Operations Research Models Applied for Allocation of Public Resources under the
Efficiency-Effectiveness-Equity (3E) Perspective
THESIS
Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in
the Graduate School of The Ohio State University
By
Melissa Pizarro Aguilar
Graduate Program in Industrial and Systems Engineering
The Ohio State University
2012
Master's Examination Committee:
Ramteen Sioshansi, Advisor
Jerald Brevick
Copyright by
Melissa Pizarro Aguilar
2012
Abstract
The main difference among operation research models for the private sector and the
public sector is the inclusion of the equity criteria. In this sense, one of the objectives of
this project was identifying proposed models to solve allocation problems in the public
sector, under the efficiency, effectiveness and equity (3E) principles as the providers of
public services need to guarantee non-discriminatory access to the services to all of the
population, and at the same time make the best use of the limited resources. We identified
various models that include the equity perspective, but we focused on the data
envelopment analysis (DEA) resources allocation model developed by Goaz Golany and
Eran Tamir, as it integrates the 3E. This integration is important as the tradeoffs among
certain decisions at each dimension could affect the performance in other dimension.
The second objective of this thesis is validating a systematic budget allocation
methodology for the primary health care sector in Costa Rica. The institution evaluated is
the area of health Jiménez Núñez (Goicoechea 2) which is composed of 10 Basic Teams
for Comprehensive Care in Health (EBAIS). The EBAIS were selected as the decision
making units for the validation. We applied the model under study progressively, as the
first evaluation done was the DEA to obtain the efficiency scores per EBAIS: four
EBAIS out of ten were found efficient. Then, to demonstrate how the DEA allocation
model improves the efficiency of the DMUs under study, we applied the model for the
ii
multi-input and multi-output scenario. As a result of the proposed allocation, now eight
EBAIS were found efficient. Finally, we include the equity perspective into the analysis.
The DEA proved to be a useful management tool as it performs the efficiency assessment
from a comparison between DMUs through data and among data by generating the
efficiency frontier. To run this operations research model, various analytical choices must
be made. When these choices depend heavily on the analysts, the appropriateness of the
model depends on the consistency between the analytical choices and the judgment of
experienced decision makers. There may exist generally accepted standards in the
efficiency and equity literature, but the mathematical model developed will always have
to address different factors depending on the object of study.
The DEA model has been widely used in health institutions and many other public sector
organizations in the world, but this is the first application at the primary health system at
Costa Rica and additionally it is including the equity perspective, as most of the literature
shows DEA case studies for efficiency assessment only. We can also think of
standardizing and systematizing the decision making at the health areas by disseminating
the use of the DEA-RAM with equity model and then applying the extensions along the
model to be broadened to incorporate resource allocation at the national level as it would
be valuable to develop this analysis for the 103 health areas at Costa Rica as DMUs.
iii
Dedication
This thesis is dedicated to my parents, my inspiration.
iv
Acknowledgments
I would like to express my deepest gratitude to my advisor, Dr. Ramteen Sioshansi for his
interest, guidance and support through the development of this project. I thank also the
director of the area of health Jiménez Nuñez, Dr. Pedro González and the chief of the
EBAIS at this area of health, Esteban Avendaño for their interest in this thesis. Special
thanks to Dr. Brevick, member the supervisory committee.
v
Vita
2001................................................................Methodist High School, Costa Rica
2007................................................................B.S. Industrial Engineering, University of
Costa Rica
2008 to 2011 ..................................................Faculty, Department of Industrial
Engineering, University of Costa Rica
2011................................................................Fulbright Faculty Development Program
Grant Award
Fields of Study
Major Field: Industrial and Systems Engineering
vi
Table of Contents
Abstract ............................................................................................................................... ii
Dedication .......................................................................................................................... iv
Acknowledgments............................................................................................................... v
Vita..................................................................................................................................... vi
List of Tables ..................................................................................................................... ix
List of Figures ................................................................................................................... xii
Chapter 1: Introduction ...................................................................................................... 1
Chapter 2: Literature Review .............................................................................................. 4
The 3E Perspective in the Public Sector ......................................................................... 5
Operation Research Models for the Allocation of Public Resources .............................. 6
The Data Envelopment Analysis (DEA) ....................................................................... 13
Equity Assessment in Health Care ................................................................................ 20
Chapter 3: The Allocation Model ..................................................................................... 24
Chapter 4: Case Study....................................................................................................... 30
The National Health System of Costa Rica................................................................... 30
Health Area Jiménez Núñez (Goicochea 2) .................................................................. 33
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DEA-RAM Equity Model for the Multi Output Scenario ............................................ 36
Chapter 5: Conclusions and Future Work ......................................................................... 50
References ......................................................................................................................... 52
Appendix A: Input Data .................................................................................................... 55
Appendix B: DEA Weight Results ................................................................................... 59
Appendix C: DEA RAM Results ...................................................................................... 60
Appendix D: Calculating the Gini Measure...................................................................... 63
Appendix E: DEA RAM with Equity Results .................................................................. 66
viii
List of Tables
Table 1.Number of users at each EBAIS .......................................................................... 34
Table 2. Inputs for each EBAIS, year 2011, area of health Goicoechea 2 ....................... 38
Table 3. Outputs for each EBAIS, year 2011, area of health Goicoechea 2 ..................... 39
Table 4. Districts served at area of health Goicoechea 2 .................................................. 40
Table 5. District assignment per EBAIS at area of health Goicoechea 2 ......................... 41
Table 6. Summary of the NBI incidences per district and grouping. ............................... 43
Table 7. Efficiency scores for the EBAIS obtained by the DEA methodology. ............... 44
Table 8. Comparison of the results of the DEA-RAM model between the previous
period’s budget and the current budget to allocate for the next period ............................ 45
Table 9. Equity constraints for each EBAIS, year 2011, area of health Goicoechea 2 .... 46
Table 10. Gini measure per equity unit ............................................................................. 47
Table 11. Optimal solutions per scenario, DEA-RAM with equity.................................. 48
Table 12. Annual costs in colones for each of the EBAIS at area of health Jimenez
Nunez, 2011 ...................................................................................................................... 55
Table 13. Annual costs in colones for each of the EBAIS in area of health Jimenez
Nunez, 2011 ...................................................................................................................... 56
Table 14. Annual costs in dollars for each of the EBAIS at area of health Jimenez Nunez,
2011................................................................................................................................... 57
ix
Table 15. Total time in hours per month programmed and used for medical appointments
at area of health Jimenez Nunez, 2011 ............................................................................. 58
Table 16. Total time in hours per year programmed and used for medical appointments at
area of health Jimenez Nunez, 2011 ................................................................................. 58
Table 17. DEA weights for the inefficient EBAIS at Goicoechea 2 ................................ 59
Table 18. DEA-RAM weights for the inefficient EBAIS at Goicoechea 2, with previous
period’s budget.................................................................................................................. 60
Table 19. DEA-RAM weights for the inefficient EBAIS at Goicoechea 2, with budget to
allocate .............................................................................................................................. 61
Table 20. DEA-RAM input allocation for the EBAIS at Goicoechea 2 ........................... 61
Table 21. DEA-RAM output allocation for the EBAIS at Goicoechea 2 ......................... 62
Table 22. Gini measure for the budget allocation in the year 2011, using the users with
hypertension as the equity unit. ........................................................................................ 64
Table 23. Gini measure for the budget allocation in the year 2011, using the users with
diabetes as the equity unit. ................................................................................................ 64
Table 24. Gini measure for the budget allocation in the year 2011, using the users over 65
years with diabetes as the equity unit. .............................................................................. 65
Table 25. Input and output allocation, E1, with previous period’s budget ...................... 67
Table 26. Input and output allocation, E1, with current budget to allocate ..................... 67
Table 27. Input and output allocation, E2, with last period’s budget ............................... 68
Table 28.Input and output allocation, E2, with current budget to allocate ...................... 68
Table 29. Input and output allocation, E3, with previous period’s budget ....................... 69
x
Table 30. Input and output allocation, E3, with current budget to allocate ...................... 70
Table 31. Weights, E1, with previous period’s budget ..................................................... 71
Table 32. Weights, E1, with current budget to allocate. ................................................... 71
Table 33. Weights, E2, with previous period’s budget ..................................................... 72
Table 34. Weights, E2, with current budget to allocate .................................................... 72
Table 35. Weights, E3, with previous period’s budget ..................................................... 73
Table 36.Weights, E3, with current budget to allocate ..................................................... 73
xi
List of Figures
Figure 1. Illustration of a Lorentz curve (Mandell, 1991) ................................................ 11
Figure 2. Example to illustrate the concept of a reference set. ......................................... 19
xii
Chapter 1: Introduction
Today, there is not much available literature on the equity, efficiency and effectiveness
tradeoffs and their application in the public sector, from the operations research
perspective. This is an important need as the managers of public services require reliable
tools that can provide a clear idea of what would happen if they make certain choices
while allocating public services and/or resources. The decision makers in the public
sector may be under a great pressure when managing, most of the time, very limited
resources, as the stakeholders in this case are all the tax payers and even further, the
society. In an effort to find which models have been used to allocate resources in the
public sector, the existing literature was surveyed. From this first inquiry, the conclusion
is that the difference among resource allocation models between the private and the
public sector is the incorporation of the equity perspective in the latter. Also as a result of
the research done, we centered our attention in three specific works that use three
interesting types of models. The first, a linear integer goal programming model
formulated to support a state-level resource allocation process with geographic equity,
developed by Tingley and Liebman. The second is Marvin Mandell’s model, which
examines the tradeoff between effectiveness and equality using the Gini measure of
inequality, under the structure of an absolute goal program. Finally, Golany and Tamir
propose a model based on data envelopment analysis (DEA) which is formulated as a
linear program. The appealing characteristic about these three models is that one author
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builds from what the previous author has done, improving the model by eliminating its
deficiencies.
Golany’s mathematical program with equity was found to be the most appropriate model
for the allocation of resources in the public sector, as it considers the three perspectives,
efficiency, effectiveness and equity, and can be used in a multi-input, multi-output
scenario. In our thesis we explain in detail the DEA-based resource allocation model
(DEA-RAM), and then we clarify how it should be modified to include the equity
constraints (based on those proposed by Mandell), to finish with the description of the
case when various outputs are analyzed. We also validated this model by using it in a
case study at the primary health care services of Costa Rica. The reason for our interest in
the health sector is better explained by Jacobs et al. (2006): “The international explosion
of interest in measuring the inputs, activities and outcomes of health systems can be
attributed to heightened concerns with the costs of health care, increased demands for
public accountability and improved capabilities for measuring performance”.
The institution chosen to validate the model is the area of health Jiménez Núñez
(Goicoechea 2). Health areas were defined to provide comprehensive health services to
the population located in a defined territorial space, and they are responsible for the
primary care level. The primary care at Costa Rica includes basic health services that
perform actions of health promotion, disease prevention, and curing and rehabilitation of
lower complexity diseases. These actions are carried out by the members of the support
teams and the Basic Teams for Comprehensive Care in Health (EBAIS). The area of
health Jiménez Núñez is composed of 10 EBAIS. For the application of the DEA-RAM
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equity model at the health area Goicoechea 2 we choose the main components of the
model by using references in the literature but also considering the most experienced
criterion of the decision makers at the area of health. The EBAIS where chosen as the
decision making units of the model as they capture the entire production process of
interest. We applied the model under study progressively, as the first evaluation done was
the data envelopment analysis (DEA) to obtain the efficiency scores per EBAIS. Then, to
demonstrate how the DEA allocation model improves the efficiency of the DMUs under
study by calculating the appropriate amounts of inputs to be used and the outputs
obtained from this assignment, we applied the model for the multi-input and multi-output
scenario. Finally, we include the equity perspective into the analysis.
This thesis is organized into five chapters. Chapter 2 includes the literature review done
to develop the thesis. This will help the reader to understand the concepts of efficiency,
effectiveness, and equity, which are the focus of the models that will be reviewed. A
summarized explanation of how the models of Kim Tingley and Judith Liebman, Marvin
Mandell, and Boaz Golany and Eran Tamir operate is also included, and finally to
complete the section a small survey on how the concepts of efficiency and equity are
assessed in health care is presented. Chapter 3 explains in depth the methodology to be
followed for the application of the Efficiency-Effectiveness-Equity (3E) allocation
model. Chapter 4 includes the case study of primary health care in Costa Rica as an
application and its results, Chapter 5 concludes the paper.
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Chapter 2: Literature Review
One of the objectives of this thesis is to adapt operations research methodologies for the
public sector. Through a bibliographical review, we came across various applications
related to the allocation of resources in the public sector, and specifically those who seek
to link the concepts of efficiency, effectiveness and equity in their execution. The first
section of this chapter will explain the very basic concepts that must be understood as
they are going to be used throughout this document. These are the concepts of public
service, efficiency, effectiveness and equity. On the second section, we are going to
describe the three operations research models chosen to illustrate the allocation of
resources in the public sector under the 3E (efficiency, effectiveness, and equity)
perspective. As the main component of one of these models is data development analysis
(DEA), we focus more on explaining its use in the third section of this chapter. In the
fourth section, we describe how the concept of equity has been assessed by different
authors, as the most important difference among the allocation models for the private
sector and the public sector is the inclusion of the equity criteria. In this last section, the
reader will notice that the focus is on the application of equity of health care allocation.
This is because, as we explained in the introduction, the case study developed in this
thesis is in the health sector.
4
The 3E Perspective in the Public Sector
A public service refers to any of the common, everyday services provided by federal,
state, and local governments (Savas, 1978). Examples of these services include health
care, education, water supply, wastewater collection and treatment, among others. The
majority of public services are used by citizens simultaneously and users cannot be
excluded. Another important aspect that the provider of these services needs to consider
is how to guarantee non-discriminatory access to the services to all of the population.
Therefore, market mechanisms that are adequate to provide private goods and services
are inadequate for supplying public goods and services (Savas, 1978).
Continuing with the characterization of public services, they are either provided at
facilities that are geographically distributed or delivered throughout an area by resources
allocated to that area. Consequently, the performance of public services is determined to
a large extent by the spatial distribution of resources that provide the service (Savas,
1978) and the allocation of public resources should be guided by the maximization of
social welfare (Athanassopoulos, 1998). To assess this performance three measures are
identified as the key indices for public services: efficiency, effectiveness, and equity (the
3E). Efficiency and effectiveness, as they have being used widely in the private sector,
are insufficient performance measures for public services. For these kinds of services
equity is of identical importance, and the three measures interact.
It is important to have a clear notion of the measures mentioned, as stated by Savas and
Golany. Efficiency measures the ratio of service outputs to service inputs; it seeks to
achieve “more for less” (Golany & Tamir, 1995). Unfortunately in the public sector, in
5
the majority of cases inputs are easier to define and measure than outputs. On the other
hand, effectiveness measures how well the need for the service is satisfied. In simple
words, effectiveness measures the difference between observed outputs and the desired
goals (Golany et al., 1995). It is a measure of the adequacy of service relative to need,
and incorporates the notion of service quality. Once again, although it is difficult to
measure effectiveness in public services it is not impossible. Also, a service can be
efficient but ineffective and vice versa. Finally, equity refers to the fairness, impartiality,
or equality of service. It measures the degree of fairness in the allocation of resources or
the distribution of outputs among the units that are evaluated. A service can be efficient
and effective but it could be perceived as inequitable if it fails to reach or treat all
segments of the population similarly. Considering this, we can notice that there exist
tradeoffs among the equity, effectiveness, and efficiency of a service.
According to Savas, different equitable formulae can be used for the allocation of public
services. They are based on four broad principles: equal payments, equal outputs, equal
inputs, and equal satisfaction of demand. Each of the corresponding formulae are clearly
equitable in a certain aspect, but inequitable in terms of other competing principles or
even specific aspects. Considering this phenomenon, it is said that equity is a matter of
values and each analyst or decision maker will have different approaches to the same
problem.
Operation Research Models for the Allocation of Public Resources
In the search for operation research models applied in the public sector, we came across
three allocation models. The interesting thing about these three models is that they are
6
related, in the sense that one author took the model of the previous one and improved it.
The first one is a linear integer goal programming model formulated to support a statelevel resource allocation process with geographic equity, developed by Tingley and
Liebman. The second is Marvin Mandell’s model which examines the tradeoff between
effectiveness and equality using the Gini measure of inequality, under the structure of an
absolute goal program. Finally, a model based on Data envelopment analysis(DEA)
proposed by Golany and Tamir, which is formulated as a linear program.
The first allocation model for public services we considered was developed in 1984 by
Kim Tingley and Judith Liebman. They develop a linear integer goal programming model
to support a state-level resource allocation process with geographic equity in the United
States Department of Agriculture Special Supplement Food Program for Women, Infants,
and Children (WIC). WIC was authorized in 1972 in the United States with the goal of
reducing infant mortality and promoting proper physical and mental development of
children through better nutrition (Tingley & Liebman, 1984). Three problems were
identified by Tingley and Liebman and solved through the mentioned mathematical
program: the lack of a systematic approach to allocating funds to local agencies, the
exclusion of total numbers of potentially eligible participants in the decision process, and
the need to consider the priority subgroups (the federal regulations specify six subgroups
within the target population of women, infants and children, that are ranked in order of
priority for WIC services) when allocating funds.
A special characteristic of their approach was incorporating the subjective attitudes of
state-level WIC administrators, the experts, who have a strong familiarity with the WIC
7
program in their states. This was achieved by allowing the administrators, as the future
users of the model, to choose the relative importance of each goal as weights at the
objective function. This peculiarity is important as then the model can change as the
goals change in time. This also allows evaluating different weights for each goal and
considering how the final allocation varies. The model proposed is useful in all public
sector programs characterized by multiple objectives and hierarchical decision making
(Tingley et al.,1984). The authors defined extensions along the model to be broadened to
incorporate resource allocation at the federal-aggregate level, and to be used in the case
of allocating budget cuts.
The model created by Tingley and Liebman was developed to allocate funds for
expanding local WIC agencies and starting up new local agencies through an integer goal
programming formulation. Through this model the service level of each priority group
across the state is seen as a separate goal, with specified target value and individually
assigned weights that serve as penalties for deviating from that target (Tingley et
al.,1984). To do so, the model requires several inputs such as the estimation of target
values (desired number of additional participants to be served in each priority group
across the entire state) for each priority group, the weights to be assigned to each priority
group, and the estimation of the cost of serving the proposed number of additional
participants in each priority group at each local agency (Tingley et al.,1984).
The allocation process is modeled as a series of decisions to fund or not fund each
priority group (binary variables) at each existing WIC agency and to fund or not fund
each proposed new WIC agency (Tingley et al., 1984). These decisions are subject to a
8
limitation on the state budget for expanding the WIC coverage. The equity dimension is
considered as geographic equity is incorporated into the model by dividing the state into
regions and setting lower bounds on the number of agencies receiving additional funds
within each region. This will prevent a disproportionate amount of funds from flowing
into a large city at the expense of neglecting smaller cities and rural areas or vice versa
(Tingley et al.,1984).
Seven years later, in 1991, Mandell developed a model that can be employed to examine
the tradeoff between effectiveness and equality, the latter defined in terms of the Gini
measure, which we will describe later. For Mandell, public sector’s management science
models have frequently ignored equity considerations entirely, and the most significant
flaw found in those approaches that try to incorporate equity into the mentioned models is
the violation of the principle of transfers. The principle of transfers requires that a
transfer of service units from a subgroup to any relatively worse-off subgroup result in an
improvement in the measure (Mandell, 1991). Mandell summarizes in three categories
the commonly used approaches to incorporate equity which violate the principle of
transfers. The first category is specifying equity in terms of the minimum and/or
maximum levels of service that can be provided to any population group. This is where
the author mentions the model proposed by Tingley and Liebman as an example of this
flaw. In this sense, the geographical equity defined in the model for the WIC violates the
principle of transfers. The second category is defining equity as the range between the
maximum and minimum levels of service, and the third is calculating equity in terms of
the sum of absolute deviations from the mean level of service. In conclusion, although
9
Tingley’s model has special characteristics as incorporating the expertise of the users,
defining priority groups and the possibility of aggregation to be applied at other decision
levels, the approach to incorporate equity is not the best and we should look to
incorporate this dimension with an approach that fulfills the principle of transfers.
Mandell then proposes two related bicriteria mathematical programming models to
identify the tradeoffs between effectiveness and equity that result from alternative
allocations of service resources among different service delivery sites. Mandell uses an
objective function for each the overall output and equity, and expresses equity in terms of
the Gini coefficient. The Gini measure can address inequality in inputs as well as
inequality in the outputs, as a result of resource allocation. For understanding the Gini
coefficient, using a Lorenz curve (see Figure 1) is the clearest approach. A given point on
a Lorenz curve, (x, y), indicates the cumulative proportion of service received by the
most disfavored percentage of the population (in terms of the service). The diagonal line
connecting the points (0, 0) and (1, 1) represents the Lorenz curve that would be obtained
if services were distributed perfectly equitably, this means each group with the same level
of services received per equity unit (Mandell, 1991). The Gini coefficient is defined as
the ratio of the area between the Lorenz curve and the diagonal line (the shaded area in
Figure 1) to the total area below the diagonal line (Mandell, 1991).
10
Figure 1. Illustration of a Lorentz curve (Mandell, 1991)
Mandell uses an absolute goal programming structure to solve the model. The target
function of inequality is transformed into a constraint that ensures that its level does not
exceed an arbitrary small parameter. The largest value this parameter can have is
calculated by solving the problem without the equity constraints. Running the model with
different parameter values creates a set of possible optimal combinations of output and
11
inequality levels that constitute a curve or frontier from which a suitable combination is
to be selected by the decision makers (Mandel, 1991).
In 1995, Golany and Tamir developed a resource allocation model based on data
envelopment analysis(DEA) to evaluate the relative efficiency of decision making units
(DMUs) in the public sector. The model extends the original DEA methodology from
measuring efficiency to include the evaluation of effectiveness and equality measures as
well. Golany and Tamir include the Gini coefficient as an important part of their model to
incorporate the equity measure, but they identify some drawbacks that needed to be
corrected from Mandell’s proposal. We summarize these downsides in three aspects, as
stated by the authors. First, Mandell’s model basically ignores the efficiency criterion by
assuming all units are operating at full efficiency. Second, he defines as an important
input of the model a certain production function to describe the behavior of the units
studied; all units are then assumed to be IID (independent and identically distributed).
Finally, Mandell's model is limited to single output scenarios. DEA can include multiple
inputs as well as multiple outputs. The model will choose a set of weights that achieves
the highest efficiency rating for each DMU, while assuring that these weights do not
cause any other DMU to have an efficiency rating higher than 1 (Golany et al., 1995).
This model will be described extensively in Chapter 3, as we chose to apply this model to
the case study. The DEA approach will be explained in the next section.
12
The Data Envelopment Analysis (DEA)
According to Jacobs, Smith & Street, there exist two analytic efficiency measurement
techniques: the stochastic frontier analysis (SFA) and the Data envelopment analysis
(DEA). As the author describes, the SFA is parametric method, similar to the
conventional regression analysis, but decomposes the unexplained error in the estimated
function into inefficiency and a two-sided random error. The SFA defines the efficient
behavior by specifying a stochastic (or probabilistic) model of output and maximizing the
probability of the observed outputs given the model. The DEA analysis on the other
hand, is a non-parametric method that uses linear programming methods to infer a
piecewise linear production possibility frontier, seeking those efficient observations that
dominate (or envelop) the others. DEA is a data-driven approach as the location and
shape of the mentioned efficiency frontier is determined by the data, using the simple
notion that an organization that employs less output than another to produce the same
amount of output can be considered more efficient (Jacobs et al., 2006).
Emmanuel Thanassoulis prefers to classify the modeling methods of comparative
performance measurement in two categories: parametric and non parametric. Among the
parametric he describes two approaches: modeling with no allowance for any inefficiency
in production using a regression method and modeling with the allowance for
inefficiency using the stochastic frontier methods. The stochastic frontier methods
estimate average rather than efficient levels of input for given outputs, and they attribute
all differences between estimated and observed levels of input to inefficiency. From the
non parametric perspective, the main method is DEA, where there is no need to
13
hypothesize a functional form linking input to outputs; instead a production possibility set
from the observed input-output correspondences is used.
To support the validity of DEA approaches for the case study elaborated in the present
thesis, we found two major surveys on the topic of efficiency assessment in health care
that have been developed. Jacobs et al. mention a literature survey by Hollingsworth
(2003). In this survey, Hollingsworth identified 189 relevant studies in terms of
efficiency assessment in health and health care. Of these, about 50% are in the hospital
sector, reflecting its central policy importance and the ready availability of data (Jacobs et
al., 2006). Other efficiency studies were applied in physicians, pharmacies, primary care
organizations, nursing homes and purchasers. Hollingsworth finds that DEA is the
dominant approach to efficiency measurement in health care and in many other sectors of
the economy. Also, a study of the United States prepared by the Southern California
Evidence-based Practice Center by McGlynn in 2008 examined 158 articles describing
measures of health care efficiency in the United States. Of these 158 articles surveyed, 93
articles (59%) measured the efficiency of hospitals, followed by studies of physician
efficiency as the second most common (21%). In this more recent review, the two most
frequent approaches used to measure health care efficiency were DEA and SFA. The
great majority of studies have used DEA and its variants, which demonstrates the
credibility of the method.
Data envelopment analysis, in its most fundamental form, is a method to assess the
comparative efficiency of homogeneous operating units. The basic DEA model defines
efficiency as the ratio of a weighted sum of outputs and a weighted sum of inputs.
14
Efficiency analysis is centrally concerned with measuring the competence with which a
set of inputs are converted into a set of valued outputs (Jacobs et al., 2006). To measure
performance of a unit, we need to estimate the input and output levels at which a unit
could be operated if efficient. Mathematically, and using the nomenclature presented by
Jacobs et al., if an organization 0 consumes a vector of M inputs
vector of S outputs
, it’s overall efficiency (
and produces a
) is measured by applying weight
vectors U and V (weights U and V indicate the relative importance of an additional unit
of input or output) to yield
∑
∑
Where
is the amount of the sth output produced by organization 0,
given to the sth output,
and
(1)
is the weight
is the amount of the mth output consumed by organization 0
is the weight given to the mth input (Jacobs et al., 2006).
To construct a DEA model, the components need to be well defined. First, the
appropriate unit of analysis must be chosen. According to Jacobs et al., three criteria must
be evaluated to make this choice: the unit of analysis should capture the entire production
process of interest, they should be decision making units (DMU) the function of which is
to convert inputs into outputs, and finally, the units comprising the analytical sample
should be comparable (for example, they produce the same set of outputs). Second the
outputs and inputs must be identified, and this depends entirely on the DMU under
analysis. For example, in the health care area, they are categorized diversely depending
on the author. Jacobs et al. uses health outcomes (additional health conferred to the
15
patient and patient satisfaction) and health care activities (for example patients treated) as
groupings for the outputs, and for the inputs, labor inputs (hours of labor or costs of
labor) and capital inputs (investment). Third, the analyst should evaluate if there exist
significant environmental constraints that are faced by any of the DMUs studied. An
example of an environmental constraint is the case of mortality rates that are heavily
dependent on the demographic structure of the population under consideration (Jacobs et
al.,2006). If they are found, Jacobs et al. proposes three ways in which environmental
factors can be taken into account in the efficiency analysis. The evaluator may restrict
comparison only to organizations within a similarly constrained environment (by
clustering, comparing DMUs with similar exogenous influences). Another way is
modeling the constraints explicitly, treating them as inputs, and finally the analyst can
adjust the organizational outputs for differences in circumstances before they are
deployed in an efficiency model.
Having defined the components, the DEA formulation can be applied1. Suppose there
exist n DMUs to be analyzed, each of which uses m inputs to produce s outputs. Let
be the amount of input i used by DMU j, and
the amount of output r
produced by DMU j. The decision variables for the problem are the weights that
correspond to each of the outputs and inputs, in order to maximize the efficiency of the
specific DMU k. The variable
is a positive weight placed on input i by DMU k and
is a positive weight attached to output r, by DMU k. A constraint is included so that
1
For this description we are using the nomenclature proposed by Thomas R. Sexton, included in the
publication edited by Richard Silkman, Measuring Efficiency: An Assessment of Data Envelopment
Approach, 1986.
16
DMU k chooses weights so that no other DMU would have efficiency greater than one if
it used the same set of weights. Also, the selected weights cannot be negative. The DEA
computes the technical efficiency by solving for each DMU k the following mathematical
program:
(
∑
∑
)
(2)
Subject to:
∑
∑
(3)
(4)
(5)
Now, the non linear problem is transformed to a linear program as follows:
∑
(6)
Subject to:
∑
(7)
∑
(8)
∑
(9)
(10)
17
The dual of this linear program is also preferred among the analysts as the value of the
objective function obtained will be the efficiency score for the evaluated DMU, and we
also obtain the coefficients of the linear combination of the efficient DMUs to create a
third hypothetical efficient DMU. In other words, in case we obtain an inefficient score
from a specific DMU from DEA, we can obtain the desired output level of a hypothetical
efficient DMU. Next, the dual of the linear program, using Thomas Sexton’s
nomenclature:
(11)
Subject to:
(12)
∑
(13)
∑
(14)
By linear programming duality theory, the optimal value of the
variable equals the
value. Also, if the DMU is efficient, then it will be the only DMU in its reference set
(frontier), and the corresponding dual variable
will be equal to one (Sexton, 1986). If
DMU k has an efficiency score less than one, it is inefficient. In other words, for those
inefficient DMUs the values of the weights
will indicate their efficient reference set.
As in the primal program, the dual linear program must be solved separately for each
18
DMU in the sample. For a better understanding, let’s take a look at the example2 in
Figure 2. This is a single input, multi-output case where a regional health authority
manages 6 hospitals, the decision making units. The number of patients is scaled, and the
axis in the graphic represents the quantity of patients per $1000 operating expenditure
treated. The efficient DMUs A, B, C and D compose the reference set. For inefficient
DMU E, E’ is that reference point that can be calculated through a linear combination of
points B and C.
Figure 2. Example to illustrate the concept of a reference set.
Finally is important to discuss that although DEA is widely used, as evaluators it is
important to have in consideration the shortfalls of this methodology that can drive the
study into incorrect results (Jacobs et al., 2006). As DEA is deterministic and based on
outlier observations, it assumes that all variables are measured accurately. In this sense,
2
Thanassoulis, 2001.
19
data that is feed into the model has to be reliable. The decision maker has to be careful
interpreting the results, as they are sensitive to small samples and outlier observations as
well. Another challenge is choosing which inputs and outputs to include in the model, as
the inclusion or exclusion of certain variables can bias efficiency estimates. A good
approach to test this is preparing different models, applying the DEA and comparing the
efficiency results to see if they change depending on the choices of the inputs and
outputs. Another important thing to consider is that the larger the number of input and
output variables used in relation to the number of DMUs in the model, the more DMUs
will be assigned as fully efficient. So the last recommendation of testing different models,
in terms of quantities of inputs and outputs included, is also applicable in this case.
Equity Assessment in Health Care
As mentioned before, the difference among an allocation model for the private sector and
a model for the public sector is the addition of the equity perspective in the latest. On the
other hand, the case study to be developed in this thesis will be applied in the health care
area of Costa Rica, specifically in primary health care attention. Having this in mind it is
important to know how equity has been included in the analysis of health care systems.
Most of the surveyed literature on the measurement of equity of health care and health is
more oriented towards creating monitoring systems. Although this is not the objective of
our research, it gives us insights on which indices and strategies are commonly used to
assess equity.
Some of the more helpful publications on this matter are from Paula Braven, who
proposes eight steps in policy-oriented monitoring of equity in health and its determinants
20
on the article Monitoring Equity in Health and Healthcare: A Conceptual Framework
(2003). In another publication for the World Health Organization (1998), she provides
practical suggestions regarding data sources and indicators for monitoring health equity.
She considers as central the inclusion of indicators of the determinants of health status
apart from health care and also states that the first step to make the system work is the
identification of the social groups of a priori concern.
The article by Anand, Diderichsen, Evans, Shkolnikov, and Wirth included in the
publication Challenging Inequities in Health: From Ethics to Action, summarizes more
theoretical concepts in terms of the existent measures of health equity. They divide these
measures into two families, intergroup differentials and interindividual differentials. This
last group includes the Gini coefficient. Anand et al. describe that there exist four uses of
the measures of health equity, namely, describing differences between groups and
between individuals, calculating the public health impact, attributing causality, and
assessing interventions. The authors also define the various issues that must be addressed
when constructing a measure of health equity. These include the measure of health status,
the population grouping across which health inequalities are assessed, identifying the
reference group or norm against which differences are measured, and the tradeoffs
between absolute versus relative measures.
Yukiko Asada, in his article included in the compilation Tackling Health Inequities
through Public Health Practice proposes a three-step framework for measuring health
inequity. The first is defining when a health distribution becomes inequitable; in this
matter the author highlights the importance of the determinants of health. The second is
21
deciding on measurement strategies to operationalise a chosen concept of equity, this
decision may need the analyst to answer questions such as why does health distribution
cause moral concern and within what time period should health equity be sought, to then
choose between an individual or group approach. The third step is quantifying health
inequity information, and this means choosing between measures such as the range
measures, the concentration index, and the Gini coefficient. In this sense, the author
recommends taking into consideration the basis of the comparison, if we are going to
look at differences absolutely or relatively, how should the differences be aggregated, if
the assessment of health inequity will be sensitive to the population’s mean health and
population size, and subgroup considerations. When a comparison is made, the range
measures compare the worst off with the best off. Other measures compare everyone to a
norm or to a mean, whereas the Gini coefficient compares everyone with everyone.
As a result of this review, we recognize the importance of analyzing the population that
will be addressed to determine if they are significant differences within it in terms of the
determinants of health. For this matter, we recall the Strategic and Conceptual Model
from the Ministry of Health of Costa Rica, the institution that has the steering role in
terms of health in the country. On the mentioned model, updated in 2011, the
determinants of health (adapted from Lalonde, 1974) include the biological determinants
(all those elements, both physical and mental, that develop within the human body as a
result of the biological and organic basic aspects of the individual), the environmental
determinants (environment in general and the human habitat, for example the water
quality, natural events, and working conditions), socioeconomic and cultural
22
determinants (for example income and equity in its distribution, education, employment,
and political participation), and the determinants of health-related services (access,
coverage, quantity, quality, nature, timing, use, relationship with users, and availability of
resources). These determinants are considered on our further analysis.
23
Chapter 3: The Allocation Model
The main objective of this thesis is to determine the most appropriate model for the
allocation of resources in the public sector. When this model is identified, it will be
validated by applying it in the primary health services at Costa Rica. As mentioned
before, for a model to be applied in the public sector it must include the equity
dimension. We also found that there are recommended characteristics that the equity
measures should satisfy, for example fulfilling the principle of transfers, as stated by
Mandell. Taking this into consideration, Golany extends the use of the data envelopment
analysis by creating a resource allocation model that that can be used to analyze tradeoffs
among efficiency, effectiveness, and equity measures. This last model satisfies our main
objective and in this chapter, we are going to describe it as it was developed by Boaz
Golany and Eran Tamir. It is important to stress that this improved resource allocation
model assesses the needs from Mandell’s model described in the literature review. We
are going to start explaining the new DEA-based resource allocation model (DEA-RAM),
and then we will clarify how it should be modified to include the equity constraints, to
finish with the description of the case when various outputs are analyzed.
24
The new DEA-based resource allocation model (DEA-RAM3) can be applied at any
organization with n decision making units (DMUs), each using m inputs (resources) to
produce a single output. The analyst then has to define two matrices to represent the past
performance of the DMU’s in a particular period: the matrix of inputs
vector of outputs
limited amounts,
and the
. Also, the analyst has to consider the fact that there are probably
, of resources available to all the DMUs in each i є C, where C
represents the set of controllable resources. The purpose of the model is maximizing the
overall effectiveness of the organization (objective function). This will be measured by
the sum of the potential outputs
the allocation of resources
across all DMUs (j=1,…, n) that can be achieved with
, (i=1,...,m). The model will choose the set of weights that
achieves the highest efficiency rating for each DMU, while assuring that these weights do
not cause any other DMU to have an efficiency rating higher than 1 (Golany et al. 1995).
The allocation has to take into account the empirical production function (this is the databased efficiency frontier as observed in the chosen period) as well as the limited amounts
of resources. The DEA-based allocation model stated by Golany and Tamir is then:
∑
(15)
Subject to:
∑
(16)
3
For this description we are using the nomenclature proposed by Boaz Golany and Eran Tamir, published
in their article “Evaluating Efficiency-Effectiveness-Equality Trade-Offs: A Data Envelopment Analysis
Approach” in 1995.
25
∑
(17)
∑
(18)
∑
(19)
(20)
In the primal DEA mathematical program, the objective is to maximize the efficiency
score. In the DEA-RAM case, the objective function assesses the effectiveness criterion.
Constraints (16), (17) and (18) are the same as in the DEA model, or in other words,
these constraints are enveloping each DMU. The combination of the objective function
with output constraint (16) ensures efficient output targets. Constraints (17) and (19)
guarantee that for any given output level, a minimal feasible resource allocation is
identified (Golany et al, 1995). Other differences from the DEA model are considering
the resource limitations in the equation (19) and having the outputs and inputs as decision
variables and no longer only parameters. The non-negativity of the decision variables is
enforced by constraint (20).
The equity measure has not yet been described within the DEA-RAM model. To assess
inequality in the model, Golany and Tamir include two new constraints, based on those
proposed by Mandell. The new constraints (21) and (22), assume that the equity
constraint is imposed on a particular input dimension, i*, and requires the Gini coefficient
to be no greater than . This is considered by adding variables
26
and
which represent
the positive and negative absolute differences respectively, between the allocations of the
input at the DMUs under study.
∑∑
(21)
∑
=0,
j=1,...,n-1 , k
(22)
>j
In other words, the analysts may choose which of the m inputs has to be allocated with
equity. An example could be choosing the investment per period on each of the DMUs.
Now the allocation of inputs will consider constraints (17), (19), (21) and (22). It is also
important to specify what  mean, in order to understand what these new constraints do.
We begin by explaining how the Gini coefficient, G, is calculated. Let
the number of
units of input i* units allocated in DMU j, where j=1,..,n. The variable
is the
proportion of equity units contained in DMU j. The equity units will be selected by the
analyst as the factor over which equity is assessed. For example, the priority of a health
care program could be focusing on the population over 65 years. Then, the total number
of persons over 65 years which receive services at each DMU is the number of equity
units. As Mandell describes, the Gini coefficient will give an average “perceived net envy
level” associated with the allocation of resources by calculating the following:
∑ ∑
|
(23)
|
∑
In this equation, the service level received by each DMU is compared to every other
DMU. If there exists no inequity among the DMUs, then G=0. So,

, and G as
an upper limit can be calculated by solving the DEA-RAM without the equity constraints
27
in equations (21) and (22). Equation (22) is the result of a simple transformation which
linearizes equation (23).
Some other changes must be done to the maximization problem presented to include the
equity constraints. The non-negativity constraint is modified to include the new variables,
equation (26). Then, trade-offs between the effectiveness and equality criteria are
examined by decreasing the value of  in equation (21). Also as the equity constraint is
tightened, a new trade-off arises as maintaining feasibility of equations (21), (22), (17)
and (19), is not always possible (Golany et al., 1995). To address this situation, a new set
of variables
is defined, allowing deviations from the observed efficient performance.
These new variables are included as shown in constraints (24), (25) and (26). In addition,
M is defined as a large penalty term. Because of M, these deviations will be used only to
avoid potential infeasibility caused by the inclusion of the equity constraints.
(24)
∑∑
∑
Subject to (2), (4), (5), (7), (8) and
(25)
∑
(26)
As a final point, to make the extension to the multioutput scenario we must change the
objective function. The effectiveness is measured by the sum of the outputs, so if various
are analyzed, subjective weights must be applied on the different output dimensions
(r=1,..,s) in order to allow their aggregation in the objective function (Golany et al, 1995).
28
DEA-RAM is modified by repeating constraint (16) for every output and modifying
objective function (24) as following:
∑∑
(27)
∑∑
As we can see, the model can be employed for resource allocation purposes without
requiring a priori assumptions on the form of the underlying production function, in
contrast to Mandell’s approach. To feed the single output DEA-RAM model the analyst
will need to define the decision making units under study, the inputs, the output and if
there exist limited amounts of inputs (in the case that they can be controlled). For the
DEA-RAM model including the equity constraints the equity units, equity proportion and
the  value will be the new parameters to add to the analysis. To run the model in the
multioutput case, all of the above mentioned have to defined, plus the subjective weights.
29
Chapter 4: Case Study
After finding an allocation model that can be applied to the public sector, we wanted to
validate it by using it in a case study. For this matter, we chose the primary health care
services of Costa Rica. The reason for this is better explained by Jacobs et al.: the
international explosion of interest in measuring the inputs, activities and outcomes of
health systems can be attributed to heightened concerns with the costs of health care,
increased demands for public accountability and improved capabilities for measuring
performance. The first section of this chapter is intended to describe the structure of the
health system in Costa Rica. In section two, the organization chosen for the application of
the model, the Health Area Goicochea 2, is characterized. Section three then develops the
DEA-RAM equity model for the multi output case. Finally, at section four we present a
discussion of the results obtained.
The National Health System of Costa Rica4
There are two primary institutions in the National Health System of Costa Rica, the
Ministry of Health and the Caja Costarricense de Seguro Social (CCSS). They execute
their roles all over the country. Thus to have a better understanding of their operation, in
this section we are going to describe their basic functions and how they are organized to
perform them.
4
These information was extracted various documents corresponding to the authorship of the Ministry of
Health of Costa Rica, the Caja Costarricense de Seguro Social and CENDEISS.
30
The Ministry of Health exercises the steering role over the actors involved in the social
production of health by encouraging their active participation and by guiding their actions
towards the development and constant improvement of the health levels of the
population. Managerially, the Ministry has three levels of management. The central level
is the political-strategic level of the institution. The regional level is the political-tactical
level and the link between the central and local level. Finally, the local level is the
political and operational level of the institution for the implementation of the lead roles
and provision of health services.
The Caja Costarricense de Seguro Social (CCSS) is the social security institution
responsible for the comprehensive care of people. The function of the CCSS is to provide
health services to all people according to the social security principles: solidarity,
universality, unity, equality and equity. As the CCSS has powers and responsibilities
throughout the national territory, it is organized into three administrative levels in order
to facilitate the coordination and implementation of its activities. The central or national
level is eminently political, regulatory, and financial, and it is driven by the higher
authorities. The reference point it takes for its functioning is the National Health Policy
determined by the Ministry of Health. The regional level’s function is to operationalize
and organize in their geographical area of influence the strategies, plans, programs and
budgets defined by the central level. It is also responsible for coordinating, supervising
and training the local human resources and managing the physical resources and funds
allocated to the region. At the local level, the staff is responsible for programming,
implementing and monitoring the health activities operationalized through plans and
31
programs defined by the central and regional level. This level is comprised of all the
institutions in which perform activities of health promotion, prevention and treatment of
diseases, and rehabilitation, or in other words, all health services. These institutions
include the national specialized and general hospitals, regional hospitals, peripheral
hospitals, health areas and the health sectors with their teams, the Basic Teams for
Comprehensive Care in Health (Equipos Básicos de Atención Integral en Salud, EBAIS).
The institutions at the local level of the CCSS attend care needs and health problems of
varying complexity, ranging from low to very specialized. Consequently, the local level
is also organized into three different levels of care. The first level of attention, the
primary care, includes basic health services that perform actions of health promotion,
disease prevention, and curing and rehabilitation of lower complexity diseases. These
actions are carried out by the members of the support teams and the EBAIS. The second
level of attention provides support to the first level and offers ambulatory and
hospitalized interventions for basic specialties such as internal medicine, pediatrics,
gynecology and obstetrics, psychiatry and general surgery. The third level of attention
provides inpatient and outpatient services in complex specialties and all subspecialties. It
serves hospitalized users, high complexity pathologies, and other emergencies requiring
hospitalization and specialized surgical procedures. Additionally, this level provides
support services and diagnostics and therapeutics requiring high technology and
specialization. The institutions of this level of care are regional, national general, and
specialized hospitals.
32
As the CCSS covers the vast majority of the population in Costa Rica, and the case study
for this project is developed for the primary care level of this institution, we describe the
first level structure. Costa Rica, for purposes of the CCSS, is divided into seven health
regions. In each of these regions the CCSS established the health areas to provide
comprehensive health services to the population located in a defined territorial space, as
responsible for the primary care level. The health areas serve populations between 15000
and 40000 in rural areas and between 30000 and 60000 inhabitants in urban areas. Also,
the health areas are considered the basic administrative and geographical unit of the
National Health System (management systems and institutional financing). They are
under the command of a director advised by a technical and administrative team called
the support team. Also, each of the health areas is subdivided into two or more health
sectors, which are geographical divisions in which between 4,000 and 4,500 people live,
on average. The organization of the health areas must ensure the timely delivery of
services to the population, according to two basic elements: the number of people
assigned and the dispersion and/or concentration of population. Each of the health sectors
is in charge of a team called Basic Team for Comprehensive Care in Health (EBAIS).
Each EBAIS is composed of at least one general doctor, one nursing assistant, one
assistant coach of primary health care, and a Technical Assistant in Primary Care
(Asistente Técnico de Atención Primaria, ATAP).
Health Area Jiménez Núñez (Goicochea 2)
The institution chosen to validate the model explained in the previous chapter is the area
of health Jiménez Núñez (Goicoechea 2), founded in 1966. The basic reason we decided
33
to select this health area is that various projects have been developed between this entity
and the Department of Industrial Engineering at the University of Costa Rica. In this
sense, the senior officers of Goicoechea 2 have a clearer vision of how an academic
project is structured and a clearer sense of which are the health area’s needs, as a result of
previous analysis conducted. In this section, we describe the basic characteristics of this
area of health as well as the needs we are seeking to address with the allocation model.
In terms of geographical distribution, this area of health belongs to the North Central
regional division, but from the administrative level, it is responsibility of the Central
South. Classified as a Type II health area, it provides health services for first and second
level of care (general medicine, pediatrics, psychology, internal medicine, family
medicine, psychiatry, clinical laboratory, pharmacy, diagnostic imaging,
electrocardiography, dentistry and emergency). The area of health Jiménez Núñez is
composed of 10 EBAIS, each one serving between 4000 and 7500 people. In Table 1, the
EBAIS and their corresponding number of users are presented.
Name of the EBAIS
Las Lomas
Barrio Pilar
Barrio Fátima
Centeno Guell
Divino Pastor
Santa Eduviges
Calle Blancos 1
El Encanto
Santa Cecilia
Calle Blancos 2
TOTAL
Number of Users
7473
6284
5153
4927
4875
4647
4528
4311
4284
3293
49775
Table 1.Number of users at each EBAIS
34
As it is stated in the Organizational Manual for the Health Areas developed by the CCSS,
each health area “should be provided with financial resources in accordance with the
objectives and targets set, to promote equity in the distribution and solve real needs,
taking into account the following aspects: population assigned, human development
index, implementation of special processes, epidemiological indicators, among others”.
But, the allocation of these resources has been done according to historical estimate (the
spending in the past). The absence of other criteria, such as enrolled population,
productivity levels, or efficiency, represents a suboptimality in the budget allocation
process.
Within the area of health under study, the budgeting process from the central level to the
area of health was described by its Director Pedro González as follows. The budget
management department at the central level of the CCSS determines a maximum amount
to be allocated to each area of health, according to revenue projections and the historical
behavior of the spending unit. Then, each area of health prepares a detailed budget by
budget account (Goicoechea 2 handles between 60 and 70 budget accounts). If any
differences are found between the total amount budgeted for the clinic and proposed by
the central level, it is negotiated. Continuing with the topic of the assigned budget for the
health area Goicoechea 2, as explained also by the director himself, the EBAIS under this
health area do not handle their own budget, because they have no administrative structure
to manage it. In turn, the chief of the first level (EBAIS), in collaboration with the
support team, performs a needs assessment that is integrated into the overall budget of the
area of health.
35
In conclusion, the budget allocation for the health areas, as well as for the EBAIS, is not
systematic as it lacks a clear, non empirical methodology to follow. It also fails to
consider the impact of the health areas’ activities over the target population and the
performance of the operative units. In this sense, we are going to validate the DEA-RAM
equity model developed by Golany and Tamir as a suitable systematic approach to help
the allocation decision making at the health area level.
DEA-RAM Equity Model for the Multi Output Scenario
To begin with the application of the DEA-RAM Equity Model at the health area
Goicoechea 2 we need to start by identifying and choosing the main components of the
model: the decision making units, the inputs and the outputs, and we also need to
determine if there exist environmental factors that will influence the performance of the
DMUs. To do so, we ask for the more experienced criterion at the health area by working
with the supervision of the director of the health area, Pedro González, and the chief of
the EBAIS, Esteban Avendaño.
The same structure of the first level of attention in health care of the National Health
System at Costa Rica, made the choice of the DMUs easy. The health areas are divided
into sectors, and each of the health sectors is in charge of an EBAIS. The EBAIS team is
in charge of developing the substantial processes of the health services provided at the
first level. In this sense, the EBAIS captures the entire production process of interest,
they convert inputs into outputs, and they are comparable as they produce the same set of
outputs.
36
Although the health system in Costa Rica has several data sources, not all of them are
systematized in an electronic format. For example, patient files typically paper-based.
This was an important limitation faced in the development of the model as we must settle
with information that could be easily obtained electronically. As discussed with the
decision makers in Costa Rica, this is a first approach to show the viability of applying a
DEA allocation model in their system. In the future, without the space and time
constraints we face, they might be able to improve the model by testing and including
more variables when the physical information is processed and available electronically.
Having clarified this, we describe the inputs and the outputs chosen.
Three variables are used as inputs: the annual budget, time contracted (programmed) for
medical appointments in hours per year, and the actual time used for medical
appointments in hours per year. The annual budget is used as it is the focus of the
allocation we want to develop and a need for the health area, as discussed in the previous
section. The medical consultation is the process that best describes the functioning of an
EBAIS team currently. This is why the hours planned and used for medical consultations
are satisfactory inputs for the allocation model in terms of representing the work done by
the EBAIS. When the physical data are processed into electronic data and become
available to the managers at the EBAIS, other inputs, such as the labor hours of each of
the other professionals in the team, can be included. In Appendix A, the description of
these variables and the procedures used to calculate them are broadly described. The
following table shows the information that will be included in the model.
Name of the EBAIS
Total Annual
Budget
Time contracted for
appointments (in hours
37
Time used for
appointments (in hours
(dollars)
per year)
per year)
1,241,119.09
1656
1569.12
Barrio Pilar Jiménez
948,110.87
1752
1396.56
Calle Blancos 1
842,353.79
1752
1329
Calle Blancos 2
873,531.49
1656
1245.6
Centeno Guell
624,064.84
1752
1329
Divino Pastor
1,333,918.33
1752
1684.32
El Encanto
603,752.20
1752
1245.6
Las Lomas
859,450.98
1752
1684.32
Santa Cecilia
1,894,102.13
1752
1684.32
Santa Eduviges
1,504,078.00
1752
1684.32
Total
10,724,481.71
17328
14852.16
Barrio Fatima
Table 2. Inputs for each EBAIS, year 2011, area of health Goicoechea 2
The outputs that are going to be considered for the analysis are the annual number of
medical consultations and the annual number of home visits done by the ATAPS. These
two outputs characterize both the results that best represent the functions of the doctor
and the ATAP at an EBAIS team. But, as a downside, these measures cannot reflect the
quality of the service offered. As mentioned by Jacobs et al. the measure of a health
outcome should indicate the value added to the health as a result of the contact with the
health system. In this sense, better outputs should measure the additional health conferred
to the patient or even the patient’s overall satisfaction. Other authors mention the
difficulty on measuring the real effectiveness of a public health care system as it is
subject to a vast heterogeneity of users. From the cost benefit perspective, it may also be
too expensive to obtain these measures. In this sense, we are facing the same issues as
other analysts have encountered when trying to assess the efficiency of health care
systems. Therefore, it is important that the decision makers consider formulating careful
interpretations of the results given in this case study. We recommend including in the
38
future other measures that are being analyzed for the ASIS (Análisis de Situación en
Salud, Health Situation Analysis) of the health area, but they are not yet available for
each of the EBAIS, for example: the degree of satisfaction of users served by EBAIS, the
percentage of complaints resolved by EBAIS, the number of successful projects
developed by EBAIS, total queries resolved by EBAIS, quantity of days a user has to
wait for an appointment at an EBAIS, among others. The outputs used are detailed in the
next table.
Name of the EBAIS
Medical Consultations
(annual)
21,483
Home Visits by ATAPS
(annual)
763
Barrio Pilar Jimenez
18,785
688
Calle Blancos 1
17,020
810
Calle Blancos 2
15,797
771
Centeno Guell
12,439
675
Divino Pastor
25,355
872
El Encanto
12,936
601
Las Lomas
20,924
983
Santa Cecilia
24,040
684
Santa Eduviges
21,148
765
Total
189,927
7,612
Barrio Fatima
Table 3. Outputs for each EBAIS, year 2011, area of health Goicoechea 2
Also, we needed to determine if there exist significant environmental constraints that
affect the DMUs studied. In other words, we were looking to evaluate the determinants of
health at the health sectors that correspond to each of the EBAIS. We also faced some
limitations to find the appropriate information. In Costa Rica, there exist seven provinces,
which are divided into 81 cantons, and each of these cantons is divided into districts. The
39
EBAIS and the health areas do not obey the same political division over their areas of
influence. The area of health Goicoechea 2 includes 5 different districts, but only two of
them completely, as shown in the following table.
District
Guadalupe
San Francisco
Calle Blancos
Mata de
Plátano
Purral
Percentage of Assignment to
the Area of Health (%)
100
100
84
30.8
11.6
Table 4. Districts served at area of health Goicoechea 2
Costa Rican institutions arrange and analyze most of the education, housing, and income
indices by cantons. So, it is a challenge to find information by districts, considering also
that each of the EBAIS under study serves only a percentage of the five mentioned
districts5, as shown in the following table.
5
Source: Caja Costarricense de Seguro Social. (2011) Inventario de Áreas de Salud, Sectores, EBAIS,
Sedes y Puestos de Visita Periódica en el Ámbito Nacional con Corte al 31 de Diciembre del 2010.
40
Name of the
EBAIS
Divino Pastor
Fátima
Santa Eduviges
Santa Cecilia
Districts assigned
Guadalupe
Guadalupe
Guadalupe
Guadalupe
Guadalupe
Las Lomas
Mata de Plátano
Purral
Guadalupe
Barrio Pilar
Calle Blancos
Guadalupe
Centeno Güell
San Francisco
Calle Blancos 1
Calle Blancos
San Francisco
Calle Blancos 2
Calle Blancos
El Encanto
Calle Blancos
Table 5. District assignment per EBAIS at area of health Goicoechea 2
One indicator that is disaggregated to the district level is the unsatisfied basic needs
(NBI, necesidades básicas insatisfechas) methodology. The NBI was proposed by the
Economic Commission for Latin America (CEPAL) in the seventies. The main objective
of this indicator is to identify households and individuals that fail to satisfy a set of
requirements deemed necessary according to welfare standards accepted as universal,
using census information available (INEC,2000). The NBI used for this thesis has as a
data source the national census developed by the INEC6 in the year 2000. Four
6
Entity with the technical lead and steering role of the National Statistics System at Costa Rica. It
coordinates the country's statistical production in order to meet the needs of information requirements at a
national level.
41
dimensions were defined by the experts at the INEC for the calculation of the critical
gaps indicator of the NBI, and each of them has different components:

Access to shelter: includes three components that express different degrees of
privation, these components are the quality of housing (slum), the quality of the
housing materials, if there exist overcrowding, and if the house lacks electricity.

Access to healthy living: evaluates the sanitary infrastructure in terms of
drinkable water availability and the existence of a waste disposal system.

Access to knowledge: considers school attendance and schooling lag for the
population between 7 and 17 years. It is considered absent for the household if
there is at least a member between 7 and 17 years that is not attending school or if
he/she attends but with more than two years’ lag.

Access to goods and services: measures the economic capacity, it is considered
absent when the head of the household was aged 50 or more and completed
primary school or less.
Every household in Costa Rica is analyzed for each of the four dimensions, and using the
critical gaps indicator, the households are classified as having one, two, three or four
critical gaps. For a household to have a gap in some dimension, it has to meet at least one
of the criteria defined for each component. It is important to mention that the four
dimensions are equally weighted. To appreciate gaps at the district level, the INEC made
a grouping of districts by percentage of critical gaps. A clustering technique is applied
according to the percentage of incidence (percentage of households with one or more
critical deficiencies by district), resulting in five groups of districts. The results for the
42
five districts served at the area of health Goicoechea 2 is almost the same, with four of
the districts clustered in group 5. Only the Purral district, is classified within a different
group (group 4). Table 6 summarizes the classifications on a scale of 1 through five, 1
represents the population with biggest needs and 5 the population with best quality of
life. It is important to remember, however that only a 11.6% of the population of the
Purral district is served by the area of health studied.
Table 6. Summary of the NBI incidences per district and grouping.
This means all the districts served by Goicoechea 2 are classified as having higher levels
of quality of life (in). Based on these findings, we assume that there are no significant
differences among the EBAIS studied due to environmental factors.
With the DMUs, inputs, and outputs defined and the environmental constraints analysis
done, the first evaluation to be applied at the area of health Goicoechea 2 is the data
envelopment analysis (DEA)7 to obtain the efficiency scores per EBAIS. We find that
only four out of the ten EBAIS at Goicoechea 2 are efficient based on the inputs and
outputs defined, these efficient DMUs are: Calle Blancos 1, Calle Blancos 2, Divino
7
For the calculations we use the program described by equations 11 through 14.
43
Pastor, and Las Lomas. The efficiency scores generated are shown in the next table and
the weights are detailed in Appendix B.
Name of the
EBAIS
Barrio Fatima
Barrio Pilar
Jimenez
Calle Blancos 1
Calle Blancos 2
Centeno Guell
Divino Pastor
El Encanto
Las Lomas
Santa Cecilia
Santa Eduviges
Efficiency Score
0.924434
0.960945
1
1
0.945674
1
0.880071
1
0.948136
0.85908
Table 7. Efficiency scores for the EBAIS obtained by the DEA methodology.
With the actual allocation of resources, only 4 EBAIS are efficient. To demonstrate how
the DEA allocation model improves the efficiency of the DMUs under study by
calculating the appropriate amounts of inputs to be used and the outputs obtained from
this assignment, we apply the model for the multi-input and multi-output scenario. We
apply the model under two settings: one uses the budget of the previous period (year
2011) and the budget8 to allocate as the resource constraints. For the other inputs (time
contracted, and time used for the appointments) the sum of resources used in year 2011
define the resource constraint. As this is a multi-output case, the formulation requires a
weight for each of the outputs analyzed, as they need to be aggregated in the objective
8
The estimation of the budget for the next period is 13,941,826.23 dollars according to the managers of the
area of health Goicoechea 2. This includes the growth and also the inflation; together they represent
approximately a 30% more of the previous period’s expenditure.
44
function. Following the recommendation of the decision makers at the health area
Goicoechea 2, the same weights are assigned to each output (0.50 -0.50).
As shown in table 8, when using the 2011 budget constraint the allocation of inputs given
by the DEA-RAM model doubles, the number of efficient EBAIS from 4 to 8. With the
current budget to allocate, this number rises to 9 efficient DMUs. In both cases the two
outputs considered (total medical consultations and the home visits by the ATAPS) show
an improvement. It is important to clarify that there is no specified target of outputs per
EBAIS or for the health area as a whole. This is the reason we have similar
improvements despite the increased budget in the second case. The inclusion of such
constraints may result in a decrease in the number of efficient DMUs, but could be more
realistic in terms of meeting the decision maker’s expectations for each EBAIS. The
weights, input allocation, and output results per EBAIS are detailed in Appendix C.
Results for previous
period's budget
(10,724,481.71 dollars)
Optimal Solution
Total medical
consultations
(annual)
Total home visits by
ATAPS (annual)
Efficient EBAIS
107,619.6
206,879.5 (16.95%
improvement)
Results for current
budget to allocate
(13,941,826.23
dollars)
109,217.3
210,172.9 (20.25%
improvement)
8,359.5 (9.82%
improvement)
8
8,261.6 (8.53%
improvement)
9
Table 8. Comparison of the results of the DEA-RAM model between the previous period’s
budget and the current budget to allocate for the next period
45
After applying the DEA model and the DEA resource allocation model and identifying
the advantages of these methodologies, we include the equity perspective into the
analysis. As discussed in Chapter 3, the decision makers need to define the equity unit,
the factor used by the model to assess equity. Taking into consideration the expertise of
the managers of the health area, we discussed which factors are prioritized in terms of the
service at the primary level. As a result, we include three different equity units (E) for the
analysis: the number of users with hypertension, the number of users with diabetes, and
the number of users over 65 years of age with diabetes. We justify this selection as these
users may require more specialized care, control and treatment than the rest of the users
and this reflects on the resources used by each of the EBAIS. The quantities per DMU for
each of the three equity units9 are specified in the next table.
Name of the
EBAIS
Barrio Fatima
Users with
hypertension
651
Users with
diabetes
304
Users over 65 years
with diabetes
147
Barrio Pilar
Jimenez
Calle Blancos 1
554
193
108
471
182
95
Calle Blancos 2
446
139
55
Centeno Guell
361
145
79
Divino Pastor
699
243
114
El Encanto
363
136
36
Las Lomas
640
256
98
Santa Cecilia
709
8
7
Santa Eduviges
558
303
168
Total general
5452
264
144
Table 9. Equity constraints for each EBAIS, year 2011, area of health Goicoechea 2
9
It is important to clarify that the DEA-RAM-equity model, considers only one equity unit per analysis,
having three different equity units means the program has to be run for each of the three scenarios.
46
Having the equity units defined and quantified the next step is calculating the Gini
coefficient (G). To do so, we choose the previous period’s budget allocation to determine
the maximum value of , the upper limit for the value of G in the model. This means that
we want to improve the equity level by having the previous period’s Gini measure as our
upper limit. The calculations are specified in Appendix D and the Gini measure values
obtained for each of the three different scenarios are summarized in the next table.
Equity unit (E)
Users with Hypertension
Users with Diabetes
Users over 65 years with
diabetes
Gini
measure
0.1155505
0.2029391
0.2189966
Table 10. Gini measure per equity unit
We have all the necessary parameters to run the DEA-RAM model with equity
constraints. We test three different values of  for each of the equity units, and we also
consider two different budget constraints: the previous period’s budget (10,724,481.71
dollars) and the current budget to allocate (13,941,826.23 dollars). The optimal solutions
of the runs are presented in the next table, and the weights and input and output
allocations are detailed in Appendix E.
47
Results: Total Weighted Outputs
δ=
0,01155
δ= 0,0555
δ= 0,1155
Previous period's budget
107,076.61
107,619.56
107,619.56
Current budget to allocate
109,217.31
109,217.31
109,217.31
Users with diabetes
δ= 0,06
δ= 0,08
δ= 0,2029
Previous period's budget
103,983.06
105,134.66
107,619.56
Current budget to allocate
109,217.31
109,217.31
109,217.31
Users over 65 years with
diabetes
δ= 0,07
δ= 0,1095
δ= 0,2189
Previous period's budget
102,043.91
104,413.76
107,619.56
Current budget to allocate
108,039.36
109,217.31
109,217.31
Users with hypertension
Table 11. Optimal solutions per scenario, DEA-RAM with equity
The basic idea behind the model as a managerial tool is giving the decision makers an
insight of how the equity, efficiency and the effectiveness of the allocation interconnect.
In this sense, the analyst may expect to find a tradeoff among effectiveness and equity: as
we look to minimize the inequity by decreasing the value of , the optimal solution
decreases. We see this behavior very clearly in the cases of users with diabetes and the
users over 65 years with diabetes when considering the previous period´s budget. But,
unpredictably in most of the cases where the current budget to allocate (13,941,826.23
dollars) is considered, the  value can decrease without affecting the optimal solution. In
other words, the system analyzed can be more equitable and still obtain the same outputs.
Of course, the decision makers have to remember that these results depend on the
parameters, inputs, outputs, and equity units chosen. One reason for this behavior can be
48
the absence of targets for the outputs, as the budget to be allocated is approximately 30%
more than the period of reference. In this sense, for further analysis the area of health
may define these target quantities so that they can be added as lower limit constraints for
the allocation of the output at each EBAIS.
49
Chapter 5: Conclusions and Future Work
DEA itself is a useful management tool as it performs the efficiency assessment from a
comparison between DMUs through data and among data by generating the efficiency
frontier. We find that this model can include more features, and the perspectives of
effectiveness, efficiency, and equity all be simultaneously analyzed. This integration is
important as the tradeoffs among certain decisions in each dimension could affect the
performance in other dimension.
To run this operations research model, various analytical choices must be made: the unit
of study, the inputs and outputs to be considered, the weights for the aggregation of the
outputs in the objective function, the environmental constraints and when equity is
incorporated, the units among which services should be distributed evenly (the equity
units). When these choices depend heavily on the analysts, the appropriateness of the
model depends on the consistency between the analytical choices and the judgment of
experienced decision makers. In this sense, a recommended approach (that in this case
study could not be applied due to time and space restrictions) is to develop various
models to compare results. By various models we mean changing and combining the mix
of inputs, outputs, equity units, and environmental constraints before making a definite
allocation decision. This just means that there exist certain generally accepted standards
in the efficiency and equity literature, but the mathematical model developed will always
have to address different factors depending on the object of study. Every DEA resource
50
allocation model with equity will be different depending on the DMUs analyzed, and for
each case, several models could be used as well.
The DEA model has been widely used in health institutions and many other public sector
organizations in the world (especially in the United States and Europe), but this is the
first application to the primary health system of Costa Rica. We learned that more
information should be made electronically accessible in order to strengthen the model
that considers the EBAIS as DMUs. Actually, most of the information is available (health
evaluations, census information) at the health area level, which means it would be
valuable to develop this analysis for the 103 health areas in Costa Rica as DMUs. This
would also be more interesting as environmental factors may play a more important role,
because health areas comprises cantons with different human development indices, they
can also be classified as rural or urban, introducing other environmental factors.
Basically, as happened with Tingley and Liebman’s model, we can think of standardizing
and systematizing the decision making at the health areas by disseminating the use of the
DEA-RAM with equity model and then applying the extensions along the model to be
broadened to incorporate resource allocation at the national level.
Operation research models applied to public services is still scarce. The development of
models analogous to those described in this paper that could incorporate equity indices
that satisfy the principle of transfers and other standard criteria for measures of equity is a
need, as we all know that the public resources are the majority of time limited and slight
changes in their allocation may cause great gains or losses to society.
51
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54
Appendix A: Input Data
The annual budget, the time contracted for medical appointments in hours per year, and
the actual time used for medical appointments in hours per year are the three inputs that
are included in the application of the allocation model for the area of health Goicoechea
2. This area of heath does not count with an aggregate annual budget per EBAIS. Instead
of this they determine a cost for their most important expenses: the cost per medical
consultation, the cost per home visit made by the ATAPS and the monthly cost of drugs.
The cost of the consultation and the visits already include entries such as salary of the
personnel in charge, materials, administrative support, and nursing assistants support. In
the following table, these costs are given in the Costa Rican currency (colones).
Cost per Medical
Consultation
(colones)
Cost per Home
Visit by ATAPS
(colones)
Cost of Drugs
(by month, in
colones)
Barrio Fatima
28,130.36
13,193.38
1,660,005.99
Barrio Pilar Jimenez
23,277.62
18,657.87
2,871,149.68
Calle Blancos 1
23,067.70
12,405.79
2,320,769.19
Calle Blancos 2
26,558.58
13,040.74
1,403,681.35
Centeno Guell
22,918.80
21,151.80
1,631,886.69
Divino Pastor
25,334.75
16,515.17
2,081,263.59
El Encanto
21,771.98
33,759.75
552,810.92
Las Lomas
16,045.03
27,879.46
6,343,040.70
Santa Cecilia
38,765.38
22,957.54
1,701,251.98
Name of the EBAIS
Santa Eduviges
33,559.64
21,114.91
3,569,334.36
Total
259,429.84
200,676.41
24,135,194.45
Table 12. Annual costs in colones for each of the EBAIS at area of health Jimenez Nunez, 2011
55
For the costs per medical consultation and home visits, we make an annual approximation
using the number of medical consultations and visits of the ATAPS in the year 2011. It is
important to mention that these are also the outputs used in the allocation analysis and
they are included in Appendix B. Discussing with the managers of Goicoechea 2, the cost
of medications does not vary significantly within the year. Their recommendation is to
approximate the total annual budget by multiplying the monthly budget by the 12 months
of work. The next table shows the results of the annual budgets by item and the
summation of these three for the total annual budget of the year 2011.
EBAIS
Total Annual
Expenditure for
Medical
Consultation
(colones)
Total Annual
Expenditure for
Home Visit by
ATAPS (colones)
Cost of Drugs
(colones)
Total Annual
Budget (2011
period)
Barrio Fatima
604,324,523.88
10,066,548.94
19,920,071.88
634,311,144.70
Barrio Pilar Jimenez
437,270,091.70
12,836,614.56
34,453,796.16
484,560,502.42
Calle Blancos 1
392,612,254.00
10,048,689.90
27,849,230.28
430,510,174.18
Calle Blancos 2
419,545,888.26
10,054,410.54
16,844,176.20
446,444,475.00
Centeno Guell
285,086,953.20
14,277,465.00
19,582,640.28
318,947,058.48
Divino Pastor
642,362,586.25
14,401,228.24
24,975,163.08
681,738,977.57
El Encanto
281,642,333.28
20,289,609.75
6,633,731.04
308,565,674.07
Las Lomas
335,726,207.72
27,405,509.18
76,116,488.40
439,248,205.30
Santa Cecilia
931,919,735.20
15,702,957.36
20,415,023.76
968,037,716.32
Santa Eduviges
709,719,266.72
16,152,906.15
42,832,012.32
768,704,185.19
5,040,209,840.21
151,235,939.62
289,622,333.40
5,481,068,113.23
Total
Table 13. Annual costs in colones for each of the EBAIS in area of health Jimenez Nunez, 2011
To obtain the cost in dollars we use the data base of the Central Bank of Costa Rica, with
the exchange rates of the year 2011 to obtain an average exchange rate. The annual costs
and the annual budget in dollars are presented in the next table.
56
Total Annual
Expenditure for
Medical
Consultation
(dollars)
Total Annual
Expenditure for
Home Visit by
ATAPS (dollars)
Cost of Drugs
(by month, in
dollars)
Total Annual
Budget (2011
period, in dollars)
1,182,446.04
19,696.62
38,976.43
1,241,119.09
Barrio Pilar Jimenez
855,580.52
25,116.64
67,413.70
948,110.87
Calle Blancos 1
768,201.17
19,661.68
54,490.94
842,353.79
Calle Blancos 2
820,900.62
19,672.87
32,958.00
873,531.49
Centeno Guell
557,812.78
27,935.87
38,316.19
624,064.84
Divino Pastor
1,256,872.87
28,178.03
48,867.42
1,333,918.33
El Encanto
551,072.89
39,699.48
12,979.83
603,752.20
Las Lomas
656,895.61
53,622.74
148,932.63
859,450.98
Santa Cecilia
1,823,432.21
30,725.05
39,944.87
1,894,102.13
Santa Eduviges
1,388,665.70
31,605.44
83,806.86
1,504,078.00
Total
9,861,880.41
295,914.42
566,686.89
10,724,481.71
EBAIS
Barrio Fatima
Table 14. Annual costs in dollars for each of the EBAIS at area of health Jimenez Nunez, 2011
Finally, it is important to mention that as with the cost of medications, the inputs of time
programmed and time used for medical appointments are managed by the EBAIS on a
monthly basis. Again, the managers at the area of health Goicoechea 2 determined that
there is no significant difference among the times throughout the year and recommended
to approximate the total annual time by multiplying the monthly hours by the 12 months
of work. The monthly and annual data are detailed in following tables.
57
Time programmed for
medical appointments
(contracted)
Time used for medical
appointments
Barrio Fatima
138
130.76
Barrio Pilar Jiménez
146
116.38
Calle Blancos 1
146
110.75
Calle Blancos 2
138
103.8
Centeno Guell
146
110.75
Divino Pastor
146
140.36
El Encanto
146
103.8
Las Lomas
146
140.36
Santa Cecilia
146
140.36
Santa Eduviges
146
140.36
1,444.00
1,237.68
Name of the EBAIS
Total
Table 15. Total time in hours per month programmed and used for medical appointments at area
of health Jimenez Nunez, 2011
Time programmed
for appointments
(in hours per year,
contracted)
Time used for
appointments (in
hours per year)
Barrio Fatima
1656
1569.12
Barrio Pilar Jiménez
1752
1396.56
Calle Blancos 1
1752
1329
Calle Blancos 2
1656
1245.6
Centeno Guell
1752
1329
Divino Pastor
1752
1684.32
El Encanto
1752
1245.6
Las Lomas
1752
1684.32
Santa Cecilia
1752
1684.32
Santa Eduviges
1752
1684.32
17,328.00
14,852.16
Name of the EBAIS
Total
Table 16. Total time in hours per year programmed and used for medical appointments at area of
health Jimenez Nunez, 2011
58
Appendix B: DEA Weight Results
By definition, the weights obtained for the efficient DMUs are equal to one. On the other
hand, for inefficient DMUs the values of the weights represent the efficient reference set.
Consider as an example the inefficient EBAIS Santa Eduviges. This EBAIS has Divino
Pastor and Las Lomas as its reference for efficiency. As shown in Table 17, Santa
Eduviges needs to combine 0.716001 of the performance of Divino Pastor and 0.14308 of
the performance of Las Lomas to reach the efficiency frontier. The EBAIS not listed are
efficient.
Efficiency Frontier
Reference Set
Reference
Weight
Calle Blancos 1
0.0611
1 Barrio Fatima Divino Pastor
0.7771
Las Lomas
0.0388
Barrio Pilar Divino Pastor
0.4769
2
Jimenez
Las Lomas
0.3198
3 Centeno Guell Las Lomas
0.6866
4 El Encanto
Las Lomas
0.6182
5 Santa Cecilia
Divino Pastor
0.9481
Divino Pastor
0.7160
6 Santa Eduviges
Las Lomas
0.1431
Table 17. DEA weights for the inefficient EBAIS at Goicoechea 2
Inneficient
EBAIS
59
Appendix C: DEA RAM Results
The DEA-RAM for the multi-input and single output case is described in Chapter 2
(equations 15 through 20). For the multi-input, multi-output case we need to change the
objective function, adding subjective weights for the different output dimensions as in
(28) and we also repeat constraint (16) for every output, as shown in (29).
∑∑
(28)
∑
(29)
In the next tables, the model results for the weights, input allocation and output allocation
are displayed for both the previous period’s budget (10,724,481.71 dollars) and the
current budget to allocate (13,941,826.23 dollars).
Inneficient
EBAIS
Efficiency Frontier
Reference
Weight
Calle Blancos 1
0.1341
Barrio Pilar
1
Calle Blancos 2
0.5427
Jimenez
Las Lomas
0.3231
Calle Blancos 2
0.4572
2 Divino Pastor
Divino Pastor
0.5428
Table 18. DEA-RAM weights for the inefficient EBAIS at Goicoechea 2, with previous period’s
budget
60
Inneficient
EBAIS
Efficiency Frontier
Reference
Weight
Calle Blancos 2
0.5383
Centeno Guell
Divino Pastor
0.4617
Table 19. DEA-RAM weights for the inefficient EBAIS at Goicoechea 2, with budget to allocate
Results for last period's budget (13941826.23 dollars)
Total Annual
Name of the EBAIS
Budget (dollars)
Calle Blancos 1
Calle Blancos 2
Santa Cecilia
Centeno Guell
Divino Pastor
Santa Eduviges
El Encanto
Barrio Fatima
Las Lomas
Barrio Pilar Jimenez
Results for budget to allocate (13941826.23 dollars)
Time programmed Time used for
Time programmed Time used for
Total Annual
for appointments
appointments
for appointments
appointments
Budget (dollars)
(hours per year) (hours per year)
(hours per year) (hours per year)
842353.79
1752
1329
873531.49
842353.79
1752
1329
873531.49
1333918.33
1752
1684.32
1333918.33
842353.79
1752
1329
1086095.217
1123414.437
1708.105666
1483.722894
1333918.33
1333918.33
1752
1684.32
1333918.33
1333918.33
1752
1684.32
1333918.33
873531.49
1656
1245.6
3565544.893
1333918.33
1752
1684.32
1333918.33
864801.0928
1699.894334
1398.557106
873531.49
Table 20. DEA-RAM input allocation for the EBAIS at Goicoechea 2
61
1656
1656
1752
1700.323851
1752
1752
1752
1899.676149
1752
1656
1245.6
1245.6
1684.32
1448.16
1684.32
1684.32
1684.32
1245.6
1684.32
1245.6
Results for last period's budget
Results for budget to allocate
(13941826.23 dollars)
(13941826.23 dollars)
Medical
Medical
Home Visit by
Home Visit by
Name of the EBAIS
Consultations
Consultations
ATAPS (annual)
ATAPS (annual)
(annual)
(annual)
Calle Blancos 1
17020
810
15797
771
Calle Blancos 2
17020
810
15797
771
Santa Cecilia
25355
872
25355
872
Centeno Guell
17020
810
20209.99344
817.6323851
Divino Pastor
20984.77038
825.8195028
25355
872
Santa Eduviges
25355
872
25355
872
El Encanto
25355
872
25355
872
Barrio Fatima
15797
771
15797
771
Las Lomas
25355
872
25355
872
Barrio Pilar Jimenez
17617.7978
844.7381165
15797
771
Total
206879.5682
8359.557619
210172.9934
8261.632385
Table 21. DEA-RAM output allocation for the EBAIS at Goicoechea 2
62
Appendix D: Calculating the Gini Measure
Tables 22 through 24 summarize the values obtained for the Gini measure for each of the
three equity units evaluated, using formula (23).
∑ ∑
|
|
(23)
∑
Is important to clarify also that the allocated budget in dollars for each of the EBAIS in
the year 2011 is
and the proportion of equity units q is calculated as follows:
(30)
∑
63
DMU i
1
2
3
4
5
6
7
8
9
10
Barrio Fatima
Barrio Pilar Jimenez
Calle Blancos 1
Calle Blancos 2
Centeno Guell
Divino Pastor
El Encanto
Las Lomas
Santa Cecilia
Santa Eduviges
TOTAL
Ei
qi
651
554
471
446
361
699
363
640
709
558
5,452
0.11941
0.10161
0.08639
0.08180
0.06621
0.12821
0.06658
0.11739
0.13004
0.10235
1
Absolute differences
Barrio
Barrio
Calle
Calle
Centeno
Divino
Xij
El Encanto Las Lomas
Fatima
Pilar
Blancos 1 Blancos 2
Guell
Pastor
1,241,119.09
0
0
0
0
0
0
0
0
948,110.87
12,905.32
0
0
0
0
0
0
0
842,353.79
6,638.81
3,687.41
0
0
0
0
0
0
873,531.49
2,775.11
11,203.04
6,556.04
0
0
0
0
0
624,064.84
7,662.84
635.34
1,862.65
6,788.69
0
0
0
0
1,333,918.33
153.81
13,987.76
7,239.59
2,874.35
8,313.13
0
0
0
603,752.20
10,543.57
1,776.51
3,926.47
8,770.81
1,573.92
11,406.74
0
0
859,450.98
43,069.26 23,964.62 24,634.08 32,234.96 16,349.91 46,396.09 13,650.17
0
1,894,102.13
64,766.52 69,171.31 54,089.01 41,349.18 44,260.62 69,374.41 47,596.99 110,578.62
1,504,078.00
52,569.76 55,798.49 43,724.75 33,636.87 35,719.73 56,314.03 38,350.44 88,598.00
10,724,481.71 201,084.99 180,224.49 142,032.59 125,654.86 106,217.31 183,491.28 99,597.59 199,176.61
Santa
Cecilia
0
0
0
0
0
0
0
0
0
1,739.24
1,739.24
Santa
Eduviges
0
0
0
0
0
0
0
0
0
0
GINI MEASURE
0.00
0.115550475
Table 22. Gini measure for the budget allocation in the year 2011, using the users with hypertension as the equity unit.
1
2
3
4
5
6
7
8
9
10
DMU i
Barrio Fatima
Barrio Pilar Jimenez
Calle Blancos 1
Calle Blancos 2
Centeno Guell
Divino Pastor
El Encanto
Las Lomas
Santa Cecilia
Santa Eduviges
TOTAL
Ei
304
193
182
139
145
243
136
256
8
303
1,909
qi
0.15925
0.10110
0.09534
0.07281
0.07596
0.12729
0.07124
0.13410
0.00419
0.15872
1
Absolute differences
Barrio
Barrio
Calle
Calle
Centeno
Divino
Santa
Santa
El Encanto Las Lomas
Xij
Fatima
Pilar
Blancos 1 Blancos 2
Guell
Pastor
Cecilia Eduviges
1,241,119.09
0
0
0
0
0
0
0
0
0
0
948,110.87
12,267.34
0
0
0
0
0
0
0
0
0
842,353.79
17,743.79
4,795.86
0
0
0
0
0
0
0
0
873,531.49
13,935.07 19,728.32 14,130.36
0
0
0
0
0
0
0
624,064.84
19,753.53
8,600.93
10,068.66 15,298.43
0
0
0
0
0
0
1,333,918.33
1,293.23
14,858.18
8,012.92
2,072.40
8,886.06
0
0
0
0
0
603,752.20
16,327.70
6,195.10
7,852.19
12,841.83
4,482.32
17,623.36
0
0
0
0
859,450.98
63,812.72 39,810.88 38,712.77 46,834.74 26,780.22 68,690.55 23,740.98
0
0
0
1,894,102.13 220,965.51 188,494.23 160,102.05 151,286.04 122,801.27 237,252.50 123,581.22 218,743.40
0
0
1,504,078.00
17,397.19
2,349.61
3,762.17
15,607.67
538.65
18,884.39
4,314.46
40,147.24 105,039.03
0
GINI MEASURE
10,724,481.71 383,496.07 284,833.11 242,641.13 243,941.10 163,488.51 342,450.79 151,636.66 258,890.63 105,039.03
0.00
0.202939135
Table 23. Gini measure for the budget allocation in the year 2011, using the users with diabetes as the equity unit.
64
1
2
3
4
5
6
7
8
9
10
DMU i
Barrio Fatima
Barrio Pilar Jimenez
Calle Blancos 1
Calle Blancos 2
Centeno Guell
Divino Pastor
El Encanto
Las Lomas
Santa Cecilia
Santa Eduviges
TOTAL
Ei
147
108
95
55
79
114
36
98
7
168
907
qi
0.16207
0.11907
0.10474
0.06064
0.08710
0.12569
0.03969
0.10805
0.00772
0.18523
1
Absolute differences
Barrio
Barrio
Calle
Calle
Centeno
Divino
Santa
Santa
El Encanto Las Lomas
Xij
Fatima
Pilar
Blancos 1 Blancos 2
Guell
Pastor
Cecilia Eduviges
1,241,119.09
0
0
0
0
0
0
0
0
0
0
948,110.87
34,574.99
0
0
0
0
0
0
0
0
0
842,353.79
29,414.07 13,710.98
0
0
0
0
0
0
0
0
873,531.49
29,043.85 31,270.16 24,384.77
0
0
0
0
0
0
0
624,064.84
33,584.97 19,166.99 19,456.14 25,033.36
0
0
0
0
0
0
1,333,918.33
3,282.36
16,377.70
9,362.95
672.40
9,886.24
0
0
0
0
0
603,752.20
22,829.85 23,717.98 18,724.23 14,718.26 15,207.05 24,462.03
0
0
0
0
859,450.98
31,477.71 15,109.64 16,766.83 24,076.53 10,521.39 33,937.82
8,011.36
0
0
0
1,894,102.13 216,587.99 185,150.17 157,131.00 148,205.03 120,600.14 232,547.67 121,451.74 215,712.05
0
0
1,504,078.00
50,292.03 22,779.29 26,088.07 38,759.90 16,001.67 54,238.80 11,687.49 17,368.20 155,240.65
0
GINI MEASURE
10,724,481.71 451,087.82 327,282.93 271,913.99 251,465.46 172,216.50 345,186.31 141,150.59 233,080.25 155,240.65
0.00
0.218996551
Table 24. Gini measure for the budget allocation in the year 2011, using the users over 65 years with diabetes as the equity unit.
65
Appendix E: DEA RAM with Equity Results
We consider:

Input 1: Total Annual Budget (dollars)

Input 2: Time programmed for appointments (hours per year)

Input 3: Time used for appointments (hours per year)

Output 1: Medical Consultations (annual)

Ouput 2: Visit by ATAPS (annual)

E1: environmental constraint considered: users with hypertension.

E2: environmental constraint considered: users over 65 years and with
diabetes.

E3: environmental constraint considered: users with diabetes.
Tables 25 through 36 display the input and output allocation per EBAIS and per δ value
used.
66
Name of the EBAIS
Barrio Fatima
Barrio Pilar Jimenez
Calle Blancos 1
Calle Blancos 2
Centeno Guell
Divino Pastor
El Encanto
Las Lomas
Santa Cecilia
Santa Eduviges
Input 1
1291817.7
1099251.3
934596.22
884940.05
732362.75
1333918.3
736455.4
1269964.7
1333918.3
1107256.9
Input 2
1743.2212
1752
1668.7332
1712.6447
1752
1752
1752
1738.6644
1752
1704.7365
Barrio Fatima
Barrio Pilar Jimenez
Calle Blancos 1
Calle Blancos 2
Centeno Guell
Divino Pastor
El Encanto
Las Lomas
Santa Cecilia
Santa Eduviges
1151754.1
1164448.7
990027.75
937426.43
842353.79
1236633.4
842353.79
1132270.4
1254284.4
1172929
1734.8991
1752
1752
1669.3234
1752
1752
1752
1709.9523
1735.3947
1718.4305
δ= 0.011552
Input 3
Output 1
1644.2008
24481
1514.6945
21376
1303.7909
17065
1346.5932
17343
1290.5541
15137
1684.32
25355
1291.9846
15207
1623.3762
24027
1684.32
25355
1468.3258
20649
δ= 0.0555
1533.7744
21954
1561.8214
22481
1546.5741
20692
1306.4879
17124
1329
17020
1613.999
23705
1329
17020
1492.1621
21169
1608.4338
23702
1530.9072
22013
Output 2
863
842
784
813
714
872
717
858
872
822
Input 1
1267860.3
1078865.1
917263.64
868528.37
793279.89
1333918.3
797712.96
1246412.5
1333918.3
1086722.2
Input 2
1738.2256
1752
1665.119
1730.4475
1752
1752
1752
1733.7533
1752
1700.4546
841
851
884
785
810
860
810
828
855
837
873531
1301640
1106670
1047870
848162
1154510
852902
1057080
1170990
1311120
1656
1745.27
1752
1752
1734.11
1714.59
1752
1752
1718.03
1752
δ= 0.02
Input 3
Output 1
1621.3708
23984
1499.9586
21030
1287.274
16705
1399.9833
17870
1311.8469
16180
1684.32
25355
1313.3964
16256
1600.9324
23538
1684.32
25355
1448.7575
20223
δ= 0.1155
1245.6
15797
1653.56
24685
1520.06
21502
1588.39
21673
1313.46
16792
1513.35
21630
1336.62
17199
1484.21
20661
1529.06
21972
1667.84
24969
Output 2
858
840
781
841
767
872
771
853
872
818
771
865
843
891
803
833
811
837
836
869
Table 25. Input and output allocation, E1, with previous period’s budget
Name of the EBAIS
Barrio Fatima
Barrio Pilar Jimenez
Calle Blancos 1
Calle Blancos 2
Centeno Guell
Divino Pastor
El Encanto
Las Lomas
Santa Cecilia
Santa Eduviges
Barrio Fatima
Barrio Pilar Jimenez
Calle Blancos 1
Calle Blancos 2
Centeno Guell
Divino Pastor
El Encanto
Las Lomas
Santa Cecilia
Santa Eduviges
δ= 0.011552
Input 3
1245.6
1353.21545
1575.70967
1513.94307
1245.6
1684.32
1245.6
1684.32
1684.32
1619.53181
δ= 0.0555
1519608.98 1899.67615
1245.6
1640669.81
1656
1245.6
1394916.49
1752
1684.32
1320802.98 1700.32385
1448.16
1069075.37
1656
1245.6
1631597.58
1752
1684.32
873531.49
1656
1245.6
1493902.51
1752
1684.32
1654886.12
1752
1684.32
1342834.91
1752
1684.32
Input 1
1686231.35
1434871.18
1219944.11
1155127.08
934975.11
1810499.3
940200.012
1657706.21
1836341.39
1265930.48
Input 2
1899.67615
1679.54824
1728.23406
1714.7184
1656
1752
1656
1752
1752
1737.82315
Output 1
15797
18142
22989
21643
15797
25355
15797
25355
25355
23944
Output 2
771
796
847
833
771
872
771
872
872
857
15797
15797
25355
20210
15797
25355
15797
25355
25355
25355
771
771
872
818
771
872
771
872
872
872
δ= 0.02
Input 1
Input 2
Input 3
1575417.54 1995.67615
1684.32
1340575.97
1752
1684.32
1139773.23 1711.51681 1499.3118
1079215.77 1698.88935 1441.60431
873531.49
1656
1245.6
1691518.99
1752
1684.32
878413.028
1656
1245.6
1548766.98 1697.9177 1437.16389
1715662.81
1656
1245.6
1350339.04
1752
1684.32
δ= 0.1155
877445.93 1899.67615
1245.6
1790099.14
1656
1245.6
1521963.04
1656
1245.6
1441099.39
1752
1684.32
1166444.88 1700.32385
1448.16
1682118.84
1752
1684.32
873531.49
1656
1245.6
1540160.13
1752
1684.32
1706128.49
1752
1684.32
1342834.91
1752
1684.32
Output 1
25355
25355
21324
20067
15797
25355
15797
19970
15797
25355
Output 2
872
872
829
816
771
872
771
815
771
872
15797
15797
15797
25355
20210
25355
15797
25355
25355
25355
771
771
771
872
818
872
771
872
872
872
Table 26. Input and output allocation, E1, with current budget to allocate
67
Name of the EBAIS
Barrio Fatima
Barrio Pilar Jimenez
Calle Blancos 1
Calle Blancos 2
Centeno Guell
Divino Pastor
El Encanto
Las Lomas
Santa Cecilia
Santa Eduviges
Input 1
1613040
1185090
1042440
639778
866871
1250930
603752
1075360
603752
1843470
Input 2
1752
1720.97
1691.22
1752
1701.41
1710.24
1752
1744.17
1752
1752
Barrio Fatima
Barrio Pilar Jimenez
Calle Blancos 1
Calle Blancos 2
Centeno Guell
Divino Pastor
El Encanto
Las Lomas
Santa Cecilia
Santa Eduviges
1613038.62
1185089.6
1042439.92
639778.365
866871.098
1250927.91
603752.2
1075359.08
603752.2
1843472.71
1752
1720.96619
1691.22084
1752
1701.41012
1710.23518
1752
1744.16767
1752
1752
δ= 0.07
Input 3
Output 1
1684.32
25355
1542.5
22265
1406.56
19304
1307.41
14062
1453.12
18222
1634.2
23273
1245.6
12936
1648.53
22511
1245.6
12936
1684.32
25355
δ= 0.218997
1684.32
25355
1542.49548
22265
1406.55922
19304
1307.41257
14061
1453.12425
18222
1634.20222
23272
1245.6
12936
1648.52625
22511
1245.6
12936
1684.32
25355
Output 2
872
839
808
655
871
835
601
915
601
872
Input 1
1417480
1258710
1107200
722074
920726
1328640
603752
1142170
603752
1619970
Input 2
1752
1736.32
1704.73
1750.2
1665.84
1750.9
1752
1712.02
1752
1752
δ= 0.109498
Input 3
Output 1
1684.32
25355
1612.66
23794
1468.27
20648
1439.93
16528
1290.57
16777
1679.29
25246
1245.6
12936
1501.59
21374
1245.6
12936
1684.32
25355
Output 2
872
856
822
773
781
871
601
830
601
872
872
839
808
655
871
835
601
915
601
872
Table 27. Input and output allocation, E2, with last period’s budget
Name of the EBAIS
Barrio Fatima
Barrio Pilar Jimenez
Calle Blancos 1
Calle Blancos 2
Centeno Guell
Divino Pastor
El Encanto
Las Lomas
Santa Cecilia
Santa Eduviges
Barrio Fatima
Barrio Pilar Jimenez
Calle Blancos 1
Calle Blancos 2
Centeno Guell
Divino Pastor
El Encanto
Las Lomas
Santa Cecilia
Santa Eduviges
δ= 0.07
Input 1
Input 2
Input 3
2095236 1869.382 1448.16
1539357
1752
1684.32
1354064
1752
1684.32
873531.5
1656
1245.6
1126011
1656
1245.6
1624877
1752
1684.32
873531.5
1656
1245.6
1396824
1752
1684.32
663839.4 1730.618 1245.6
2394555
1752
1684.32
δ= 0.2189
873531.5 1899.676 1245.6
1704459 1718.494 1531.197
1499292
1752
1684.32
873531.5
1656
1245.6
1246780 1733.83 1601.283
1799151
1752
1684.32
873531.5
1656
1245.6
1546638
1752
1684.32
873531.5
1656
1245.6
2651380
1752
1684.32
Output 1 Output 2
20210
818
25355
872
25355
872
15797
771
15797
771
25355
872
15797
771
25355
872
13573
639
25355
872
15797
22019
25355
15797
23546
25355
15797
25355
15797
25355
Input 1
1996365
1466717
1290168
1207964
1072877
1548202
873531.5
1330910
873531.5
2281560
Input 2
1995.676
1752
1742.877
1725.736
1697.567
1694.143
1656
1656
1656
1752
δ= 0.1094
Input 3
1684.32
1684.32
1642.629
1564.293
1435.563
1419.915
1245.6
1245.6
1245.6
1684.32
Output 1 Output 2
25355
872
25355
872
24447
862
22740
844
19936
815
19595
811
15797
771
15797
771
15797
771
25355
872
771
837
872
771
853
872
771
872
771
872
Table 28.Input and output allocation, E2, with current budget to allocate
68
Name of the EBAIS
Barrio Fatima
Barrio Pilar Jimenez
Calle Blancos 1
Calle Blancos 2
Centeno Guell
Divino Pastor
El Encanto
Las Lomas
Santa Cecilia
Santa Eduviges
Input 1
1588517.3
1043319.6
983855.81
751406.36
783841.17
1313609.7
735188.96
1337698.8
603752.2
1583291.9
Input 2
1752
1691.4043
1679.0049
1752
1697.8256
1747.7652
1752
1752
1752
1752
Barrio Fatima
Barrio Pilar Jimenez
Calle Blancos 1
Calle Blancos 2
Centeno Guell
Divino Pastor
El Encanto
Las Lomas
Santa Cecilia
Santa Eduviges
1281846.6
1205470.2
1136764.6
868188.37
905664.13
1277663.6
849450.5
1079449.8
842353.79
1277630
1752
1725.216
1710.8895
1672.4521
1752
1752
1752
1719.1797
1752
1740.2627
δ= 0.06
Input 3
1684.32
1407.3975
1350.7322
1498.9404
1255.3339
1664.9671
1376.2288
1684.32
1245.6
1684.32
δ= 0.16
1646.6808
1561.9169
1496.4448
1259.8928
1374.7629
1684.32
1334.1297
1534.3313
1329
1630.6808
Output 1
25355
19322
18087
17549
15027
24933
16061
25355
12936
25355
Output 2
872
808
795
822
721
868
754
872
601
872
24472
22688
21262
16007
18093
24830
17140
21181
17020
24186
865
844
829
778
818
885
811
860
810
860
δ= 0.08
Input 1
Input 2
Input 3
1499707.8
1752
1684.32
1059449.5 1694.7677 1422.7683
999066.4 1682.1766 1365.227
794250.79 1748.744 1511.5149
828535
1752
1324.1698
1333918.3
1752
1684.32
777108.69 1690.3117
1245.6
1333918.3
1752
1684.32
603752.2
1752
1245.6
1494774.6
1752
1684.32
δ= 0.2029
883772.79 1658.1355 1255.3593
1261901.1 1746.5941 1626.298
1189979.3
1752
1580.2758
908830.34
1752
1377.0516
948060.43
1752
1405.4084
1333918.3
1752
1684.32
889215.31 1659.2704 1260.5457
1127006.9
1752
1534.7572
847878.85
1752
1443.8239
1333918.3
1752
1684.32
Output 1
25355
19657
18403
18231
16783
25355
14774
25355
12936
25355
Output 2
872
812
799
857
798
872
710
872
601
872
16010
24035
22914
18147
18812
25355
16123
21847
18282
25355
773
860
854
818
823
872
774
846
866
872
Table 29. Input and output allocation, E3, with previous period’s budget
69
Name of the EBAIS
Barrio Fatima
Barrio Pilar Jimenez
Calle Blancos 1
Calle Blancos 2
Centeno Guell
Divino Pastor
El Encanto
Las Lomas
Santa Cecilia
Santa Eduviges
Input 1
2086404
1324592
1249097
953981
995160
1667751
933391.5
1756972
894935.3
2079541
Input 2
1904.549
1750.055
1734.313
1672.775
1681.362
1752
1668.482
1752
1660.463
1752
Barrio Fatima
Barrio Pilar Jimenez
Calle Blancos 1
Calle Blancos 2
Centeno Guell
Divino Pastor
El Encanto
Las Lomas
Santa Cecilia
Santa Eduviges
1024084
1399924
1320135
1008235
1051756
1762598
986474.7
1856894
1333918
2197808
1899.676
1656
1749.126
1679.647
1656
1752
1679.551
1752
1752
1752
Barrio Fatima
Barrio Pilar Jimenez
Calle Blancos 1
Calle Blancos 2
Centeno Guell
Divino Pastor
El Encanto
Las Lomas
Santa Cecilia
Santa Eduviges
873531.5 1899.676
1418118
1656
1337293
1752
1021339 1686.821
1065426
1656
1785506
1752
999295.7 1669.503
1881027
1752
1333918
1752
2226372
1752
δ= 0.06
Input 3
1267.869
1675.433
1603.491
1322.263
1361.504
1684.32
1302.643
1684.32
1265.996
1684.32
δ= 0.16
1245.6
1245.6
1671.186
1353.667
1245.6
1684.32
1353.228
1684.32
1684.32
1684.32
δ= 0.22
1245.6
1245.6
1684.32
1386.451
1245.6
1684.32
1307.309
1684.32
1684.32
1684.32
δ= 0.08
Input 3
Output 1 Output 2
1245.6
15797
771
1648.456
24574
864
1578.052
23040
848
1302.835
17044
784
1245.6
15797
771
1684.32
25355
872
1283.633
16626
780
1684.32
25355
872
1495.024
21231
828
1684.32
25355
δ= 0.2029
15797
771
873531.5 1899.676 1245.6
15797
771
15797
771
1418118
1656
1245.6
15797
771
25068.85 868.9763 1337293
1752
1684.32
25355
872
18151.35 795.8786 1021339 1686.821 1386.451 18865.6016 803.4261
15797
771
1065426
1656
1245.6
15797
771
25355
872
1785506
1752
1684.32
25355
872
18141.79 795.7776 999295.7 1669.503 1307.309 17141.3918 785.2063
25355
872
1881027
1752
1684.32
25355
872
25355
872
1333918
1752
1684.32
25355
872
25355
872
2226372
1752
1684.32
25355
872
Output 1 Output 2
16282
776
25161
870
23594
853
17467
789
18322
798
25355
872
17040
784
25355
872
16241
776
25355
872
Input 1
2041814
1296283
1222402
933592.6
973891.6
1632108
913443.1
1719422
1173772
2035098
Input 2
1899.676
1744.152
1728.747
1668.524
1656
1752
1664.322
1752
1710.579
1752
15797
771
15797
771
25355
872
18865.6 803.4261
15797
771
25355
872
17141.39 785.2063
25355
872
25355
872
25355
872
Table 30. Input and output allocation, E3, with current budget to allocate
As the program also gives the corresponding efficiency weights, we include in this
appendix the matrices of weights for the three equity units evaluated at the maximum δ
value desired (this is the value of the Gini measure calculated in Appendix D). Is
important to notice that with the current budget to allocate, there exist significantly less
inefficient units after the allocation.
70
Inneficient
EBAIS
Efficiency Frontier Reference Set
Barrio Pilar
Jimenez
Calle Blancos 2
Divino Pastor
Calle Blancos 1
Divino Pastor
Calle Blancos 1
Divino Pastor
Las Lomas
Calle Blancos 1
Calle Blancos 2
Calle Blancos 2
Divino Pastor
Calle Blancos 1
Divino Pastor
Calle Blancos 1
Divino Pastor
Calle Blancos 2
Divino Pastor
Calle Blancos 1
Divino Pastor
Calle Blancos 1
Calle Blancos 2
Centeno Guell
Divino Pastor
El Encanto
Las Lomas
Santa Cecilia
Santa Eduviges
Reference
Weight
0.0701049
0.929895
0.462292
0.537708
0.269991
0.406853
0.323156
0.813694
0.186306
0.389691
0.610309
0.978541
0.0214587
0.563184
0.436816
0.353898
0.646102
0.0463741
0.953626
Table 31. Weights, E1, with previous period’s budget
Inneficient
EBAIS
Efficiency Frontier Reference Set
Reference
Weight
0.538293
Centeno Calle Blancos 2
Guell
Divino Pastor
0.461707
Table 32. Weights, E1, with current budget to allocate.
71
Inneficient EBAIS
Efficiency Frontier Reference Set
Reference
Weight
Calle Blancos 2
0.0349435
Barrio Fátima
Divino Pastor
0.6419
Las Lomas
0.323156
Calle Blancos 2
0.401666
Barrio Pilar Jiménez
Divino Pastor
0.598334
Calle Blancos 2
0.702077
Calle Blancos 1
Divino Pastor
0.297923
Calle Blancos 2
0.102064
Calle Blancos 2
Divino Pastor
0.897936
Calle Blancos 2
0.263015
Divino Pastor
Divino Pastor
0.736985
Calle Blancos 1
0.134076
Las Lomas
Calle Blancos 2
0.489595
Divino Pastor
0.376329
Calle Blancos 1
0.00664006
Santa Eduviges
Divino Pastor
0.99336
Table 33. Weights, E2, with previous period’s budget
Inneficient
EBAIS
Barrio Pilar
Jiménez
Centeno Guell
Efficiency Frontier Reference Set
Reference
Calle Blancos 2
Divino Pastor
Calle Blancos 2
Divino Pastor
Weight
0.349021
0.650979
0.189272
0.810728
Table 34. Weights, E2, with current budget to allocate
72
Inneficient
EBAIS
Efficiency Frontier Reference Set
Reference
Weight
Calle Blancos 2
0.977755
Barrio Fátima
Divino Pastor
0.022245
Calle Blancos 1
0.0937661
Barrio Pilar
Calle Blancos 2
0.0563116
Jiménez
Divino Pastor
0.849922
Calle Blancos 1
0.292818
Calle Blancos 1
Divino Pastor
0.707182
Calle Blancos 1
0.864765
Calle Blancos 2
Divino Pastor
0.135235
Calle Blancos 1
0.784959
Centeno Guell
Divino Pastor
0.215041
Calle Blancos 2
0.965933
El Encanto
Divino Pastor
0.0340666
Calle Blancos 1
0.420924
Las Lomas
Divino Pastor
0.579076
Calle Blancos 1
0.676844
Santa Cecilia
Las Lomas
0.323156
Table 35. Weights, E3, with previous period’s budget
Inneficient
EBAIS
Calle Blancos 2
El Encanto
Efficiency Frontier Reference
Set
Reference
Calle Blancos 2
Divino Pastor
Calle Blancos 2
Divino Pastor
Weight
0.678949
0.321051
0.859344
0.140656
Table 36.Weights, E3, with current budget to allocate
73