and Vidal (2015) addresses the home care districting problem in the context
of a rapid-growing city with its idiosyncrasies (security level, for instance). The model
applies to other situations of home care services that require the care of specialist
physicians and specialized health teams.
The Lin et al. (2017) model refers to a specific case: the Meals-On-Wheels service
districting (MOWSD) problem. In this model, meal packages are delivered by driving
and/or by walking.
Blais, Lapierre, and Laporte (2003), which was the first publication on districting in
community health clinics, describes a districting study for the community health clinic
(CLSC) of Côte-des-Neiges in the province of Quebec in Montreal, Canada. One
territory needs to be divided into six districts by clustering their basic areas of service.
Five district criteria must be observed: indivisibility of basic units, respecting the limits
of adjacencies, connectivity, mobility of visitation staff, and workload balance. The last
two criteria are fused into a single objective function and the solution is found using
Tabu search. The districting solution was satisfactory according to clinic management
statements after two years of implementation.
Blais, Lapierre, and Laporte (2003) built on the modelling of Bozkaya, Erkut, and
Laporte (2003) and considered the five constraints (indivisibility of basic units, respect
for neighbourhood boundaries, connectivity, mobility and workload balance) with the
first three being normal restrictions and the other two in a weighted bi-objective
function:
𝑓(𝑠) = ∝ 𝑓1(𝑠) + (1−∝)𝑓2(𝑠) (3.4)
where 𝑓1(𝑠) and 𝑓2(𝑠) evaluate the degree of mobility and the workload balance of
solution 𝑠, respectively, and α is a control parameter in the interval [0, 1]. The degree
of mobility of the solution 𝑠 is evaluated as
𝑓1(𝑠) = ∑ (∑ 𝑣𝑖𝑣𝑗𝑑𝑖𝑗𝑖,𝑗∈𝐷𝑘,𝑖<𝑗 )𝑚𝑘=1 / [(𝑛𝑘(𝑛𝑘 − 1)/2)(∑ 𝑣𝑖𝑖∈𝐷𝑘 )
2] (3.5)
where 𝑚 is the number of districts, 𝑖 and 𝑗 are the basic units of district 𝐷𝑘, 𝑑𝑖𝑗 is the
distance (or travel time) between the centres of the basic units 𝑖 and 𝑗 either using
99
public transportation or walking, 𝑛𝑘 is the number of basic units of 𝐷𝑘, and 𝑣𝑖 is the
number of visits made to unit 𝑖.
In Equation 3.5, the numerator calculates for each 𝑘 the total distance travelled within
the district 𝐷𝑘 each analysed period of time. The denominator is a scale factor, where
(𝑛𝑘(𝑛𝑘 − 1)/2) represents the quantity of pairs (𝑖, 𝑗) in the numerator. The lower the
value of 𝑓1(𝑠) the greater the degree of mobility within the district or territory.
The function of a balanced workload is given by
𝑓2(𝑠) = (∑ 𝑚𝑎𝑥{𝑊𝑘𝑚𝑘=1 − (1 + 𝛽)�̅�, (1 − 𝛽)�̅� − 𝑊𝑘 , 0}) / �̅� (3.6)
where 𝑊𝑘 is the workload in district 𝑘. In order to calculate 𝑓2(𝑠), the authors use a
piecewise linear function that considers that the workload has higher penalty if it ends
up being outside the interval [(1 − 𝛽) �̅�, (1 + 𝛽) �̅�], where �̅� is the average
workload, and 0 ≤ 𝛽 ≤ 1.
The workload 𝑊𝑘 is the sum of 𝑉𝑘 (total visit time) and 𝑇𝑘 (total travel time) in district 𝑘
in a given period (e.g. one year). Indeed, the figures 𝑊𝑘, 𝑉𝑘, and 𝑇𝑘 are dependent on
the districting solution and in turn total travel time ∑ 𝑇𝑘𝑚𝑘=1 and the total visit time ∑ 𝑉𝑘
𝑚𝑘=1
in the territory are also dependent on the solution. In other words, depending on the
solution, 𝑇𝑘 can be decreased by an optimum grouping of customers and optimum
planning of community worker visits. 𝑉𝑘 is also dependent on the solution because less
time spent on the trip permits more time in the visit.
The authors considered historical data for approximation ∑ 𝑉𝑘𝑚𝑘=1 and used a parameter
𝜆 for the relation between the total travel time 𝑇𝑘 and the total workload 𝑊𝑘 equal to
18 per cent.
3.4.1 ALGORITHMS IMPLEMENTED
Blais, Lapierre, and Laporte (2003) chose as a solution the metaheuristics Tabu search
developed by Bozkaya, Erkut, and Laporte (2003) for political districting, but with a
different objective function.
100
From an initial solution, the Tabu search of Bozkaya iteratively goes from one solution
to another in its neighbourhood by doing two kinds of movements: it either moves one
basic unit from its current district to an adjacent district (transferring), or it swaps two
basic units on the border of two different adjacent districts (swapping).
The initial solution can be any solution done by hand, or an iterative solution built using
seed basic units. By using these seeds, the districts are one at a time built by adding
to each step a basic unit adjacent to district 𝑘 that has a lower workload. Accurately,
considering 𝑆(𝑘) as the set of basic units in district 𝑘 and 𝑆′(𝑘) the set of basic units
not designated adjacent to district 𝑘, the basic unit 𝑖∗ is included in district 𝑘∗ if and
only if 𝑘∗ and 𝑖∗ satisfy
𝑚𝑖𝑛𝑘 𝑚𝑖𝑛𝑖 ∈ 𝑆´(𝑘) {𝑔 (𝑖, 𝑘)} (3.7)
and
𝑔 (𝑖, 𝑘) = ∑ 𝑣ℎ𝑣𝑗𝑑ℎ𝑗ℎ,𝑗∈ 𝑆(𝑘)∪{𝑖} (3.8)
In our study, an iterative initial solution was also considered based on the seed basic
units using formulas (3.7) and (3.8). However, as we did not have access to the code
developed by the authors mentioned above, we developed our own algorithm logic and
the application by using R code, from the formulas presented in (3.4) to (3.8). The
metaheuristics Tabu search used was from the tabuSearch package from library R
(DOMIJAN, 2012), which is more complete and developed than the one used by
Bozkaya, Erkut, and Laporte (2003), the latter being restricted to two types of
movement.
3.4.2 COMPARISON OF RESULTS
To illustrate the algorithms, a simple example was used with 10 basic units to be
assigned to two districts (Figure 3.4) presented in Blais, Lapierre, and Laporte (2003).
This example is actually a subset (Districts 1 and 2) of the 36 districts of the Côte-des-
Neiges CLSC with real data and can be considered a simple instance of validation.
101
Figure 3.4. Territory partitioned into 10 basic units
Source: Prepared by the authors through an algorithm developed in R code with an RgoogleMaps package that reads information from Google Maps based on real coordinates, adapted from Blais, Lapierre, and Laporte (2003).
The workload 𝑣𝑖 of each basic unit 𝑖 is given in Table 3.4 and the travel time chart (𝑑𝑖𝑗)
is given in Table 3.5.
102
Table 3.4. Workload 𝒗𝒊 in each basic unit 𝒊
Basic Units Workload
1 79
2 50
3 35
4 113
5 82
6 229
7 357
8 154
9 43
10 74
Source: Blais, Lapierre, and Laporte (2003)
Table 3.5. Travel time 𝒅𝒊𝒋 between basic units 𝒊 and 𝒋
j
i / j 1 2 3 4 5 6 7 8 9 10
1 0 7.4 13.9 17.8 25.2 12.1 7.5 13.6 21.5 21.9
2 7.4 0 7.6 13.8 21.2 19.5 14.9 7.7 14.1 15.5
3 13.9 7.6 0 6.2 13.6 25.9 21.3 12 8.1 7.9
4 17.8 13.8 6.2 0 7.4 29.9 25.3 18.2 14.3 8.2
5 25.2 21.2 13.6 7.4 0 37.3 32.8 25.7 17.6 9.5
6 12.1 19.5 25.9 29.9 37.3 0 4.5 13.8 23.1 31.2
7 7.5 14.9 21.3 25.3 32.8 4.5 0 9.3 18.5 26.6
8 13.6 7.7 12 18.2 25.7 13.8 9.3 0 9.2 17.4
9 21.5 14.1 8.1 14.3 17.6 23.1 18.5 9.2 0 8.1
10 21.9 15.5 7.9 8.2 9.5 31.2 26.6 17.4 8.1 0
Source: Blais, Lapierre, and Laporte (2003)
103
In this simple example, for the initial solution were considered basic units 3 and 9 as
seeds. The adjacent units are step-by-step grouped with the seeds to minimise the
unbalance in workload between the two districts at each iteration, as shown in Table
3.6 (a and b). In this way, in iteration 2, the basic unit 2 is included in District 1 since
this action minimises 𝑔 (𝑖, 𝑘).
In the algorithm developed by Blais, Lapierre, and Laporte (2003), at last of iteration
10, District 1 includes the basic units 3, 2, 1, 4, 7, and 6 whereas District 2 includes
units 9, 10, 5, and 8. In the algorithm developed in this study, the same result is
obtained after only five iterations.
The result of the initial solution by Blais et al. (2003) produces a function 𝑓(𝑠) = 1,193
for ∝ = 0.9, 𝛽 = 0.25, and 𝜆 = 0.18. The algorithm proposed presented the value of 𝑓(𝑠)
= 1.226 for the initial solution considering the same parameters ∝, 𝛽, and 𝜆.
In the algorithm by Blais, Lapierre, and Laporte (2003), the search procedure by
Bozkaya et al. (2003) is then applied to the initial solution 𝑠 with the two movements of
transferring and swapping in order to optimise 𝑓(𝑠). In this way, transferring basic unit
4 to District 2 produces 𝑓(𝑠) = 0.814 whereas swapping basic units 4 and 8 between
Districts 1 and 2 produces 𝑓(𝑠) = 0.815. Of course, the transferring movement is
preferable in this case. The routine continues with 316 (100√10 ) iterations until there
are no further improvements in the objective function.
The two movements indicated by Blais were performed in the algorithm that we
propose, producing respectively 𝑓(𝑠) = 0.814 and 𝑓(𝑠) = 0.818, which shows a good
consistency between the results. However, the Tabu search mechanism used here
was tabuSearch from the library R (DOMIJAN, 2012), which has obtained the same
results with only 40 iterations (Figure 3.5). The optimal solution for the case presented
the basic units 1, 2, 6, 7, and 8 in District 1 and 3, 4, 5, 9, and 10 in District 2 with 𝑓(𝑠)
= 0.674.
104
Table 3.6a. Comparison of algorithms for initial solution
Iteration Basic Units in District k = 1 i ∈ S´ (1) g (i, 1)
Blais et al. (2003) Algorithm Proposed
Blais et al. (2003) Algorithm Proposed
Blais et al. (2003) Algorithm Proposed
1 3 3 - - - -
2 3 3 1 38433.5
2 2 13213 13300.0
4 4 24561 24521.0
5 39032.0
6 207588.5
7 266143.5
8 64680.0
10 10 20513 20461.0
3 3, 2 3, 2 1 1 67735 67663.5
4 4 102305 102491.0
5 125952.0
6 430863.5
7 532108.5
8 8 124417 123970.0
10 77752
4 3, 2 3, 2, 1 1 67735
4 4 102305 261391.6
6 649764.6
7 743631.0
8 8 124417 289427.6
5 3, 2, 1 3, 2, 1, 4 4 261206
6 1423487
7 7 745669 1764258
8 290361
6 3, 2, 1 3, 2, 1, 4, 6, 7 4 - 261206 -
7 - 745669 -
8 - 290361 -
7 3, 2, 1, 4 7 - 1767909 -
8 - 607947 -
8 3, 2, 1, 4 7 - 1767909 -
9 3, 2, 1, 4, 7 6 - 1794634 - 10 3, 2, 1, 4, 7, 6 - - - -
Source: Prepared by the authors, adapted from Blais, Lapierre, and Laporte (2003).
105
Table 3.6b. Comparison of algorithms for initial solution
Iteration Basic Units in District k = 2 i ∈ S´ (2) g (i, 2)
Blais et al.(2003) Algorithm Proposed
Blais et al. (2003) Algorithm Proposed
Blais et al. (2003) Algorithm Proposed
1 9 9 - - - -
2 9 9 1 73035.5
2 30315.0
4 69483.7
5 62057.6
6 227465.7
7 283993.5
8 8 61121 60922.4
10 10 25838 25774.2
3 9 9, 10 1 201062.9
4 138052.1
5 119703.6
6 756180.9
7 986712.3
8 8 61121 259212.8
10 25838
4 9, 10 9, 10, 5
4 4 138192 206620.5
5 119870
6 1456600.3
7 1946899.5
8 8 258842 583752.4
5 9, 10 9, 10, 5, 8 4 138192
5 119870
6 1943271
7 2458195
8 258842
6 9, 10, 5 9, 10, 5, 8 4 - 206853 -
8 - 582876 -
- -
7 9, 10, 5 8 - 582876 -
- -
8 9, 10, 5, 8 7 - 2457547 -
9 9, 10, 5, 8 - - - - 10 9, 10, 5, 8 - - -
-
Source: Prepared by the authors, adapted from Blais, Lapierre, and Laporte (2003).
106
Figure 3.5. Tabu search results of the algorithm proposed with 40 iterations
The proposed algorithm was run for two real instances: Côte-des-Neiges CLSC
territory in Montreal (Figure 3.6); and Assis Valente Family Clinic territory (DIOGO;
ARAÚJO, 2013; RIO DE JANEIRO, 2017a) in Rio de Janeiro (Figure 3.7). Both
territories are partitioned into 36 basic units that should be grouped in districts.
Typically, each district is formed by four to eight basic units depending on the care
workload.
For the case of the CLSC of Côte-des-Neiges we used exactly the same data
considered in the work of Blais, who kindly provided us with his dataset. In our case,
we used public transport for travel distances. The proposed algorithm presented
exactly the same results as the Blais model, however, only 100 iterations were required
against 600 of the previous model. The computation time was only 29.59 seconds for
our model, considering 600 iterations, against 300 seconds of the Blais model.
However, this comparison is not fair, once the computers performance are quite
different (Intel Core i5 M480 @ 2.67 GHz versus Sun Enterprise 10 000 @ 400 MHz).
2 4 6 8 10
10
20
30
No of times selected
variable
2 4 6 8 10
34
56
78
Most frequent moves
variable
0 10 20 30 40
35
79
Sum of included variables
iterations
0 10 20 30 40
0.0
0.5
1.0
1.5
Objective Function
iterations
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Figure 3.6. Real instance: CLSC territory partitioned into 36 basic units
108
Figure 3.7. Real instance: Assis Valente Family Clinic territory partitioned into 36 basic units
109
In the case of Assis Valente Family Clinic, we used the exact geographical coordinates
to calculate travel distances travelled by walking and/or by bus through Google Maps.
As we did not have access to the historical data of the community agent’s visits, we
used an assessed current reality, obtained by interviews with community agents. As
seen in Figure 3.7, the workload distribution is much more uniform in the automatic
solution (reduction of 81.8% in standard deviation). In addition, for travel times, except
for District F (Tubiacanga), which uses buses, all other districts have shown a great
gain in reducing travel times.
3.5 CONCLUSIONS
The work of Blais, Lapierre, and Laporte (2003) best fit the characteristics of a
community clinic among the four districting studies under evaluation.
Our implementation of R code with the local search engine tabuSearch presented
excellent results when compared with the Blais solution. The proposed algorithm
required fewer steps for the Initial Solution, which in several cases is very close to the
Final Solution. The local search mechanism (tabuSearch) is much more sophisticated
than the Bozkaya procedure used by Blais, making the process more efficient and also
contributing to a shorter computing time.
The application in real cases such as the Assis Valente Family Clinic showed
consistency and robustness of the tool. In this scenario, the automatic solution
presented a configuration of the basic units for the districts much better than the
manual solution.
We consider, in our on-going research, the potential results of applying the proposed
model to a real case with historical data. However, we are very confident in the model’s
high performance, as evidenced by our results.
After the districting phase and the appropriate partitioning of the territory into districts,
a routing phase of the visitation teams (CHAs) should be implemented. In this phase,
110
the patient differentiation should still be considered, i.e. if the patient needs to be visited
more frequently or if the visit time needs to be longer or shorter.
With an automated solution to the territory alignment problem, it will be possible to
replace the current precariously intuitive and manual FCT planning procedure, thus
increasing the efficiency and effectiveness of the Family Clinics. Family registration will
take place more quickly considering CHAs will occupy their basic areas of service in
an adequate and balanced way from the start, thereby promoting the clinic’s services,
and starting the routine visitations at the right times and frequency.
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4 3RD PAPER: SCHEDULING AND ROUTING PROBLEM WITH SERVICE
PRIORITY IN PRIMARY HEALTH CARE: A SOLUTION FOR FAMILY CLINICS
IN BRAZIL
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SCHEDULING AND ROUTING PROBLEM WITH SERVICE PRIORITY IN
PRIMARY HEALTH CARE: A SOLUTION FOR FAMILY CLINICS IN BRAZIL
ABSTRACT
Primary health care in Brazil is provided by basic health units, called Family Clinics,
which proactively serve an adjacent territory with up to 6,000 households. Depending
on the level of risk and vulnerability of these families, care workers’ visits should be
more or less frequent, thus setting certain priorities. This work presents a computerised
model for the problem of scheduling and routing community health agents from a
Family Clinic over a service territory. A solution based on Period Vehicle Routing
Problem with Service Priority, here called PVRP-SP, is suggested. An algorithm was
developed in R code to implement the solution method, and a classical heuristic for
Capacitated VRP (CVRP) was used as routing subroutine. Results found in tests with
known benchmark instances, as well as in a real-life case, demonstrated the practical
applicability of the computer model, with values close to optimal and computing times
of a few seconds.
Keywords: routing; scheduling, care workers; primary health care; periodic VRP;
algorithm
4.1 INTRODUCTION
Primary health care in Brazil is provided by basic health units that proactively serve an
adjacent territory with up to 6,000 households (BRASIL, 2012). These basic health
units are spread throughout the country, and in the city of Rio de Janeiro are called
Family Clinics (RIO DE JANEIRO, 2015). Care workers from these clinics, known as
Community Health Agents, are responsible for daily visitation to households in the
territory to initially register the patients of the families in each household, and then
monitor the health condition of these relatives, as well as promote health and prevent
117
disease. Depending on the level of risk and vulnerability of these families, visits should
be more or less frequent, thus setting certain priorities. Currently, the procedures for
scheduling visits and routing of health agents are done manually. Therefore, coverage
is not achieved with monthly visitation of all families, respecting their priorities, in a
satisfactory way. The present work intends to present a computerised model for the
problem of scheduling and routing community health agents from a Family Clinic over
a service territory. A solution based on Period Vehicle Routing Problem (PVRP) with
Service Priority, here called PVRP-SP, is suggested for the case of Family Clinics.
In section 4.2, a knowledge background about the Family Clinics, their services,
responsibilities, and related planning problems is presented.
Models that address the problem of scheduling and routing in home care and their
methods of solution are analysed in Section 4.3. After analysing all the constraints
presented in each model from literature, a model considered more appropriate to the
circumstances of the Family Clinics was developed, and the formulation is described
in Section 4.4. An algorithm was developed in R code to implement the solution to the
scheduling and routing problem, and a classical heuristic for CVRP (Capacitated VRP)
was used as routing subroutine. In Section 4.5, results found in tests with known
benchmark instances, as well as in a real-life application, demonstrated the practical
applicability of the computer model. These results are discussed in Section 4.6, and,
at last, in Section 4.7, conclusion and suggestions for future works are placed.
4.2 BACKGROUND
Primary health care is intended mainly for promoting health and preventing disease,
as well as monitoring chronic non-transmittable diseases, controlling endemic
diseases and zoonoses by combating disease vectors, and health surveillance
(BRASIL, 2012). The public Unified Health System in Brazil (SUS) has organised a
primary health care network through the so-called Family Health Strategy, which
involves the three spheres of government (federal, state and municipal). The
responsibility of implementing this strategy, and with it, effectively creating the
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necessary operational infrastructure was delegated to the municipalities. In the city of
Rio de Janeiro, primary health care units are referred to as Family Clinics (RIO DE
JANEIRO, 2015), which should develop health actions through multidisciplinary teams
(Family Health Teams), directed to a population in a well-defined territory (up to 24.000
inhabitants), by which it assumes the sanitary responsibilities, considering its local
characteristics, solving the health problems of greater frequency and greater relevance
in its territory.
The Family Health Team (FHT) is responsible for the knowledge, care and follow-up
of families registered in their area (territory), in a dynamic, continuous and permanent
way, taking into account local characteristics and health indicators of the population.
The assignment of the clientele enables the bonding and co-responsibility between the
team and the community increasing the capacity to respond to the health needs of the
population within the area of coverage (territory).
The Family Clinics typically have six Family Health Teams (FHT), each comprising a
physician, a nurse, a nursing assistant, six community health agents (CHA), and a
health surveillance agent. As stated in the National Policy on Primary Care (PNAB)
(BRASIL, 2012), the CHA responsibilities are: (i) to work with ascriptions of families on
a defined geographical basis (micro area); (ii) register all the people in your micro area
and keep the registers updated; (iii) guide families in the use of available health
services; (iv) carry out programmed activities and attention to spontaneous demand;
and (v) to monitor, by means of visits to patient's homes, all families in your micro area.
The visits should be scheduled taking into account the risk and vulnerability criteria so
that the families with the greatest need are visited more often, keeping as a reference
the average of one visit per family per month; (vi) develop actions that seek integration
between the health team and the population assigned to the Family Clinic, considering
the characteristics and purposes of the follow-up work of individuals and social groups
or groups; and (vii) be in permanent contact with families, developing educational
actions, aiming at health promotion, disease prevention, and monitoring of people with
health problems, as well as monitoring the requirements of the income transfer
program and coping with vulnerabilities implemented by the Federal Government
(Bolsa Família Program) or another similar program that may be implemented in the
119
spheres of federal, state or municipal government according to the team's planning.
CHA is allowed to develop other activities in the Family Clinics, as long as they are
linked to the attributions previously described.
As can be seen from the above, the routine of a Family Clinic is mainly dependent on
the daily visits made by the CHA to the homes of the residents of the designated region
(territory). In practice, the current planning for coverage of service territories of Family
Clinics does not consider certain criteria that could favour the good service delivery,
since the Family Health Teams (FHT) are distributed only taking into account the
number of households to be served (BRASIL, 2012). Criteria such as workload of
community agents (number of home visits) and contiguity and compactness of service
areas (to reduce travel time and avoid crossing routes) are not met in the current
planning of the teams.
Each FHT is responsible, on average, for the monitoring of 3,450 inhabitants, and can
reach a maximum of 4,000 inhabitants (RIO DE JANEIRO, 2015). Considering that a
typical clinic has six FHTs composed of six CHAs, and according to the metrics
adopted, each FHT serves up to 4,000 inhabitants, it is concluded that each CHA
should visit at least 165 households per month, about eight per day, if intend to visit all
the residents of their basic area at least once a month. In practice, the CHA visitation
goals range from clinic to clinic, with numbers ranging from 80 to over 165, considering
the current manual method of agent distribution by the micro areas (basic service units)
and team routing.
Currently, there are 109 Family Clinics in operation in Rio de Janeiro, within 67 of the
city’s neighbourhoods. This represents 67.25% of the total service area to be covered,
according to data from the Rio de Janeiro city government (RIO DE JANEIRO, 2017).
Field surveys conducted at Family Clinic units in 2012 (ARAÚJO, 2012; DIOGO;
ARAÚJO, 2013) and 2015 (SILVEIRA, 2015) showed that the CHA visited monthly,
respectively, only 56.9 and 52.0% of the households in the assigned territory.
4.2.1 THE FAMILY RISK SCALE OF COELHO-SAVASSI
120
In response to the perspective of establishing priorities for home care and for the
attention of the population assigned according to the principle of equity, the Family
Risk Scale of Coelho-Savassi (FRS-CS or simply CS scale) was developed as a
stratification instrument (COELHO; SAVASSI, 2004). This tool uses data available in
the routine of the Family Health Teams and the Primary Care Information System
(SIAB), which is fed by CHA at the time of registration of the family visited for the first
time and updated at subsequent visits, when necessary. The CS scale allows the team
to recognise the living conditions of people in their area of coverage, based on
demographic, socio-economic and health-disease indicators, which allow better
planning for interventions. These data were defined and classified according to
epidemiological relevance, health relevance and impact on family dynamics, as
follows: (i) bedridden; (ii) physical and mental disabilities; (iii) low sanitation conditions;
(iv) severe malnutrition; (v) drug addiction; (vi) unemployment; (vii) illiteracy; (viii) less
than six months of age; (ix) over 70 years of age; (x) systemic arterial hypertension;
(xi) Diabetes Mellitus; and (xii) resident / room ratio.
The CS scale is applied to the assigned families, which score from the Risk Score
according to Table 4.1 (COELHO; SAVASSI, 2004), and the individual risk indicators
should be multiplied by the number of individuals with the condition.
From the sum of these values, according to (SAVASSI; LAGE; COELHO, 2012) each
family is classified into R1 lower risk, R2 medium risk or R3 maximum risk (see Table
4.2). In addition, Guzella (2015) considers that families with a sum less than five were
denominated with R0 usual risk because it does not necessarily mean absence of risk.
Table 4.1. Risk indicators and risk score
Risk indicator Risk score
Bedridden 3
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Physical disability 3
Mental disability 3
Low sanitation conditions 3
Severe malnutrition 3
Drug addiction 2
Unemployment 2
Illiteracy 1
Less than six months of age 1
Over 70 years of age 1
Systemic arterial hypertension 1
Diabetes Mellitus 1
Resident / room ratio
If greater than 1 3
If equal to 1 2
If less than 1 0
Source: Coelho and Savassi (2004)
Table 4.2. Classification of family risk
Total score Family risk
Less than 5 R0 usual risk
5 and 6 R1 lower risk
7 and 8 R2 medium risk
More than 8 R3 maximum risk
Source: Adapted from Savassi, Lage, and Coelho (2012) and Guzella (2015).
4.2.2 HUMAN RESOURCE PLANNING PROCESS
Chahed et al. (2009) analysed the decision processes related to the operations
management of home care organisations, considering their horizons of time and
frequency of application and classified them in different and successive hierarchical
plans. More recently, a hierarchy of operations management decisions in home care
organisations has been also proposed in literature (MATTA et al., 2014; SAHIN;
MATTA, 2015) including healthcare planning process and related operations research
problems. Figure 4.1 summarises the information merging those concepts and
illustrates the human resource planning process issue undergoing by home care
organisations.
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Figure 4.1. Hierarchy of human resource planning process
As can be seen in Figure 4.1, the strategic level includes decisions that must be taken
in the long term over a period of one to three years, which in the case of Family Clinics
means addressing issues such as what types of care services will be provided, for
which type of patients, based on the quality of service measured over the coverage
area, as well as taking into account an estimate of overall demand (e.g., annual
volumes of patient visits). Demands for home visits may change within the time frame
considered, due to worsening or improving the health-disease conditions of patients or
by increasing or decreasing the category in the family risk and vulnerability scale.
These changes in demands will force a realignment of the territory, with a new
Districting solution (Partitioning Problem). We address the Districting Problem of
territories in home care operations in another work that will be published soon (see
Chapter 3).
Decisions at the tactical level are taken over a horizon of six to 12 months considering
the decisions made at the strategic level and addressing their implementation. For
example, the districting process simplifies the resource allocation problem, since
patients are first assigned to a district and then assigned the health teams that will
provide assistance to the families in the district, including the community health agents
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(CHAs) who will be responsible for the visitation of households within each micro area
belonging to these districts. At this stage, resource dimensioning encompasses
material and primarily human resources that will be used in each district. This task is
accomplished by solving a resource allocation problem.
Operational level decisions, with a time horizon in weeks to months, are taken so that
the flow of activities occur within the standards set at the higher hierarchical levels and
can thus be controlled. In the case of home care organisations, the main decisions at
this level refer to the assignment of care workers to patients. Especially in the case of
Family Clinics, the task is to determine the number of community health agents that
will make up the Family Health Team and the designation of the micro area for each
of them.
Decisions of detailed operational level affect the planning, coordination and
supervision of day-to-day activities. In home care organisations these decisions
include the scheduling of visits and the routing of health workers across the territory.
At this level, the major operational research issues to consider are Transportation
Problem and Sequencing Problem. Specifically for Transportation Problem it is usually
considered the Vehicle Routing Problem (VRP) and the Travelling Salesman Problem
(TSP) (SAHIN; MATTA, 2015). This article deals with the problem of scheduling and
routing of individuals of the health teams responsible for home visits in territories
attached to community health clinics, such as Family Clinics.
4.2.3 THE SCHEDULING AND ROUTING PROBLEM APPLIED TO THE FAMILY
CLINICS
Visiting routes to patients' homes to be travelled by community health agents are
usually done by walking, and eventually in some districts can be done by public
transportation, usually buses. The community health agents leave the Family Clinic
and visit the patients' homes one at a time and return to the Clinic at the end. Thus,
the mathematical modelling of the related routing problem should naturally be that of
the travelling salesman (TSP) (DANTZIG; FULKERSON; JOHNSON, 1954). However,
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TSP can be considered a special case of vehicle routing problem (VRP), involving a
fleet of a single vehicle and external carriers (VOLGENANT; JONKER, 1987), and,
due to the various constraints contained in real-life problems involving routing of health
teams, VRP has been used in place of TSP for the solution of these problems.
VRP belongs to the class of NP-hard combinatorial optimisation problems, which is
hard to solve in a polynomial time (DANTZIG; RAMSER, 1959; BODIN et al., 1983).
For NP-hard problem the time to find a solution grows exponentially with problem size.
For this reason, exact solutions are not commonly used when dealing with a
considerable number of nodes, and it is preferable to use heuristics for their solution
(LAPORTE et al., 2000; FAULIN; JUAN, 2008; AFSAR; PRINS; SANTOS, 2014;
QUINTERO-ARAUJO et al., 2017, 2019; BELLOSO; JUAN; FAULIN, 2019). For
instance, Faulin and Juan (2008) use an entropy-based heuristic with Clarke and
Wright (1964) algorithm and Monte Carlo simulation for solving a VRP problem. In turn,
Afsar, Prins, and Santos (2014) use an exact method with column generation and local
search metaheuristics in a VRP problem with flexible fleet size. Quintero-Araujo et al.
(2017) solve a location routing problem (LRP), which involves a facility location
problem (FLP) and a VRP, by using a two phase routing heuristic. In (BELLOSO;
JUAN; FAULIN, 2019), a multistart biased-randomised heuristic is proposed for solving
a fleet mixed VRP with backhauls. Quintero-Araujo et al. (2019) once again analysed
the LRP problem, this time considering horizontal cooperation scenarios, each one
being solved with the use of metaheuristics.
The most widely used VRPs are (LAPORTE, 1992; EUCHI, 2011): Capacitated vehicle
routing problem (CVRP) (LAPORTE, 1992; TOTH; VIGO, 2002; FAULIN; JUAN,
2008); Period or Multi-period vehicle routing problem (PVRP or M-VRP) (CORDEAU;
GENDREAU; LAPORTE, 1997; FRANCIS; SMILOWITZ; TZUR, 2006); Vehicle routing
problem with time windows (VRPTW) (SOLOMON, 1987; DESROCHERS;
DESROSIERS; SOLOMON, 1992); and Vehicle routing problem with pickup and
delivery (VRPPD) (SAVELSBERGH; SOL, 1995; TOTH; VIGO, 2002).
More recent than the classical VRPs, the period vehicle routing problems (PVRPs)
consider a planning horizon where a vehicle may make several routes (EUCHI, 2011).
The main purpose of this problem is divided into two parts: (i) to schedule deliveries
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for each customer over a predetermined time; and (ii) to organise the vehicles tours to
make deliveries required while optimising the total cost of transport. The PVRP can
thus break up into two problems: (i) assignment of sequences of delivery to the
customers; and (ii) resolution of a classical vehicle routing problems per day of the
horizon.
Indeed, in the case of Family Clinics, we have to use a variation of the PVRP that
considers a priority for the care service. We must plan a schedule for the set of days
on which a node will be visited. The service of a node should be characterised by the
scheduling of designated visits and the frequency of these visits. In special, we must
consider that each patient’s home demands a minimum of visits per period, according
to the Coelho-Savassi (CS) vulnerability and risk scale. Thus, a PVRP with service
priority (PVRP-SP) model is a Scheduling and Routing problem characterised by the
objective of finding a set of routes for each community health agent on a daily basis
over the period that minimises the total travel cost and complies with operational
constraints (quantity and duration of visits, and minimum visit frequency).
For the mathematical modelling of this scheduling and routing problem applied to such
a special kind of home health care, we first have to examine which models of approach
and solution methods are considered in the literature for similar cases (see next
section).
We know in advance that these automated solutions for scheduling and routing
problems show productivity and cost effectiveness benefits when compared to manual
(not computerised) decisions. As reported by experienced authors, savings varying
from 18 to 20% on the total travelling time were found and about 7% on total working
time (EVEBORN; FLISBERG; RÖNNQVIST, 2006; BARD; SHAO; JARRAH, 2014;
CISSÉ et al., 2017).
4.3 RELATED LITERATURE
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In this section, we first review the relevant efforts in home health care academic
research and then present the literature on the scheduling and routing problem applied
to home health care operations.
4.3.1 HOME HEALTH CARE ARTICLES
Although there has been an evolution of home health care (or home care) services in
both developed and developing countries given the relevance of these services, the
quantity of research papers concerning this subject within operations management is
still relatively small (39 articles found). These findings are also reinforced by Rais and
Viana (2010), Benzarti, Sahin, and Dallery (2013), Sahin and Matta (2015), and Cissé
et al. (2017). Figure 4.2 illustrates the quantity of home health care articles per each
theme.
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Figure 4.2. Quantity of home health care articles
The theme with the greatest number of articles was Scheduling and Routing with 13
documents (BEGUR; MILLER; WEAVER, 1997; CHAHED et al., 2009; BARD; SHAO;
JARRAH, 2014; AIANE; EL-AMRAOUI; MESGHOUNI, 2015; BRAEKERS et al., 2016;
CASTILLO-SALAZAR; LANDA-SILVA; QU, 2016; LIU; YUAN; JIANG, 2016; REDJEM;
MARCON, 2016; CISSÉ et al., 2017; LIN et al., 2018; XIAO; DRIDI; EL-HASSANI,
2018; YANG et al., 2018; YUAN; LIU; JIANG, 2018). Second among the most
published are four themes with four documents each: Assessment (SAHIN; VIDAL;
BENZARTI, 2013; GUTIÉRREZ et al., 2014; HOLM; ANGELSEN, 2014; SAHIN;
MATTA, 2015), which deals with examining factors that may generate complexity in
Scheduling and RoutingAssessment
DistrictingAssignmentRouting
SchedulingAssignment and RoutingStaff dimensioning
ModellingLogisticsPlanning and Routing
Planning and SchedulingAssignment, Scheduling, and Routing
Home Health Care Articles
Articles
Qu
an
tity
02
46
81
01
2
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managing home health care operations; Districting (BLAIS; LAPIERRE; LAPORTE,
2003; BENZARTI; SAHIN; DALLERY, 2013; GUTIÉRREZ; VIDAL, 2015; LIN et al.,
2017); Assignment (HERTZ; LAHRICHI, 2009; LANZARONE; MATTA; SAHIN, 2012;
LIN et al., 2016; YALCINDAG et al., 2016); and Routing (LIU et al., 2013; BASTOS et
al., 2015; EN-NAHLI et al., 2016; FATHOLLAHI-FARD; HAJIAGHAEI-KESHTELI;
TAVAKKOLI-MOGHADDAM, 2018). Third among the most published themes are two
themes with two documents each: Scheduling (EVEBORN; FLISBERG; RÖNNQVIST,
2006; AKJIRATIKARL; YENRADEE; DRAKE, 2007); and Assignment and Routing
(EN-NAHLI; ALLAOUI; NOUAOURI, 2015; ELISEU; GOMES; JUAN, 2018). Fourth,
and lastly, in the clippings of subjects found in the literature review, we have six
themes, some covering previous themes, each with only one published document:
Staff dimensioning (RODRIGUEZ et al., 2015); Modelling (MATTA et al., 2014);
Logistics (GUTIÉRREZ; VIDAL, 2013); Planning and Routing (HEWITT; NOWAK;
NATARAJ, 2016); Planning and Scheduling (LIU et al., 2018); and Assignment,
Scheduling, and Routing (CAPPANERA; SCUTELLÀ, 2015).
In the next subsection we will discuss in more detail, in chronological order, the 24
most relevant articles found in the literature review that somehow involve the problem
of scheduling and routing in home health care operations.
4.3.2 HOME HEALTH CARE SCHEDULING AND ROUTING PROBLEM ARTICLES
Begur, Miller, and Weaver (1997) tackle the issue of developing an integrated spatial
decision support system (DSS) for scheduling and routing home health care nurses in
the United States. The problem is modelled as a Mixed-integer programming (MIP)
with an objective function for minimising total travel distance/time travelled/spent by
care workers when visiting patients. The authors use Geographic Information System
(GIS) for travel time estimations and Clarke and Wright (1964) heuristic for greedy
routes. The model considers the following particular inputs/settings: number of visits
per time window (weeks) assigned for patients; visits allocations for a specific day of
week; patients’ locations; and care workers (nurses) available. The model´s constraints
are: patient’s time window; time lag between visits; work time; holidays; qualification /
skill; workload balancing; and multiple depots. The decision variables consider which
available nurse to see which patient and when, and also what travel route to use. Real
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data with up to seven care workers (nurses) and a maximum number of 200 services
compose the case study.
Eveborn, Flisberg, and Rönnqvist (2006) treat the Home Care (HC) service in its strict
sense of caring elderly and disabled people in Sweden. The problem here is to plan
visiting schedules for caregivers with some hard restrictions and soft objectives, such
as travel time, travel costs, scheduled hours, inconvenient working hours, patient
preferences etc. The constraints include: time window for visits, a set of skills for staff
members, and client preferred staff members. The authors describe a decision support
system (DSS) called Laps Care to assist planners with the issue in hands. The
approach used is the set partitioning model solved with a repeated matching algorithm.
Practical impact of the DSS system in health organisations in terms of time savings
and measured quality for the clients is reported.
The work presented by Akjiratikarl, Yenradee, and Drake (2007) also treats the same
problem of HC, considering community care provided by the local authorities in UK.
However, here optimisation routes for each care worker are accomplished minimising
the travelled distance ensuring that the capacity and time windows constraints are not
violated. The authors introduce a collaborative population-based metaheuristic named
Particle Swarm Optimisation (PSO) to solve the scheduling problem. The Earliest Start
Time Priority with Minimum Distance Assignment (ESTPMDA) heuristic is used for
generating an initial solution, and then the PSO metaheuristic is applied to improve the
solution quality. The method is tested on real-life instances.
Chahed et al. (2009) address the home health care scheduling and routing problem
regarding the production-delivery (drug supply chain) process related to chemotherapy
care at home. The authors use a modelling approach of exact solution with Integer
Programming and Branch and bound algorithm as solution method. The model
considers a manifold objective function in order to optimise the production and/or the
delivery/administration processes. Considering the production process, the objective
function intends to: (i) minimise setup total costs; and (ii) maximise satisfied demand
(produced drugs, emergency demands). For the distribution/administration process,
the objective function aims to: (i) minimise travel distance; (ii) minimise costs with drug
obsolescence; (iii) maximise profits; and (iv) maximise total time slacks. Particular
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inputs/settings encompasses the number of patients to be visited each day, the
number of drugs to be produced each day, production and administration durations,
patients locations, and care workers (nurses) available. The model considers many
constraints, such as drug production start times, patient requirements, coupling
restriction (visit time at the patient and the production starting time of the drug); time
limits for the production of the drugs; and visit time windows. The decision variables
are: (i) drug delivery date; (ii) nurse visit date; (iii) sequence of drugs’ production; and
(iv) nurses’ routes. The model was tested in a fictitious example considering a coupled
production-delivery problem to anti-cancer drugs.
The problem presented in Liu et al. (2013) deals with the home delivery of medicines
and medical devices from the pharmacy of the health care company. It also includes
the delivery of special medicines directly from the hospital to patients in their homes,
and collecting material for examination and unused medicines and devices. The
approach is a special case of VRP with time windows (VRPTW). Two MIP models are
proposed, and the solution method uses a Genetic Algorithm (GA) and a metaheuristic
Tabu Search (TS). These approaches were tested on known VRPTW benchmark
instances.
Bard, Shao, and Jarrah (2014) focus on the problem of providing rehabilitative services
to both in-clinic and home-based patients. The aim is to construct weekly tours for a
set of multi-skilled care workers that minimise the travel, care, and administrative costs
while ensuring that all geographically dispersed patients are visited within their time
preferences and contractual agreements are observed. The modelling approach
considers a Mixed-integer linear program (MILP) solved with adaptive sequential
greedy randomised adaptive search procedure (GRASP). The model was extensive
tested with both real data and datasets provided by a U.S. rehab organisation and
demonstrated very good results.
Aiane, El-Amraoui, and Mesghouni (2015) focus on the problem of optimising routes
and rosters for staffs, satisfying specific constraints. The issue is treated as a multiple
travelling salesman problem with time windows (MTSPTW) and modelled as a MILP.
The model’s objective function intends to minimise travel time of each resource until
his return to origin node. The problem is solved by using ILOG/CPLEX (IBM
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optimisation software package). The model sophistication requires several particular
inputs/settings, such as: quantity of patients; quantity of resources (nurses,
physicians, physiotherapists); patients-resources assignments; travel times; visit
times; time window data for patients; and dates and times without availability for each
patient. Regarding the constraints, the model imposes two concerns: patient’s time
window and dates of unavailability. The paper presents results on many instances
supplied by a real living-lab in Bourges, France.
In (BASTOS et al., 2015), the authors present a web based application for optimisation
of home care professionals visits with patient’s priority. The task is considered as a
vehicle routing problem (VRP) and modelled as a Mixed-integer programming (MIP),
being that the solution method used is the classical Clarke and Wright (1964)
algorithm. The modelling approach uses as particular inputs/settings the patient’s
location, patient priority, and travel times between different nodes in minutes. The
objective function minimises the total time of routes visiting all patients. The decision
variable treats the accumulated time flow between two nodes in a graph. Example of
application is presented, where the patient's data are artificial, but the addresses are
real places in Coimbra, Portugal.
A multifaceted issue involving a combined assignment, scheduling and routing
problem applied to a home care service dedicated to palliative and terminal patients is
tackled by Cappanera and Scutellà (2015). The modelling approach uses Integer linear
programming (ILP) with two balancing functions maxmin and minmax, which,
respectively, maximise the minimum operator utilisation factor, and minimises the
maximum of such factor (total workload of the skilled care worker divided by maximum
possible workload). The problem is treated as Skill VRP and solved by using
ILOG/CPLEX. The modelling approach uses as particular inputs/settings skill levels for
both patients and operators; care plan for each patient; and a set of patterns defines
schedules for eventual skilled visits. Results are shown in both palliative home care
instances based on actual data, as well as in two real-life data sets from the literature.
En-Nahli, Allaoui, and Nouaouri (2015) address the assignment and routing problem
for Home Health Care Services (HHCS) as an extension of the multiple TSP with time
windows (MTSPTW). The authors use a multi-objective optimisation modelled as a
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MILP with weighted linear aggregation to minimise the travel time between patients
and the sum of arrival time of caregivers, and maximise the operability factor of
caregiver, as defined in (CAPPANERA; SCUTELLÀ, 2015), and the satisfaction of
patients. The model presents many inputs/settings, such as: the earliest and the latest
service time for patients; care durations; travel times between patients; skill of
resources; resource workloads; and skill levels required by patients. The solution is
achieved by using ILOG/CPLEX. The model is tested on existing travelling salesman
problem benchmark instances and extra random data.
Braekers et al. (2016) analyse the trade-off between costs and patient inconvenience
for home care providers. The challenge is modelled by the authors as the Bi-objective
Home Care Routing and Scheduling Problem (BIHCRSP) and solved by using a
metaheuristic algorithm, embedding a large neighbourhood search heuristic. The bi-
objective function intends to minimise the total cost (routing plus overtime costs) and
minimise patient inconvenience, calculated as the deviation from the patient’s
preferred visit time and also how disliked the assigned nurses are. Particular
inputs/settings are considered: maximum regular working time duration for each nurse;
maximum allowed daily working time; hard availability time window; cost for exceeding
working times; services duration; preferred starting times; and preferences for nurses
for each job. The paper presents results from tests with benchmark instances based
on real-life data.
Castillo-Salazar, Landa-Silva, and Qu (2016) focus on scenarios of workforce
scheduling and routing problems (WSRP) to guarantee that care workers arrive on
time at the locations where care service is demanded. The paper concerns on the
computational difficulty of solving these types of problems. The modelling approach
uses mathematical programming, being that for the problem solution a Gurobi solver
is used, and the model is tested comparing benchmark instances.
Home Health Care Services (HHCS) are once again approached by En-Nahli et al.
(2016) this time with the concern in optimising the routing of the health teams. The
routes have to comply with synchronisation constraints for the case where a patient
demands simultaneously more than one caregiver. The model is formulated as a VRP
with time windows and synchronisation constraints (VRPTWSyn) and is solved with an
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initial solution achieved by using a constructive heuristic, and then a metaheuristic
Iterated local search (ILS) with a Random Variable Neighbourhood Descent (RVND)
method is used to improve the outcomes of the initial solution. The paper presents
computational results regarding 15 well-known benchmark instances.
Hewitt, Nowak, and Nataraj (2016) examine appropriate planning horizon length and
the routing cost of planning, advocating that a long horizon can have significant
potential for savings in terms of transportation costs and staffing levels. The authors
modelled the problem as a Consistent VRP (ConVRP) with Stochastic Customers
solved using algorithms.
Redjem and Marcon (2016) analyse the scheduling and routing problem in home care
services, where patients receive multiple caregivers with precedence and coordination
constraints. The authors developed a new method called Caregivers Routing Heuristic
(CRH), which improves the outcomes of the classical MILP approaches and solves
real world problems. The model was tested using several instances.
Liu, Yuan, and Jiang (2016) approach the home care worker scheduling and routing
problem with lunch break constraints. The problem is modelled as a three-index MIP,
which is resolved by a branch-and-price (B&P) method with a CPLEX solver. A label-
correcting algorithm is also used to treat lunch break constraints in pricing sub-
problems. The paper presents tests results from VRPTW benchmark instances and
real-life examples.
A study of significant routing and scheduling problems related to home care operations
is presented in Cissé et al. (2017). The authors show an overview of methods to solve
the health care routing and scheduling problem (HHCRSP), addressing exact and
approximate solutions. At last, the paper discusses future research directions.
More recently there has been much academic production on issues involving the
problem of routing and scheduling in home health care services, with new modelling
approaches and solution methods. For instance, Fathollahi-Fard, Hajiaghaei-Keshteli,
and Tavakkoli-Moghaddam (2018) are concerned about the problem of environmental
pollution and discuss the impact of Green House Gas (GHG) emissions for the home
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health care system. The mathematical approach uses a bi-objective green home health
care routing model. The function aims to minimise the cost of transport due to travelled
distance for the nurses by a predefined transportation system, and also to minimise
the amount of CO2 emission released by the same system. The solution considers
recent and old metaheuristics and introduces four heuristics based on two different
strategies. In (YANG et al., 2018), the authors tackle the stochastic home health care
scheduling and routing problem (SHHCSRP) in dense communities, where travelling
distance is shortened and waiting cost is likely higher. Constraints still include multi-
appointment, mixed time window and skill-demand matching. For the solution chance
constrained programming model is used and solved by Best-Worst Ant Colony
Optimisation heuristic. Liu et al. (2018) study the medical team planning and
scheduling problem in home health care within a weekly horizon. A bi-objective
function formulated as a MIP is solved with an ϵ -constraint method to obtain exact
non-dominated solutions, and also heuristics. The method was tested in a medium
scale instance with significant impact on the solution. Yuan, Liu, and Jiang (2018)
address caregiver scheduling and routing uncertainties in home care service providers
on a daily basis, due to sudden changes in road traffic and patient health conditions.
The work uses a set partitioning model solved with a branch-and-price (B&P) method.
Simulated instances were carried out to validate the model. Once again the Home
Care in its strict sense is approached in (ELISEU; GOMES; JUAN, 2018). Therein the
classical VRPTW is used for modelling an assignment and routing issue with some
specific characteristics. The solution model uses a biased-randomised heuristic, and
was tested in small but real-case instances with good results. Xiao, Dridi, and El-
Hassani (2018) focus on the home health care scheduling and routing problem with
daily planning horizon. The problem is formulated as a variant of VRPTW with MILP
model. In addition to the regular constraints of VRPTW the model requires the
consistency of the serving time including (or not) lunch break between successive
visits. The solution method uses the Gurobi commercial solver. At last, Lin et al. (2018)
address the nurse rostering problem (NRP) and also the VRPTW, which are usually
treated separately, since each alone is an NP-hard problem. The solution method uses
harmony search algorithm (HSA) improved by two strategies. Results from
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experiments show that the method presents good performance and can conform to
changes caused by sudden incidents.
4.4 MATHEMATICAL MODEL APPLIED TO FAMILY CLINICS
As we can see from the literature review presented above, the problem of routing and
scheduling in home health care is modelled by VRP variants, which consider several
constraints peculiar to each situation. Three classes of constraints were distinguished:
temporal, assignment, and geographic (CISSÉ et al., 2017). Constraints can also be
seen from three perspectives: HHC organisation, patient, and care worker (see Table
4.3).
Table 4.3. Classification scheme based on constraints
Actors Temporal constraints Assignment constraints Geographic constraints
HHC organisation Planning horizon
Frequency of decision
Continuity of care Sectors/districts
Types of services provided
Patient Frequency of visits
Time windows
Temporal dependency
Non-simultaneous services
Preferences Type of network between home locations
Care worker Contract type
Capacity/working hours
Qualification/skill
Workload balancing
Location of care workers
Source: Cissé et al. (2017)
In the specific case of Family Clinics, it is verified that several items of the classification
scheme based on constraints, presented in Table 4.3, can be suppressed. For
example, in this case, from the point of view of HHC organisation, there are no different
types of services to be provided, summarising the geographical constraints only to
sectors (micro areas) and districts. From a patient's perspective, considering time
constraints, there are no time windows, no time dependence or non-simultaneous
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services, since the focus is primary health care, usually dealing with health promotion
and disease prevention, and eventually monitoring chronic non-communicable
diseases. That is, only frequency of visits should be considered as a temporal
restriction under patient’s perspective. From the point of view of the care worker,
considering the temporal constraints, in the case of Family Clinics, there is no
restriction other than that of the working hours. Still from the perspective of the care
worker, regarding the assignment constraints, there is no differentiation of skills or
qualifications in the Family Clinics problem, and the workload balance is the only
restriction.
Thus, the home health care scheduling and routing problem related to the Family
Clinics becomes simpler than most of the models analysed in the literature, and can
be seen as a variant of the period vehicle routing problem with service choice (PVRP-
SC) (FRANCIS; SMILOWITZ; TZUR, 2006), which we are calling here as PVRP with
service priority (PVRP-SP), where there is an upper limit for the total time of each route
and the set of nodes has four cohorts according to patient’s priorities. These priorities
correspond to the Coellho-Savassi scale of vulnerability and risk. The indices,
parameters, and decision variables for the modelling approach are presented in Table
4.4 below.
Table 4.4. Indices, parameters and variables
Indices Description
𝑖, 𝑗
𝑙, 𝑚
𝑠
Index of nodes (family homes), 𝑖, 𝑗 ∈ {2,3, … , 𝑁}, 1is the origin node (Family Clinic)
Index of CS risk scale, 𝑙, 𝑚 ∈ {0,1,2,3}
Index of schedules, 𝑠 ∈ {1,2,3, … |𝑆|}
𝑑 Index of days in the period, 𝑑 ∈ {1,2,3, … , |𝐷|}
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Parameters Description
𝑉 The set of nodes (family homes)
𝐴
S
𝐷
𝑇
The set of arcs (𝑖, 𝑗) between each pair of nodes 𝑖, 𝑗 ∈ 𝑉 ∪ {1}
The set of service schedules
The set of days
The upper limit for total time of each route
𝑡𝑖𝑗
𝑟𝑖
𝑉𝑙
𝑉𝑠
Travel time from 𝑖 to 𝑗, for 𝑖, 𝑗 ∈ 𝑉 and 𝑖 ≠ 𝑗
Duration of visit at family home 𝑖, for 𝑖 ∈ 𝑉 (assuming that 𝑟1 = 0)
The set of nodes with CS risk scale equal to 𝑙. 𝑉 = 𝑉0 ∪ 𝑉1 ∪ 𝑉2 ∪ 𝑉3 and 𝑉𝑙 ∩ 𝑉𝑚 = 0; 𝑙, 𝑚 ∈ {0,1,2,3}
The set of nodes in schedule 𝑠 ∈ 𝑆. 𝑉𝑠 is a subset of 𝑉
Variables Description
𝑥𝑖𝑗𝑑
𝑦𝑖𝑗𝑠
1 𝑖𝑓 𝑎𝑟𝑐(𝑖, 𝑗) 𝑏𝑒𝑙𝑜𝑛𝑔𝑠 𝑡𝑜 𝑡ℎ𝑒 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑜𝑛 𝑑𝑎𝑦 𝑑 ∈ 𝐷 𝑓𝑜𝑟 𝑖, 𝑗 ∈ 𝑉𝑠 𝑎𝑛𝑑 𝑖 ≠ 𝑗, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 0
Accumulated time flow on schedule 𝑠 ∈ 𝑆 after travelling through arc (𝑖, 𝑗) and before starting the
service at 𝑗, for 𝑖, 𝑗 ∈ 𝑉𝑠 and 𝑖 ≠ 𝑗.
As a starting point for the formulation of the problem, we consider in general the
directed graph
𝐺 = (𝑉 ∪ {1}, 𝐴) (4.1)
where 𝑉 is the set of nodes (family homes); 1 is the origin node (Family Clinic); and
𝐴 is the set of arcs (𝑖, 𝑗) between each pair of nodes 𝑖, 𝑗 ∈ 𝑉 ∪ {1} . The parameter
𝑡𝑖𝑗 describes the travel time from 𝑖 to 𝑗, for each arc (𝑖, 𝑗); and 𝑟𝑖 is the duration of visit
at family home 𝑖, for each node 𝑖 ∈ 𝑉. All the routes start and end at node 1, and a
route is determined by a series of arcs linked together. The total time of a route is the
summation of the travel times of the arcs that compose it (𝑡𝑖𝑗) together with the times
of visitation of each node (𝑟𝑖), having as an upper limit 𝑇.
For solving the problem in hands, we have to design a set of routes, guaranteeing the
visitation of each node (family home), respecting the Coelho-Savassi risk scale and
the upper limit for each route, with the objective of minimising the total time of all the
routes. Scheduling typically has a period of one month or four weeks, with each week
typically consisting of five business days. Daily routes will be constructed, attending
visits to the patients' homes according to their priorities. For each schedule 𝑠, the
model considers variables and parameters related to the arcs (𝑖, 𝑗) with 𝑖, 𝑗 ∈ 𝑉𝑠, 𝑠 ∈
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𝑆. The set of nodes 𝑉𝑠 is, in fact, a subset of 𝑉, composed of the combination of sets
𝑉𝑙 or a partition of these, for 𝑙 = {0,1,2,3}, 𝑉𝑙 being the set of nodes with corresponding
Coelho-Savassi (CS) risk scale 𝑙, which varies from 0 to 3 (𝑅0 to 𝑅3).
For calculating the travelling times, it is possible to consider the problem in a symmetric
sense. However, we will use the asymmetric sense in the mathematical formulation in
order to consider priorities among the family homes in an easier manner. In this case,
each arc (𝑖, 𝑗) can be substituted by the pairs (𝑖, 𝑗) and (𝑗, 𝑖).
Indeed, the problem can be adapted from a capacitated VRP (CVRP) (LAPORTE;
SEMET, 2001; LAPORTE, 2009), remembering that in this classical case the flow
variables represent the goods accumulated/released in the graph’s nodes, whereas in
the present model they mean the summation of times in the arcs and nodes through
the route.
The mathematical formulation for the problem becomes:
(PVRP-SP) 𝑚𝑖𝑛 𝑧 = ∑ ∑ 𝑦𝑖1𝑠
𝑖∈𝑉𝑠 𝑑∈𝐷 (4.2)
subject to:
∑ 𝑥𝑖𝑗𝑑
𝑖∈𝑉𝑠∪{1}
= 1, 𝑗 ∈ 𝑉𝑠, 𝑑 ∈ 𝐷 (4.3)
∑ 𝑥𝑖𝑗𝑑
𝑗∈𝑉𝑠∪{1}
= 1, 𝑖 ∈ 𝑉𝑠, 𝑑 ∈ 𝐷 (4.4)
∑ 𝑦𝑗𝑖𝑠
𝑖∈𝑉𝑠∪{1}
− ∑ 𝑦𝑖𝑗𝑠
𝑖∈𝑉𝑠∪{1}
− ∑ 𝑡𝑗𝑖
𝑖∈𝑉𝑠∪{1}
𝑥𝑗𝑖𝑑 = 𝑟𝑗 , 𝑗 ∈ 𝑉𝑠, 𝑑 ∈ 𝐷 (4.5)
(𝑡𝑖𝑗 + 𝑟𝑖)𝑥𝑖𝑗𝑑 ≤ 𝑦𝑖𝑗
𝑠 ≤ (𝑇 − 𝑟𝑗)𝑥𝑖𝑗𝑑 , 𝑖, 𝑗 ∈ 𝑉𝑠 ∪ {1}, 𝑑 ∈ 𝐷 (4.6)
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𝑥𝑖𝑗𝑑 ∈ {0,1}, 𝑦𝑖𝑗
𝑠 ≥ 0, 𝑖, 𝑗 ∈ 𝑉𝑠 ∪ {1}, 𝑑 ∈ 𝐷 (4.7)
The objective function (4.2) minimises the accumulated time flow on each schedule s
∈ S. Constraints (4.3) and (4.4) ensure that to each node (family home) arrives one
and only one agent (route) and that just one agent (route) leaves from each node
(family home). Constraint (4.5) ensures the added time of each route. The equations
4.3 to 4.5 prevent the formation of sub-paths among the nodes in 𝑉𝑠. Inequality (4.6)
relates 𝑥 to 𝑦 and ensures that the total time on any route does not surpass the upper
limit 𝑇.
Section 4.4.1 provides an overview of the solution method chosen to solve the
problem.
4.4.1 CLARKE AND WRIGHT ALGORITHM
In the development of a solution to the PVRP-SP problem, we searched for a route
planning heuristic that presented a good result when compared to the optimal value
and at the same time presented a fast response in terms of computing time. We chose
the Clarke and Wright algorithm (1964), which met these objectives, and was also used
by some authors (BEGUR; MILLER; WEAVER, 1997; FAULIN; JUAN, 2008; BASTOS
et al., 2015), whose works were studied in the literature review. In addition, an
experienced researcher said that “the savings heuristic put forward by Clarke and
Wright (1964) is easy to describe and to implement, and yields reasonably good
solutions. This explains its on-going popularity” (LAPORTE, 2009). And on an earlier
occasion, the same author with the collaboration of another experienced researcher
had said that “the Clarke and Wright (1964) algorithm is perhaps the most widely
known heuristic for the VRP” (LAPORTE; SEMET, 2001).
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According to (LAPORTE; SEMET, 2001), this algorithm takes decisions supported by
the calculation of savings, i.e., considering that two routes (1, . . . , 𝑖, 1) and (1, 𝑗, . . . , 1)
can be merged into a single route (1, . . . , 𝑖, 𝑗, . . . ,1), a distance/travel time saving 𝑠𝑖𝑗 =
𝑡𝑖1 + 𝑡1𝑗 − 𝑡𝑖𝑗 is generated.
A pseudocode of the algorithm in both parallel and sequential versions is shown below.
Clarke and Wright Algorithm
1: compute distance matrix 𝑖𝑛: 𝑛𝑜𝑑𝑒𝑠, 𝑐𝑜𝑜𝑟𝑑_𝑥, 𝑐𝑜𝑜𝑟𝑑_𝑦 𝑜𝑢𝑡: 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒𝑠/𝑡𝑟𝑎𝑣𝑒𝑙 𝑡𝑖𝑚𝑒𝑠 2: compute the savings 𝑠𝑖𝑗 = 𝑡𝑖1 + 𝑡1𝑗 − 𝑡𝑖𝑗 for 𝑖, 𝑗 = 2, … , 𝑛 and 𝑖 ≠ 𝑗
3: create 𝑛 − 1 vehicle routes (1, 𝑖, 1) for 𝑖 = 2, … , 𝑛 4: order the savings in a nonincreasing way 5: // parallel version 6: start from the top of the savings list 7: choose a saving 𝑠𝑖𝑗
8: check if there are two routes, one containing arc(1, 𝑗) and the other containing arc(𝑖, 1), that can feasibly be merged
9: if so, combine these two routes by deleting (1, 𝑗) and (𝑖, 1) and introducing (𝑖, 𝑗) 10: repeat the operations until no further improvement is possible, then stop 11: // sequential version 12: consider each route (1, 𝑖, … , 𝑗, 1) 13: determine the first saving 𝑠𝑔𝑖 or 𝑠𝑗ℎ that can feasibly be used to merge the current
route with another route containing arc(𝑔, 1) or containing arc(1, ℎ) 14: execute the merge and repeat this operation to the current route 15: if no feasible merge exists, go to the next route and reapply the operations above 16: stop when no route merge is feasible
Several numerical results reported in the literature signalise that the parallel version of
the algorithm dominates to a large advantage the sequential one (LAPORTE; SEMET,
2001).
4.4.2 CW_VRP ALGORITHM
In order to implement the solution method to solve the scheduling and routing problem
applied to Family Clinics, we developed an algorithm (PVRP-SP) in the programming
language code R. Specifically, to solve the routing problem with the solution formulated
in the previous section, we went in search for heuristics that were at the same time
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accurate, with resulting cost values close to the optimal solution, as well as fast enough
in computational time, to be able to be embedded in a mobile web platform application
in the future.
We then select the Naveen Kaveti (2017) algorithm developed in R code and called
CW_VRP, which implements the Clarke and Wright (1964) heuristic in both parallel
and sequential versions to find greedy routes. The author strongly recommends using
the parallel version in case of building more than one route. In our implementation, this
algorithm in its parallel version is triggered as a subroutine for each schedule 𝑠
determined by the PVRP-SP model.
4.5 COMPUTATIONAL RESULTS
This section shows computational results from tests considering well-known instances
as a benchmarking to compare cost values of greedy routes, and also outcomes from
a real-life case regarding an existing Family Clinic.
4.5.1 COMPARISON ON BENCHMARK INSTANCES
This section presents the performance results of the CW_VRP algorithm when
submitted to the instances for CVRP (CVRPL, 2017) and compared to the optimum
solution values. We used the well-known benchmark instances for the CVRP from the
literature (CVRPL, 2017), named as set A from Augerat et al. (1998) and set E from
Christofides and Eilon (1969). In order to understand the instances notation, we have,
for example, in A-n32-k5, an instance of Augerat et al. (1998) with 32 nodes (𝑛),
including the depot, and a minimum of five vehicles (𝑘). Table 4.5 shows the
percentage above the optimal solution value and the computing time, in seconds, for
each chosen instance with number of nodes compatible with the application in the case
of Family Clinics. Considering that the community health agent should typically visit at
least 165 and a maximum of 220 households per month (GUZELLA, 2015), the number
of nodes per week should vary between 37 and 55.
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Table 4.5. Comparison on benchmark instances
Instance CW_VRP Optimal value % above Computing time (s)
A-n32-k5 852.13 784 8.7 0.9290531
A-n33-k5 701.26 661 6.1 1.627093
A-n36-k5 860.89 799 7.7 1.549088
A-n39-k5 893.57 822 8.7 8.329476
A-n45-k7 1229.62 1146 7.3 4.593263
A-n53-k7 1110.97 1010 9.9 3.769215
E-n22-k4 388.77 375 3.7 0.69504
E-n23-k3 660.93 569 16.2 0.5250299
E-n30-k3 603.40 534 12.9 1.281073
E-n33-k4 843.10 835 1.0 1.714099
E-n51-k5 605.57 521 16.2 4.225242
4.5.2 APPLICATION TO THE REAL CASE OF A FAMILY CLINIC
The developed PVRP-SP model was tested in a real-life example for a Family Clinic.
The primary health care unit chosen was the Assis Valente Family Clinic, located in
Governador Island, Rio de Janeiro. This clinic serves a territory of 24,000 inhabitants,
divided into six districts, each subdivided into six micro areas. Each of the 36 micro
areas is served by a community health agent responsible for home visits. For the
present application example, one of these micro areas, called Ema, was chosen.
Studies by the School of Nursing at the University of São Paulo (USP), in 2015
(GUZELLA, 2015), indicate that according to the Coelho-Savassi family risk scale
(COELHO; SAVASSI, 2004; SAVASSI; LAGE; COELHO, 2012), 12% of the population
of São Paulo city, on average have Risk 3 (high), 13% with Risk 2 (medium), 23% with
Risk 1 (low) and 52% with Risk 0 (usual). As we do not have data collected for Rio de
Janeiro city, we used the same proportions to estimate the number of households at
each level of the CS scale. These proportions were randomly distributed among the
165 households within the micro area under analysis (Ema), as shown in Table 4.6.
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To test the model, we used the exact geographical coordinates of 165 addresses of
households served in the Ema micro area. In addition, different visit times (𝑟𝑖) were
also considered according to the Coelho-Savassi risk scale, specifically 20 minutes for
R0, 25 for R1, 30 for R2 and 33 for R3. Daily routes were then planned for each week,
considering five working days per week. Table 4.7 shows the planned routes and
schedule for the four weeks, the resulting computation times being, respectively, 1.98,
1.47, 3.15, and 4.09 seconds. As an illustrative example of the results found by the
PVRP-SP algorithm, Figure 4.3 shows the routes for the first week. The model is
implemented in R code and was executed in a computer with Intel CORE i5 processor,
2 GB RAM, and 2.67 GHz.
Table 4.6. Visit frequency and schedule
Visit frequency R0 R1 R2 R3 Total
30 days 86 38 21 - 145 15 days - - - 20 20
Schedule R0 R1 R2 R3 Total
1st week - - 21 20 41
2nd week - 38 - - 38
3rd week 33 - - 20 53
4th week 53 - - - 53
Table 4.7. Planned routes and schedule
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Schedule 1st day 2nd day 3rd day 4th day 5th day Computing
time (s) 1st week 1-135-143-
147-151-155-159-141-139-123-1
1-91-99-115-107-111-119-131-127-103-1
1-51-63-71-79-87-83-75-67-59-1
1- 9-13-19-31-39-95-55-47- 1
1-35-43-27-23-15-5-1
1.981113
2nd week 1-101-149-163-153-145-137-133-129-1
1-57-61-77-85-93-65-69-73-1
1- 97- 89-109-117-121-125-113-105-1
1-11-29-45-37-33-25-21-17-1
1- 3-7-41-49-53-81-1
1.468084
3rd week 1-60-62-95-111-119-141-155-147-135-127-103-55-1
1-56-64-66-71-79-87-63-58-54-50-39-36-40-34-31-1
1-2-4-9-6-14-8-10-16-18-22-24-28-26-23-20-15-12-1
1-30-32-1 1-38-42-48-46-52-47-44-1
3.151181
4th week 1-130-132-138-142-156-161-164-166-165-162-160-124-122-1
1-116-136-140-144-150-157-152-158-154-148-146-134-128-1
1-76-82-96-98-106-110-102-94-100-90-86-80-84-1
1-68-92-108-104-114-126-120-118-112-88-78-72-1
1-70-74-1 4.098235
145
Figure 4.3. Routes for the first week
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4.6 DISCUSSION
From the performance results presented for the CW_VRP algorithm (see Table 4.5),
we consider that they were satisfactory, with values reasonably close to the optimum,
taking into account the simplicity of the heuristic used. Better cost values could be
achieved by applying in sequence local search metaheuristics. However, this strategy
would greatly increase the computational time, and the gain achieved would not be so
much perceived by the real cases of the Family Clinics. On the other hand, the
computational time found in the tests was excellent, enabling the algorithm to be
embedded in a mobile web platform in the near future, which was one of the objectives
of the routing heuristic choice.
The schedule presented in Table 4.6 is, in fact, the result of the planning algorithm,
which considers the Coelho-Savassi CS scale of risk and vulnerability of patients
(GUZELLA, 2015). The planning considers attending patients in the first week of most
risk (CS equal to 2 and 3). Since patients with CS equal to 3 (greatest risk) should be
visited every fortnight, so these visits will be repeated in the third week. In the second
week, households with a CS risk of 1 are considered. In the third week, in addition to
the patients of CS risk 3, patients of the cohort CS zero will be visited. The algorithm
makes an equal distribution for the third and fourth weeks in order to have the same
number of visits in the workload of the community health agent.
The outcomes from Assis Valente Family Clinic showed that the PVRP-SP model not
only presents a great solution to the scheduling and routing problem in question, as it
did with an excellent computational time (less than 4 seconds) in a median
performance computer (Intel CORE i5 processor, 2 GB RAM, 2.67 GHz). The results
also showed that it is possible to carry out all planned visits within the considered
periods (four weeks), respecting the priorities of the risk and vulnerability scale, and
the working hours of the community health agent (CHA) dedicated to this task.
Regarding this last issue, the PNAB (BRASIL, 2012) stipulates a maximum of 104
hours per month for the CHA to carry out home visits, which has proven to be perfectly
feasible since micro areas can typically be walked on and the households are very
147
close to one of the others, characteristic of low-income neighbourhoods attended by
the Family Clinics, and the average length of visits being 30 minutes.
4.7 CONCLUSION AND FUTURE WORK
Family Clinics in Brazil have a great challenge to differentiate themselves from a
conventional health clinic, whose work is reactive, attending the patients who seek it,
precisely by providing a proactive service, which seeks to care for their patients
through home visits periodically.
Recently, in Rio de Janeiro city, some Family Clinics were transformed into
conventional health clinics because they did not perform well in relation to the home
visits of their community health agents. This means that the patients living in the
assisted area have to go to these conventional clinics in the event of illness. When this
happens, the health centre usually has no way of attending and directs the patient to
another unit of urgency and emergency. In this scenario, we see that health promotion
and disease prevention, bases of primary health care, do not occur. If there were better
management of the operations of these Family Clinics, this would not be happening.
In order to contribute to a better management of the operational processes of the
Family Clinics, it was thought to develop this work, and present an automatic model
for the scheduling of home visits and routing of the health teams. The results presented
for the real case of Assis Valente Family Clinic showed that the objective of developing
an automated and functional tool to assist the health teams' work planning was
achieved.
The authors of this work are very confident of the applicability of the presented
management tool and liked to see this or similar models being used in the operational
processes of the Family Clinics, since they are still currently performed manually, with
no regulatory recommendations of how to carry out these tasks.
As a next step, the implementation of a version of the PVRP-SP model for a mobile
web platform should be carried out, as well as the consideration of coupling a GPS
148
(Global Positioning System) application to the community agent, in order to verify the
proper care of the route designed for him.
For future work, it is clear that consideration of other heuristics for the routing solution
should be an issue. Perhaps in addition to the heuristics, one can also consider the
possibility of aggregating a local search metaheuristic for better performance, but
always taking into account the trade-off of computational time.
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5 CONCLUSION
The districting problem in home care operations was mainly addressed by four group
of authors: Blais, Lapierre, and Laporte (2003); Benzarti, Sahin, and Dallery (2013);
Gutiérrez and Vidal (2015); and Lin et al. (2017). The first group model is the one that
best suits the case of community clinics because it does not differentiate between
types of patients or types of health professionals for their care. Although the model has
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been presented some time ago, its logic of grouping for building districts, followed by
a local search engine is still considered very satisfactory and referenced in dozens of
works. The algorithm developed in R code and the use of a more sophisticated
metaheuristic updated the model, making it even more interesting and applicable. The
performance of the implemented algorithm was excellent when compared to real and
fictitious examples, showing a very good applicability to the case of Family Clinics.
For further work on the districting model, some sophistication can be introduced,
especially given the difficulties encountered by CHAs with violence in the communities
they serve. In this case, some aspects of (GUTIÉRREZ; VIDAL, 2015) can be included
in the model, such as security levels in each basic unit.
Regarding the scheduling and routing processes, the PVRP-SP model not only
presents a great solution to problem in question, as it did with an excellent computing
time. The results also showed that it is possible to carry out all planned visits within the
considered periods, respecting the priorities of the risk and vulnerability scale, and also
the working hours of the community health agent (CHA) dedicated to this task. The
simplicity and efficiency of the chosen routing algorithm makes it possible to be
embedded in a mobile web platform in a future application, eventually making the
process operational on a daily basis.
The products of this work can be extended to other applications in the health sector,
such as home care, in its strict sense. To do this, some modifications to modelling
approaches must be implemented. For example, in the districting model the constraints
imposed by (BENZARTI; SAHIN; DALLERY, 2013) to accommodate different patient
profiles and their needs must be taken into account. In the case of scheduling and
routing processes, some sophistication to the model proposed here should be
considered. For instance, consider as in (AIANE; EL-AMRAOUI; MESGHOUNI, 2015;
EN-NAHLI; ALLAOUI; NOUAOURI, 2015; EN-NAHLI et al., 2016) several settings,
such as resource skill (nurses, physicians, physiotherapists), patients-resources
assignments, time window for patients, dates and times without availability for each
patient, and synchronisation of care workers.
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This chapter closes the present work with the final considerations in Section 5.1. In
sequence, Section 5.2 outlines the resulting papers and abstracts that have been
published in conference proceedings.
5.1 FINAL CONSIDERATIONS
The World Health Organisation (WHO) has ambitious targets for 2019 and beyond.
Among the goals of the United Nations agency is expanding access and health
coverage to serve one billion more people compared to current numbers. The
institution also wants to ensure that one billion individuals are protected from health
emergencies. Furthermore, the organisation hopes to improve the well-being of one
billion people on Earth (UN, 2019). To take these resolutions from paper, WHO has
set ten priorities for the coming years. The list includes combating environmental
pollution and climate change, communicable infections, chronic diseases and other
public health challenges.
Regarding the chronic non-communicable diseases such as diabetes, cancer and
cardiovascular diseases, it is worth remembering that monitoring them is one of the
responsibilities of Family Clinics. In addition to disease monitoring, Family Clinics must
carry out actions to prevent diseases and promote health and well-being. In 2019,
WHO will work with governments to achieve the global goal of reducing physical
inactivity by 15% by 2030. This will be done through a series of policies that encourage
people to be more active every day. Family Clinics have a key role to play in these
initiatives. But to do so, they must present effective coverage in terms of home visits.
In October 2018, WHO co-hosted a major global conference in Astana, Kazakhstan,
where all countries renewed their commitment to primary health care (WHO, 2018).
This position had already been made official in the Declaration of Alma-Ata in 1978
(WHO, 1978). In 2019, WHO will work with partners to revitalise and strengthen
primary health care in countries and follow up the specific commitments made in the
Astana Declaration.
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In Brazil, the current health minister said that he will expand health promotion and
disease prevention actions, and mentioned the creation of a new secretariat for
Primary Care, also emphasising the need to strengthen health regionalisation, in a
model in which states and municipalities to organise themselves in networks to better
serve the population (CONASS, 2019).
All these news are auspicious and cause urgent actions of organisation, planning and
governance of Family Clinics to be carried out. The present work can contribute to
increase the productivity and efficiency of these basic health units. The methodologies
and tools developed herein showed that it is possible to fully cover the service territory
of the clinics, within the daily and weekly workload required of their community health
agents. The proposed algorithms have proved their applicability according to the
requirements of strategic planning (districting) and operational planning (scheduling
and routing). Although the contribution may be small at this time, the work opens a
new field for the development of the Family Clinic operations management processes,
as well as other home care organisations.
5.2 RESULTING WORKS
This section shows the work produced during the research, which supported and
provided feedback for the improvement and evolution of articles submitted to journals.
5.2.1 FULL PAPER PRESENTED IN CONFERENCE
Work nominated to the ANPAD Award as Best Operations and Logistics Management
Article, in XL EnANPAD:
DIOGO, O. A.; DE VARGAS, E. R.; WANKE, P. O problema de alinhamento de territórios: uma possível aplicação às clínicas da família. In ENCONTRO CIENTÍFICO DE ADMINISTRAÇÃO – EnANPAD, 40., 2016, Costa do Sauípe. Anais do XL EnANPAD. Costa do Sauípe, BA: ANPAD, 2016.
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5.2.2 ABSTRACTS APPROVED FOR PRESENTATION IN CONFERENCES
DIOGO, O. A.; DE VARGAS, E. R.; WANKE, P. A districting solution for community health clinics. In: ANNUAL MEETING OF THE DECISION SCIENCES INSTITUTE - DSI, 48., 2017, Washington, D.C. Proceedings of annual meeting of the decision sciences institute. Washington, D.C.: DSI, 2017.
______; ______; ______. A multi-period traveling salesman problem with service priority solution for community health clinics. In: ANNUAL MEETING OF THE DECISION SCIENCES INSTITUTE - DSI, 49., 2018, Chicago. Proceedings of annual meeting of the decision sciences institute. Chicago,Illinois: DSI, 2018.
______; ______; ______. A multi-period vehicle routing problem with service priority solution for community health clinics. In: ANNUAL CONFERENCE OF THE PRODUCTION AND OPERATIONS MANAGEMENT SOCIETY - POMS, 2019, Washington, D.C. Proceedings of annual conference of the production and operations management society. Washington, D.C.: POMS, 2019.
5.3 REFERENCES
AIANE, D.; EL-AMRAOUI, A.; MESGHOUNI, K. A new optimization approach for a home health care problem. In: INTERNATIONAL CONFERENCE ON INDUSTRIAL ENGINEERING AND SYSTEMS MANAGEMENT, 2015, Seville. Proceedings of the international conference on industrial engineering and systems management. Seville: Springer, 2015, p. 285-290.
BENZARTI, E.; SAHIN, E.; DALLERY, Y. Operations management applied to home care services: analysis of the districting problem. Decision support systems, 2013. v. 55, n. 2, p.587-598.
BLAIS, M.; LAPIERRE, S. D.; LAPORTE, G. Solving a home–care districting problem in an urban setting. Journal of the operational research society, 2003. v. 54, n. 11, p. 1141–1147. Disponível em: <http://dx.doi.org/ 10.1057/palgrave.jors.2601625>.
CONASS – Conselho Nacional de Secretários de Saúde. Com foco no fortalecimento e ampliação da atenção primária à aaúde e na regionalização, ministro confirma criação de nova secretaria de atenção básica. Notícias CONASS, 2019. Disponível em: <http://www.conass.org.br/com-foco-no-fortalecimento-e-ampliacao-da-atencao-primaria-saude-e-na-regionalizacao-ministro-da-saude-confirma-criacao-de-nova-secretaria-de-atencao-basica/>. Acesso em: 27 jan. 2019.
EN-NAHLI, L. et al. Local search analysis for a vehicle routing problem with synchronization and time windows constraints in home health care services. IFAC papersonline, 2016. v.. 49, n. 12, p. 1210-1215.
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EN-NAHLI, L.; ALLAOUI, H.; NOUAOURI, I. A multi-objective modelling to human resource assignment and routing problem for home health care services. IFAC papersonline, 2015. v. 48, n. 3, p. 698-703.
GUTIÉRREZ, E.; VIDAL, C. A home health care districting problem in a rapid-growing city., Ingeniería y universidad, 2015. v. 19, n. 1, p. 87-113.
LIN, M. Y. et al. An effective greedy method for the Meals-On-Wheels service districting problem. Computers & industrial engineering, 2017. v. 106, p. 1-19.
UN – United Nations. OMS define 10 prioridades de saúde para 2019. Nações Unidas no Brasil, 2019. Disponível em: <https://nacoesunidas.org/oms-define-10-prioridades-de-saude-para-2019/amp/>. Acesso em: 27 jan. 2019.
WHO – World Health Organisation. Declaration of Alma-Ata. In: INTERNATIONAL CONFERENCE ON PRIMARY HEALTH CARE, 1978, Alma-Ata, USSR. Proceedings of international conference on primary health care. Alma-Ata: WHO, 1978, p. 1-3. Disponível em: <https://www.who.int/publications/almaata_declaration_en.pdf>. Acesso em: 24 jan. 2019.
WHO – World Health Organisation. Declaration of Astana. In: GLOBAL CONFERENCE ON PRIMARY HEALTH CARE, 2018, Astana, Kazakhstan. Proceedings of global conference on primary health care. Astana: WHO, 2018, p. 1-12. Disponível em: < https://www.who.int/docs/default-source/primary-health/declaration/gcphc-declaration.pdf>. Acesso em: 27 jan. 2019.
APPENDIX A – LONGITUDINAL BIBLIOMETRIC ANALYSIS
The methodology used for building the Social Network of Authors is depicted in Section
A.1.
In Section A.2 an approach to carry out the analysis of the evolution of a specific
research field is shown, being in the present case the mathematical models to solve
the territory alignment problem.
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Section A.3 describes the methodology for mapping change analysis.
A.1 BIBLIOMETRIC MAPPING
Two types of maps can be distinguished that are commonly used in bibliometric
research: distance-based maps and graph-based maps. Distance based maps are
maps in which the distance between two items reflects the strength of the relation
between the items. A smaller distance generally indicates a stronger relation. In many
cases, items are distributed quite unevenly in distance-based maps. Graph-based
maps are maps in which the distance between two items need not reflect the strength
of the relation between the items. Instead, lines are drawn between items to indicate
relations. Items are often distributed in a fairly uniform way in graph-based maps.
Distance-based and graph-based maps both have advantages and disadvantages.
The present work uses a computer program called VOSviewer, a program developed
for constructing and viewing bibliometric maps. VOSviewer can for example be used
to construct maps of authors or journals based on co-citation data or to construct maps
of keywords based on co-occurrence data. To construct a map, the program uses the
VOS (visualisation of similarities) mapping technique (VAN ECK; WALTMAN, 2007b),
which supports only distance-based maps.
VOSviewer constructs a map based on a co-occurrence matrix. The construction of a
map is a process that consists of three steps. In the first step, a similarity matrix is
calculated based on the co-occurrence matrix. In the second step, a map is
constructed by applying the VOS mapping technique to the similarity matrix. And
finally, in the third step, the map is translated, rotated, and reflected.
In Step 1, a similarity matrix can be obtained from a co-occurrence matrix by
normalising the latter matrix, that is, by correcting the matrix for differences in the total
number of occurrences or co-occurrences of items. The program uses a similarity
measure known as the association strength (VAN ECK; WALTMAN, 2007a; VAN ECK
et al., 2006). This way, the similarity 𝑠𝑖𝑗 between two items 𝑖 and 𝑗 is calculated as
160
𝑠𝑖𝑗 = 𝑐𝑖𝑗
𝑤𝑖𝑤𝑗 (A.1)
where 𝑐𝑖𝑗 denotes the number of co-occurrences of items 𝑖 and 𝑗 and where 𝑤𝑖 and 𝑤𝑗
denote either the total number of (statistically independent) occurrences of items 𝑖 and
𝑗 or the total number of co-occurrences of these items.
In Step 2, the program uses the VOS mapping technique to construct a map based on
the similarity matrix obtained in Step 1. Let 𝑛 denote the number of items to be mapped.
The VOS mapping technique constructs a two-dimensional map in which the items
1, … , 𝑛 are located in such a way that the distance between any pair of items 𝑖 and 𝑗
reflects their similarity 𝑠𝑖𝑗 as accurately as possible. Items that have a high similarity
should be located close to each other, while items that have a low similarity should be
located far from each other. The idea of the VOS mapping technique is to minimise a
weighted sum of the squared Euclidean distances between all pairs of items. The
higher the similarity between two items, the higher the weight of their squared distance
in the summation. To avoid trivial maps in which all items have the same location, the
constraint is imposed that the average distance between two items must be equal to
1. In mathematical notation, the objective function to be minimised is given by
𝑉(𝑥1, … , 𝑥𝑛) = ∑ 𝑠𝑖𝑗 𝑖 < 𝑗 ||𝑥𝑖 − 𝑥𝑗||2
(A.2)
where the vector 𝑥𝑖 = (𝑥𝑖1, 𝑥𝑖2) denotes the location of item 𝑖 in a two-dimensional
map and where || • || denotes the Euclidean norm. Minimization of the objective
function is performed subject to the constraint
2
𝑛(𝑛−1) ∑ ||𝑥𝑖𝑖<𝑗 − 𝑥𝑗 || = 1 (A.3)
The constrained optimisation problem of minimising Equation (A.2) subject to Equation
(A.3) is solved numerically in two steps. The constrained optimisation problem is first
converted into an unconstrained optimisation problem. The latter problem is then
solved using a so-called majorisation algorithm.
The optimisation problem discussed in Step 2 does not have a unique globally optimal
solution. This is because, if a solution is globally optimal, any translation, rotation, or
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reflection of the solution is also globally optimal (BORG; GROENEN, 2005). Therefore,
in Step 3, the software applies three transformations to the solution in order to produce
consistent results: (i) translation: the solution is translated in such a way that it
becomes centred at the origin; (ii) rotation: the solution is rotated in such a way that
the variance on the horizontal dimension is maximised. This transformation is known
as principal component analysis; and (iii) reflection: if the median of 𝑥11, … , 𝑥𝑛1 is larger
than 0, the solution is reflected in the vertical axis. If the median of 𝑥12, … , 𝑥𝑛2 is larger
than 0, the solution is reflected in the horizontal axis.
A.2 LONGITUDINAL ANALYSIS OF A RESEARCH FIELD
The construction of maps from bibliometric information (GARFIELD, 1994) is a
technique used to show the different themes or topics treated by a scientific field in a
given time. Different bibliometric information can be used in order to build a bibliometric
map. Depending on the information used, different aspects of the research field can
be studied. Co-word analysis and co-citation analysis are tools widely used to do this.
Whereas co-citation is used to analyse the structure of a scientific research field, co-
word analysis is used to analyse the conceptual structure. That is, co-word analysis
allows us to discover the main concepts treated by the field and it is a powerful
technique for discovering and describing the interactions between different fields in
scientific research. Although both techniques are useful for mapping science, the aim
of our approach is to discover the conceptual evolution of a research field, and,
therefore, co-word analysis is more suitable.
Formally, the methodological foundation of co-word analysis is the idea that the co-
occurrence of key terms describes the content of the documents in a file (CALLON;
COURTIAL; LAVILLE, 1991).
With a list of the important keywords of the research field a graph can be built, where
the keywords are the nodes and the edges between them represent their relationships.
Two nodes (keywords) are connected if they are presented in the same documents.
We can add to each edge a weight representing how important the associated
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relationship in the whole corpus is (i.e., the set of documents belonging to the research
field under study.)
As result of the co-word analysis, a set of detected themes is obtained for each
subperiod studied. In order to represent the results in a visual way, different
visualisation techniques can be used. In the proposed approach the results are
visualised by means of strategic diagrams and the conceptual evolution is shown
through thematic areas.
To sum up, the stages carried out by our approach are: (i) to detect the themes treated
by the research field by means of co-word analysis for each studied subperiod; (ii) to
layout in a low dimensional space the results of the first step (themes); (iii) to analyse
the evolution of the detected themes through the different subperiods studied, in order
to detect the main general thematic areas of the research field, their origins and their
inter-relationships; and (iv) to carry out a performance analysis of the different periods,
themes and thematic areas, by means of quantitative and impact measures. The
approach indicated above was implemented in a step-by-step process by using
SciMAT 1.1.03 version software.
In Step 1, the software reads the raw data from the researched bases, with all included
items such as authors’ names, document title, journal, year, abstract, citations,
references, and keywords. This complete data set will form the knowledge base. Then,
the individual keywords should be aggregated in groups of keywords. In addition,
should be selected the subperiods for the longitudinal analysis.
Step 2 is the selection of the unit of analysis. As the unit of analysis can be selected
any of the groups existing in the knowledge base: Authors, Words (Author’s words,
Source’s words, and/or Extracted words), References, Authors-reference, and
Sources-reference. In the present case, Words should be selected with its three
options marked.
Step 3 is the data reduction. For each selected subperiod, a minimum frequency
threshold must be selected. That is, only the item that appears in almost 𝑛 documents
in a given subperiod will be taken into account.
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Step 4 is the selection of the kind of matrix: co-ocurrence or coupling. The options are:
Co-ocurrence, Basic coupling, Aggregated coupling based on authors, and
Aggregated coupling based on journals. In this proposed approach, the relevant
information consists of the co-occurrence frequencies of keywords. The co-occurrence
frequency of two keywords is extracted from the corpus of documents by counting the
number of documents in which the two keywords appear together.
Step 5 is the network reduction. The network has to be filtered using a minimum edge
value threshold. For each selected subperiod, a threshold value must be set. That is,
only the edges with a value greater or equal to 𝑛 in a given subperiod will be taken into
account.
Step 6 is the choice of similarity measure used to normalise the network. Similarities
between items are calculated based on frequencies of keywords’ co-occurrences.
Different similarity measures can be selected: Association strength, Equivalence
index, Inclusion index, Jaccard’s index and Salton’s cosine (HAMERS et al., 1989).
For instance, the Equivalence index 𝑒𝑖𝑗 is defined as:
𝑒𝑖𝑗 = 𝑐𝑖𝑗
2
𝑐𝑖𝑐𝑗 (A.4)
where 𝑐𝑖𝑗 is the number of documents in which two keywords 𝑖 and 𝑗 co-occur and 𝑐𝑖
and 𝑐𝑗 represent the number of documents in which each one appears. When the
keywords always appear together, the equivalence index equals unity; when they are
never associated, it equals zero.
Step 7 is the choice of the clustering algorithm used to locate subgroups of keywords
that are strongly linked to each other. Different clustering algorithms can be used:
Simple centres algorithm (COULTER; MONARCH; KONDA, 1998; COBO et al., 2011),
Single link clustering algorithm (SMALL; SWEENEY, 1985), Complete link clustering
algorithm, Average link clustering algorithm, and Sum link clustering algorithm.
Step 8 is the selection of the documents mapper used in the performance analysis for
co-occurrence networks. Different document mappers are available: Core mapper
(COBO et al., 2011); Intersection mapper, which adds the documents that have all the
164
items of the cluster; 𝑘-core mapper, which adds the documents that have at least 𝑘
items (to be selected) in common with the cluster; Secondary mapper; and Union
mapper, which adds documents that have at least one item in common with the cluster
(this is the union of the documents associated with the core and secondary mappers).
Step 9 is the performance and quality analysis. SciMAT uses the number of documents
as performance measure by default. For selecting of the quality measures there are
several indices: h-index (ALONSO et al., 2009; HIRSCH, 2005), g-index (EGGHE,
2006), q2-index (CABRERIZO et al., 2010), hg-index (ALONSO et al., 2010), Average
citations, Sum citations, Max citations, and Min citations.
At last, the Step 10 is the selection of the similarity measure used to build the
longitudinal map. For the evolution map there are some options: Association strength,
Equivalence index, Inclusion index, Jaccard’s index, and Salton’s cosine. For the
overlapping map we have the same possibilities of choice.
A.3 MAPPING CHANGE ANALYSIS
In this subsection a methodology as stated by Rosvall and Bergstrom (2010) of how to
generate significance clusters and alluvial diagrams for mapping change in networks
is presented. However, as the mentioned methodology is suitable for large networks,
some modifications were implemented to adapt to small and medium networks.
Because this method assesses how much confidence we should have in the clustering
of a network, we can detect, highlight, and simplify the significant structural changes
that occur over time in small, medium or large networks, for example, citation networks,
and co-word networks.
This approach may be applied for any clustering algorithm. The choice of algorithm
depends on the network type (undirected, directed, unweight, weighted) and the scope
of the study. Here we focus on the general case of weighted directed networks. We
also assume that the weight of the links can be described by a Poisson-like process.
That is, the weights represent, or can be modelled by, independent events in time. This
can be generalised to other distributions of link weights.
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The method consists of four steps: (i) cluster the original networks observed at each
subperiod; (ii) generate and cluster the bootstrap replicate networks for each
subperiod; (iii) determine significance of the clustering for at each subperiod; and (iv)
generate an alluvial diagram to illustrate changes between subperiods.
A.3.1 CLUSTER REAL-WORLD NETWORK
In Step 1, we partition the network 𝐺 into the modular description 𝑀. In the modular
description, each node is assigned to one and only one module. The number of
modules depends on the network and the objective function of the clustering algorithm.
To capture the dynamics across the links and nodes in directed weighted networks,
we use the map equation as the objective function, as described in Appendix S1 of
(ROSVALL; BERGSTROM, 2010):
𝐿(𝑀) = 𝑞⃕ 𝐻(𝑄) + ∑ 𝑝↻𝑖𝑚
𝑖=1 𝐻 (𝑃𝑖) (A.5)
The first term of this equation gives the average number of bits necessary to describe
movement between modules, and the second term gives the average number of bits
necessary to describe movement within modules. In the first term, 𝑞⃕ is the probability
that the random walker switches modules on any given step, and 𝐻(𝑄) is the entropy
of the module names, i.e. the frequency-weighted average length of codewords in the
index codebook. In the second term, 𝐻(𝑃𝑖) is the entropy of the within-module
movements - including an “exit code” to signify departure from module 𝑖, i.e. the
frequency-weighted average length of codewords in module codebook 𝑖 – and the
weight 𝑝↻𝑖 is the fraction of within-module movements that occur in module 𝑖, plus the
probability of exiting module 𝑖 such that ∑ 𝑝↻𝑖𝑚
𝑖=1 = 1 + 𝑞⃕ .
For a given network partition 𝑀, the map equation specifies the theoretical limit 𝐿(𝑀)
of how concisely we can describe the trajectory of a random walker on the network.
The underlying code structure of the map equation is designed such that the
description can be compressed if the network has regions in which the random walker
tends to stay for a long time. Therefore, with a random walk as a proxy for real flow,
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minimising the map equation over all possible network partitions reveals important
aspects of network structure with respect to the dynamics on the network.
To efficiently describe a random walk using a two-level code of this sort, the choice of
partition 𝑀 must reflect the patterns of flow within the network, with each module
corresponding to a cluster of nodes in which a random walker spends a long period of
time before departing for another module. To find the best such partition, we therefore
seek to minimise the map equation over all possible partitions 𝑀.
The map equation can be performed by cluster_infomap function of igraph package in
R library (CSARDI; NEPUSZ, 2006).
A.3.2 GENERATE AND CLUSTER BOOTSTRAP-WORLD NETWORKS
Step 2 is the process of generating and clustering bootstrapped networks from the
original networks at each subperiod.
The bootstrap is a statistical method for assessing the accuracy of an estimate by
resampling from the empirical distribution. This method is particularly powerful when
the variance of the estimator cannot be derived analytically or when the underlying
distribution is not accessible. Because the cluster assignments are a result of a
computational method and the network is idiosyncratic by nature, the bootstrap is
indispensable for the process described here.
To generate a single bootstrap replicate network 𝐺𝑏∗, we resample every link weight
𝑤𝛼𝛽 (directed link from node 𝛼 to node 𝛽 with weight 𝑤) of the original network 𝐺 from
a Poisson distribution with mean equal to the original link weight 𝑤𝛼𝛽. That is, 𝑤∝𝛽∗
~ 𝑃𝑜𝑖𝑠(𝑤𝛼𝛽) for each link in the bootstrap network. Because of the parametric
resampling of the link weights, this method formally falls under parametric
bootstrapping. If the link weights cannot be modelled by a Poisson process, or if the
links are unweight, the Poisson resampling should be replaced by an appropriate
alternative resampling procedure (KARRER; LEVINA; NEWMAN, 2008; GFELLER;
CHAPPELIER; RIOS, 2005).
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To perform a parametric bootstrap, we first need to define the bootstrap function (which
in our case corresponds to a Poisson distribution) and then call the boot function of
boot package in R library (CANTY; RIPLEY, 2016).
In order to verify if there is a good fit in the structures of both the real-world network
and bootstrap-world networks, we use the ergm package of statnet family in R library
(HUNTER et al., 2008; HANDCOCK et al., 2016). The ergm package allows us to fit
exponential-family random graph (ERG) models to network data sets (HUNTER;
GOODREAU; HANDCOCK, 2008).
Subsequently we partition the bootstrap replicate network with the same clustering
method we used on the original network; this yields the bootstrap modular description
𝑀𝑏∗. This procedure - generating a bootstrap replicate network and clustering it into
modules - is repeated to generate a large number 𝐵 ~ 1000 of bootstrap modular
descriptions 𝑀∗ = {𝑀1∗, 𝑀2
∗, . . . , 𝑀𝐵∗ }.
A.3.3 IDENTIFY SIGNIFICANT ASSIGNMENTS
In Step 3, the basic idea behind significance clustering is that we can look at the
bootstrap replicates to see which aspects of the modular description of the original
network are best supported by the data. Features of the original network that occur in
all or nearly all of the bootstrap replicates are well-supported by the data; features that
occur in only some of the bootstrap replicates are less well-supported.
First, we consider as a feature the assignment of each node to a module. By looking
at the set of bootstrap modular descriptions, we can assess which of these
assignments are strongly supported by the data, and which node assignments are less
certain. To identify the nodes that are significantly assigned to a module, we search
for the largest subset of nodes in each module of the original modular description
𝑀 that are also clustered together in at least 95% of all bootstrap modular descriptions
𝑀∗. To pick the largest subset, we need some measure of size. The size of a subset
could simply correspond to the number of nodes in the subset, but in line with our
general clustering philosophy, we use the volume of flow through the subset. This is
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the total PageRank of the cluster, which corresponds to the steady-state flow of
random walkers that we use in the information theoretic clustering algorithm. The
PageRank of the cluster can be performed by page_rank algorithm of igraph package
in R library.
Then, we follow the local search with Tabu search instead of the standard simulated
annealing scheme originally used by Rosvall and Bergstrom (2010). The Tabu search
scheme can be performed by tabuSearch package in R library (DOMIJAN, 2012).
A.3.4 CONSTRUCT ALLUVIAL DIAGRAM
In Step 4, to reveal change over time or between states of real-world networks, we
summarize the results of the significance clusterings of the different states 𝐺1, 𝐺2, . .. in
an alluvial diagram. The diagram is constructed to highlight the significant changes,
fusions, and fissions that the modules undergo between each pair of successive states
𝐺 𝑖 and 𝐺𝑖+1. Each significance clustering for a state 𝐺 𝑖 occupies a column in the
diagram and is horizontally connected to preceding and succeeding significance
clusterings by stream fields. Each block in a row of the alluvial diagram represents a
cluster, and the height of the block reflects the size of the cluster (here in units of flow
through the cluster, though other size measures, such as number of nodes, could be
used instead). The clusters are ordered from bottom to top by size, with mutually no
significant clusters placed together and separated by a third of the standard spacing.
We use the stream fields to reveal the changes in cluster assignments and in level of
significance between two adjacent significance clusterings. The height of a stream field
at each end, going from the significant or no significant subset of a cluster in one
column to the significant or no significant subset of a cluster in the adjacent column,
represents the total size of the nodes that make this particular transition. By following
all stream fields from a cluster to an adjacent column, it is therefore possible to study
in detail the mergers with other clusters and the significance transitions. To reduce the
number of crossing stream fields, the stream fields are ordered by the position of the
clusters to which they connect.
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The alluvial diagram can be constructed by alluvial_ts function of alluvial package in R
library (BOJANOWSKI; EDWARDS, 2016).
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