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Constrained Optimization Methods in Health Services Research – An Introduction:
Report 1 of the ISPOR Optimization Emerging Good Practices Task Force
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Abstract
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Health services should be organized to give the greatest possible benefit to patients. Identifying optimal
system and patient health services is the purview of mathematical optimization models. Once a range of
alternatives has been identified, it is often possible to determine if the problem has an optimal
solution. Seldom is the term ‘’optimal’ based on evidence that demonstrates such solutions are, indeed,
optimal – in a mathematical sense. This is a missed opportunity that could result in poor policy choices—
e.g., selection of policies associated with a local optimal solution, rather than a global optimal solution
for the system.
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In this report, we introduce constrained optimization methods with “optimal” defined as the best
possible solution for a given problem given the complexity of the system inputs, outputs/outcomes, and
constraints. We present definitions of important concepts and terminology, as well as the methods and
their formulation basics. We also explain how these mathematical optimization methods relate to
simulation methods, to standard health economic analysis techniques, and to the emergent fields of
analytics and machine learning.
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This task force report identifies: 1) key optimization concepts and the main steps in building an
optimization model; 2) the types of problems where optimal solutions can be determined in real world
health applications and 3) the appropriate optimization methods for these problems. While a number of
examples are included to illustrate how applying optimization works, we developed a simple model
based on a culinary production problem – “Why making pizza and maximizing health care value are the
same problem”.
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Guidance on optimization methods is important because current health economic and outcomes
evaluation methods generally simulate different outputs based on known parameter distributions, but do
not provide guidance on optimal system design and optimal care pathways given system or resource
constraints.
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1.
Introduction
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In common vernacular, the term “optimal” is often used loosely in health care applications to refer to any
demonstrated superiority among a set of alternatives in specific settings. Seldom is this term based on
evidence that demonstrates such solutions are, indeed, optimal – in a mathematical sense. By “optimal”
we mean the best possible solution for a given problem given the complexity of the system inputs,
outputs/outcomes, and constraints. Failing to identify a truly optimal solution represents a missed
opportunity that could result in a poor clinical decision or choice of policy.
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Constrained optimization methods trace their origin to the development of the simplex algorithm--the
primary approach to solving linear programming problems--in 1947 (Kirby, 2003). Since that time, a
variety of constrained optimization methods have been developed in the field of operations research and
widely applied across a broad range of industries. This creates significant opportunities to transfer
knowledge from fields outside the health care sector, to the optimization of health care delivery systems
and provide value.
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Patient scheduling, provider resource scheduling, and logistics are another large area of research in the
application of constrained optimization methods to healthcare (Burke et al., 2006; Cheang et al., 2003;
Lin et. al., 2013a; Lin et al., 2013b; Sir et al., 2015). Constrained optimization methods can also be used
by health care systems to identify the optimal allocation of resources across interventions subject to a
variety of different types of constraints (Earnshaw, 2002; Thomas et al., 2013). Constrained
optimization methods from operations research have been applied to problems of diagnosing disease
(Lee and Wu, 2009; Liberatore and Nydick, 2008) and the development of optimal treatment algorithms
(Lee et al., 2008; Ehrgott, et al., 2008).
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Constrained optimization methods may also be very useful in guiding clinical decision-making in
actual clinical practice where physicians and patients face constraints such as proximity to
treatment centers, health insurance benefit designs, and the limited availabili ty of health resources.
The potential benefits of constrained optimization may be especially evident relative to decision
making that is based upon clinical trials evidence where such constraints are explicitly removed
through study design.
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These methods are now being applied to outcomes research and health economic problems due to their
relevance in health technology assessment (Thokala P, et al. 2015). Researchers who conduct health
systems and outcomes research studies come from diverse backgrounds and some may lack
understanding of systems science and basic training in the theory and methods for operations
research. In addition, many researchers may not be aware of the range of operations research
methods available and the contexts in which they should be used most appropriately, recognizing
both their strengths and limitations. Thus, there is a need for guidance for this emerging area of
outcomes research in health care delivery and economics.
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Recently, the ISPOR Emerging Good Practices Task Force on Dynamic Simulation Modeling
Applications in Health Care Delivery Research published two reports in Value in Health (Marshall et al.
2015a, 2015b) and one in Pharmacoeconomics (Marshall et al. 2016) on the application of dynamic
Identifying optimal health system and patient care interventions is the purview of mathematical
optimization models. There is a growing recognition of the applicability of constrained optimization
methods from operations research to health care problems. In a review of the literature, Rais and Viana
(2010) note more than 200 constrained optimization and simulation studies in health care. For
example, constrained optimization methods have been applied in problems of capacity management and
location selection for both healthcare services and medical supplies (Ndiaye and Alfares, 2008; Araz et
al., 2007, Bruni et al., 2006; Verter and Lapierre, 2003).
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simulation modeling (DSM) to evaluate problems in health care systems. DSM enables the evaluation of
a wide range of intended and unintended outcomes from health care interventions or policy changes in
health care systems. These methods help decision-makers to anticipate the downstream consequences
of changes in the health care system. Decision makers are able to test ‘what if’ scenarios of a proposed
policy before implementation -- without actually having to implement the policy first. While simulation
can provide a mechanism to test/evaluate various scenarios, by design, they do not necessarily provide
optimal solutions. The Optimization Task Force emphasizes mathematical approaches to identify
optimal resource allocations in the face of constraints.
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The overall objective of the task force on optimization methods is to develop guidance for health services
researchers, knowledge users and decision makers regarding operations research methods to optimize
healthcare delivery and value in the presence of constraints. Specifically, this task force will (1)
introduce the value of constrained optimization methods in conducting research on health care systems
and individual-level outcomes research; (2) describe problems for which constrained optimization
methods are appropriate; and (3) identify good practices for designing, populating, analyzing, testing
and reporting high quality research for optimizing healthcare delivery and value at both the systems and
individual levels in the presence of constraints.
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Because this is a practice area with a substantial amount of material to cover, the Optimization Task
Force will publish two reports. In this first paper, we introduce readers to constrained optimization
methods. We present definitions of important concepts and terminology, and provide examples of
health care decisions where constrained optimization methods are already being applied. We also
describe the relationship of constrained optimization methods to health economic modelling and
simulation methods. The second paper will present a series of case studies illustrating the application of
these methods including model building, validation, and use.
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2.
Relationship of Constrained Optimization to Related Fields
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a) Constrained Optimization Methods Compared with Traditional Health Economic
Modelling in Health Technology Assessments
Constrained optimization methods differ substantially from traditional health economic modeling
methods traditionally used in health technology assessment processes. The main difference between the
two approaches is that traditional health economic modeling approaches, such as Markov models, are
built to estimate the costs and effects of different diagnostic and treatment options. If decision makers,
are basing their judgements on modeling results, they may not formally consider the constraints and
resource implications in the system. Constrained optimization methods provide a structured approach to
optimize the decision problem and to present the best alternatives given an optimization criterion, such
as constrained budget or availability of resources.
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These differences have major implications. There is an opportunity to learn from optimization methods
to improve HTA processes. Optimization is a better means of capturing the dynamics and complexity of
the health system to inform decision making for several reasons. Constrained optimization methods (a)
explicitly take budget constraints and impact into account; (b) can handle other constraints in the health
system, such as capacity; (c) take emergent behavior and evolution over time into account instead of
informing a decision of a single technology at a single point in time; and (d) inform not only about
affordability, but also about implementability and feasibility.
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i.
Informed decision making about resource allocation requires an external estimate of the
decision-maker’s willingness to pay for a unit of health outcome – the threshold. HTA then relies
on the principle that by repeatedly applying the threshold to individual HTA decisions,
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optimization of the allocation of health resources will be achieved. However, health economics
(HE) usually is about relative efficiency without directly aiming for budget optimization because
many jurisdictions do not explicitly implement a constrained budget nor do they employ
mechanisms to retrospectively evaluate cost-effectiveness of medical technologies.
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ii.
Constrained optimization methods also allow consideration of the effect of other constraints in
the health system, such as capacity or short-term inefficiencies. Capacity constraints are usually
neglected in health economic models. In HE models, the outcomes are central to decision
makers while the process to arrive at these outcomes is ignored. For health policy makers and
health care planners, such capacity considerations are critical and cannot be neglected. Likewise,
some technologies are known for short-term inefficiencies, e.g., large equipment such as PET-MR
imaging, are usually not taken into consideration. It takes a certain amount of time before a new
device operates efficiently, and such short-term inefficiencies do influence implementation (Van
de Wetering, Woertman, and Adang 2011).
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iii.
Emergent behavior and evolution over time
Health economic models with a clinical perspective, such as a whole disease model (Tappenden,
2012), or a treatment sequencing model, may allow the full clinical pathway to be framed as a
constrained optimization problem that accounts for both intended and unintended consequences
of health system interventions over time with feedback mechanisms in the system. Each
combination of decisions within the pathway can be a potential solution, constrained by the
feasibility of each decision, e.g., the licensed indication for various treatments within a clinical
pathway.
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The model can evaluate alternative guidance configurations and report the performance in terms
of an objective function (cost per QALY, net monetary benefit). There is emerging literature in
this area (Kim 2015; Tosh 2015). It may avoid piece-meal guidance development, such as the
United Kingdom’s National Institute for Health and Care Excellence’s single technology
assessment (NICE STAs) that may use different models and different evidence, resulting in suboptimal resource allocation.
iv.
Implementability
An advantage of constrained optimization is the imposition of constraints to achieve specific
performance outcomes (minimum or maximum) associated with an objective function. HE
models are not typically constrained in this way – it is assumed resources are available as
required. In some sense, classic HE cost effectiveness models are ‘hypothetical’ to illustrate the
potential value as measured by a specific outcome with respect to cost, whereas optimization is
focused on what can be achieved in an operational context. This suggests constrained
optimization methods are much better for informing decisions about implementability as they
actually consider these barriers in the modeling approach.
b) Constrained Optimization Methods as Part of Analytics
Constrained optimization as a technique falls within the area of analytics. Broadly, analytics can be
classified into descriptive, predictive and prescriptive analytics. Descriptive analytics concern the use
of historical data to describe data, patterns and test hypotheses. Research typically uses theory and
concepts to identify hypotheses, and historical data is used to test these hypotheses using statistical
methods. Examples may include natural history of aging, disease progression, evaluation of clinical
interventions, policy interventions, and many others. Traditional health services and outcomes
research for the most part falls within the area of descriptive analytics.
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Predictive analytics focus on trying to forecast or foretell the future states of disease or states of
systems. With the surge in the amount of data on health and health care, large amounts of data, also
referred to as big data, are warehoused, Predictive analytics looks for trends and patterns in data to
estimate future values for variables. Prediction can be helpful in determining what is expected as well
as help inform clinical, administrative and policy decisions.
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Prescriptive analytics uses the understanding of systems, both the historical and future based on
descriptive and predictive analytics respectively to determine future course of action/decisions.
Constrained optimization is a specialized form of prescriptive analytics, since it helps with determining
the optimal decision or course of action in the presence of constraints.
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Figure 1.
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Future states
Optimal
decision
Optimization
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Wilson, ISPOR 2014
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c) Constrained Optimization Methods Compared with Dynamic Simulation Models
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Dynamic simulation modeling methods (DSMs), such as system dynamics, discrete event simulation
and agent based modeling are used to design and develop mathematical representations, i.e., formal
models, of the operation of processes and systems. They are used to experiment with and test
interventions and scenarios and their consequences over time in order to advance the understanding
of the system or process, communicate findings, and inform management and policy design
(Marshall et al. 2015a, 2015b, 2016;, Harrison et al., 2007; Banks, 1998; Sokolowski, 2011). These
methods have been broadly used in health applications (e.g., Milstein et al 2011, Troy and Rosenberg
2009, Macal et al. 2014).
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Unlike constrained optimization methods, DSMs do not produce a specific solution. Rather they
allow for the evaluation of a range of possible or feasible scenarios or intervention options that may
or may not improve the system’s performance. Constrained optimization methods, in general, seek to
provide the answer to which of those options is the “best”. Hence, the types of problems and
questions that can be addressed with DSMs (Marshall et al 2015a, 2015b, 2016) are different from
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those that are addressed with optimization methods. However, both types of methods can be
complementary to each other in helping us to better understand systems.
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Traditionally, constrained optimization methods have served two distinct purposes in DSM
development. 1) model calibration – fitting suitable model variables to past time series is discussed
elsewhere (Marshall et al 2015a, 2015b, 2016; 2) evaluating a policy’s performance/effect relative to a
criterion or set of criteria. However, the complexity of DSMs compared to simple analytic models
may render constrained optimization cumbersome, inappropriate and potentially infeasible due to
the large search space e.g., using methods of optimal control.
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Due to this complexity, alternative approaches such as algorithmic search strategies are available.
Historically, these types of methods have been used in system dynamics and other DSMs. Due to
their heuristic nature, there is no certainty of finding the “best” or optimal parameter set rather
“good enough” solutions. Hence, the ranges assigned need careful consideration in order to get
“good” solutions, i.e., prior knowledge of sensible ranges both from knowledge about the system and
knowledge gained from model building. There are many examples that show the usefulness of
optimization in system dynamics to gain insight about policy design and strategy design, particularly
when the traditional analysis of feedback mechanisms becomes risky due to the large numbers of
loops in a model, i.e., hundreds. Similar procedures to evaluate policies and strategies can be can be
utilized in discrete event simulation (DES) and agent based modeling (ABM), e.g., simulated
annealing algorithms and genetic algorithms.
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d) Constrained Optimization Methods Compared with Machine Learning
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With the increased volume and complexity of health care data, especially medical claims and
electronic medical record data, and the ability to link to other information such as feeds from
personal devices and socio demographic data, big data methods such as machine learning are
garnering increased attention (Crown, 2015). Machine learning methods, such as predictive
modelling and clustering, have an important intersection with constrained optimization methods.
Machine learning methods are valuable for addressing problems involving classification, as well as
addressing data dimension reduction issues and identifying predictors.
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Constrained optimization would suggest developing models to predict patterns using all input
variables from big data, which could represent hundreds, thousands or millions of inputs. On the
other hand, minimal models can elegantly describe trends or patterns. However, they have high
mean squared errors that can question the predictive validity of such approaches. The concept of
optimization allows one to develop machine-learning models that select an appropriate number of
variables to create predictively valid models that are manageable and accurate.
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3.
Definition of Constrained Optimization
Constrained optimization is an essential tool to inform decision making based on quantitative
information. It is the systematic set of procedures applied to find the best among a set of feasible
solutions. It entails maximizing or minimizing an objective function that is representative of the
quantifiable measure of interest to the decision maker subject to constraints also faced by the decision
maker. Maximizing/minimizing the objective function is carried out by systematically selecting input
values from within an allowed set of values and computing the value of the objective function. Note that,
programming and optimization are often used as interchangeable terms in the literature, e.g., linear
programming and linear optimization. (Programming here refers to mathematical programming as
opposed to computer programming.)
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The components of a problem are its objective function(s), its decision variable(s) and its constraint(s).
After formulating the problem mathematically, optimization can be described as finding the minimum or
maximum of a given objective function by changing its decision variables within the feasibility space
subject to a set of constraints. Definitions of these components are elaborated below:
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4.
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At the outset, this problem seems fairly straightforward. One might decide on four regular pizzas to use
up all the baking time that is available. This feed eight people while leaving £5 as excess budget. An
alternate approach might be to make as many large pizzas as you can since each can feed one more
person than a regular pizza. You would end up with three large pizzas (totaling £15). This approach can
feed a total of nine people leaving 15 minutes extra baking time unused. There are others combinations
of regular and large pizza, such as respectively, three and one, two each, etc.
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This is graphically represented in Figure 2. The optimal solution is two regular pizzas and two large
pizzas. This approach uses the total one hour backing time as well as the total £15 budget. Since regular
and large pizzas feed two and three people, respectively, we are able to feed 10 people and still meet the
time and budget constraints.
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No other combination of pizzas is capable of feeding more people while still meeting the time and budget
constraints. Note that not all resource constraints have to be completely used to attain the optimal
solution. The pizza example is a small-scale problem with only two decision variables; the number of
regular and large pizza that can be made. Hence, they can be represented graphically with one variable
on each axis. Larger problems cannot be represented graphically, hence we turn to mathematical
approaches, such as the simplex algorithm to find the solutions.
Parameters are constant for a given optimization model and system. The values are determined based
on real world aspects of the decision-making problem being solved. Decision variables are the
constituents of the system for which decisions should be taken. They should describe the decisions that
can be taken and which will change the value for the objective function. The objective function is a
function of the decision variables, which represents the quantitative measure that the decision maker
aims to minimize/maximize. The constraints are the restrictions on decision variables. These
restrictions are defined by inequalities relating to functions of decision variables. They determine the
allowable/feasible values for the decision variables.
A Simple Illustration of a Constrained Optimization Problem -- Let’s Make Pizza!
(Why making pizza and maximizing health care value are the same problem.)
To illustrate the concept and terminology, we describe a constrained optimization problem for making
pizza. Although it is a simple example, it illustrates many of the concepts and definitions just described.
It is movie night, and you have offered to feed pizza to as many hungry people as possible. You have two
options--baking regular or large pizzas. Regular pizzas can feed two people and larges ones can feed
three people. Each pizza, irrespective of size, takes 15 minutes to bake, only one pizza can be baked at
any given point in time. You have one hour of total oven time at your disposal. Regular pizzas cost £2.50
each, and large pizzas cost £5 each. You have a total budget of £15. What is the greatest number of
people you can feed?
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Time constraint (1 hour)
3
Objective function
2
Large Pizza
Budget constraint (£15)
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0
1
2
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Regular Pizza
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The mathematical formulation of the model is as follows:
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max
subject to
fR xR + fL xL
cR xR + cL xL ≤ B
tR xR + tL xL ≤ T
xR ,xL ≥ 0 and integer
(objective function)
(budget constraint)
(time constraint)
(decision variables)
Where:
cR,cL= cost of regular and large pizza, respectively
B = total budget available
tR,tL= time to make regular and large pizza, respectively
T = total time available
fR,fL= number of people a regular and large pizza can feed, respectively
xR,xL= number of regular and large pizzas to make, respectively
In the current version of the problem, the parameters are:
fR = 2 people, fL = 3 people
cR = £2.50, cL = £5, B = £15
tR =0.25 hours, tL = 0.25 hours, T = 1 hour
So the model is as follows:
max
subject to
2 xR + 3xL
(objective function)
2.5xR + 5xL ≤ 15 (budget constraint)
0.25xR + 0.25xL ≤ 1
(time constraint)
xR ,xL ≥ 0 and integer
so, What is the greatest number of people you can feed?
Problems That Can Be Tackled with Constrained Optimization Approaches
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In this section, we discuss problems where constrained optimization approaches can shed insight. The
selected examples do not represent a comprehensive picture of this field, but provide the reader a sense
of what is possible. In Table 1, we provide a summary of the section, comparing problems using the
terminology of the previous section, with respect to decision makers, decisions, objectives, and
constraints. These terms will be defined more formally in Section 4.
Table 1 – Examples of health care decisions for which constrained optimization is
applicable
Type of health
Typical
Typical decisions
Typical
Typical
care problem
decision
objectives
constraints
makers
Resource allocation
within and across
disease programs
Health
authorities,
insurance
funds
Public health
agencies,
health
protection
agencies
Organ banks,
transplant
service centers
Radiation
therapy
providers
List of interventions
to be funded
Increase
population health
Overall health?
budget
Optimal vaccination
coverage level
Ensure disease
outbreaks can be
rapidly and cost
effectively
contained,
Matching organ
donors with
potential recipients
Minimizing the
radiation on
healthy anatomy
Availability of
medicines, disease
dynamics of the
epidemic
Disease
management
Models
Leads for a
given disease
management
plan
Identify the best
plan using a whole
disease model,
maximizing QALYs
Workforce
planning/ Staffing
/ Shift template
optimization
Hospital
managers,
all medical
departments
(e.g., ED,
nursing)
Operation
room/ ICU
planners
Clinical
department
managers
Strategic
health
planners
Best interventions to
be funded, best timing
for the initiation of a
medication, best
screening policies
Number of staff at
different hours of the
day, shift times
Detailed schedules
Minimize waiting
time
Availability of
beds, staff
Detailed schedules
Minimize over- and
under-utilization of
health care staff
Ensure equitable
access to hospitals
Availability of
appointment slots
Resource allocation
for infectious
disease
management
Allocation of
donated organs
Radiation
treatment planning
Inpatient
scheduling
Outpatient
scheduling
Hospital facility
location
Matching of organs
and recipients
Positioning and
intensity of radiation
beams
Set of physical sites
for hospitals
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Increase efficiency
and maximize
utilization of
healthcare staff
Every organ can be
received by at most
one person
Tumour coverage
and Restriction on
total average
dosage
Budget for a given
disease or capacity
constraints for
healthcare
providers
Availability of staff,
human factors,
state laws (e.g.,
nurse-to-patient
ratios), budget
Maximum
acceptable travel
time to reach a
hospital
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An important application of optimization in planning health care expenditure is the resource
allocation problem faced by a planner. A planner has a number of investment opportunities, but a
fixed budget that is not adequate to fund cover all these opportunities (Stinnett and Paltiel 1996).
Perhaps the simplest case of this is where the investment opportunities are incremental to current care
and fall in distinct categories, e.g., children’s services, cardiovascular disease, cancer, respiratory disease
and mental health (as in Airoldi, Morton et al. 2014). Evaluation of financial efficiency and resources
used for delivery of health & social services have been accomplished using constrained optimization
(Medina-Borja et al., 2007; Pasupathy and Medina-Borja, 2008).
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In this case, decisions about investments in different clinical areas can be made independently of one
another and the optimization problem – of choosing the best set of investment opportunities to fund
subject to a fixed budget constraint in order to meet an objective such as maximizing total QALYs – is
called a knapsack problem (Martello and Toth 1990). The size of the knapsack represents the budget
available, and the objective is to maximize health by filling the knapsack (spending the budget) with a set
of health interventions. The standard cost-effectiveness rule of prioritizing projects based on their
incremental cost-effectiveness ratios (ICERs) and funding only those with an ICER above a critical
threshold can be understood as an algorithm for solving (perhaps approximately) this knapsack problem
(Weinstein and Zeckhauser 1973).
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Other resource allocation problems may not be so straightforward. For example, consider the case of
allocating resources for the prevention and cure of an infectious disease such as HIV,
Hepatitis C, TB, malaria, or polio (Castillo-Chavez and Feng 1998, He, Li et al. 2015, Juusola and
Brandeau 2015). Here, there may be significant and complex interactions between different
investments: if the planner invests in vaccination, there may be fewer cases to treat in the future (and so
investment in highly capital-intensive treatment facilities may be wasted). On the other hand,
vaccination itself is costly, and if the disease has low prevalence, it may be cost-effective to target
treatment (Lee, Yuan et al. 2015).
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Unlike the knapsack problem, where the optimal solution can be identified using nothing more complex
than a spreadsheet, optimizing infectious disease programs may involve making multiple runs of a stateof-the-art simulation (Marshall, Burgos-Liz et al. 2015, Marshall, Burgos-Liz et al. 2015) of the infectious
disease dynamics, to plot out how the particular patterns of resource allocation perform against the
objective (of minimizing the total number of cases, or maximizing the probability of achieving disease
eradication). For a review of mathematical approaches for infectious disease prediction and control, see
Dimitrov and Meyers (2010).
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In other settings, the critical resources might not be money. For example, allocating donated organs
(such as kidneys) is more complex than allocating money because not every kidney will be compatible
with every donor (in contrast, money is fungible – one dollar is always worth the same, no matter who
receives it). In this case, the underlying problem becomes a matching problem (Roth and Sotomayor
1992).
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Matching problems can be thought of as trying to arrange a large number of marriages – not everyone
will get the best match, but the objective is to ensure that as few people are left on the shelf (patients
without kidneys; kidneys without patients). Bertsimas, Farias et al. (2013) have an interesting
discussion about how to incorporate fairness in such problems: some measures of prioritization (for
example, time on waiting list) may be incorporated in the objective function, but some fairness
considerations may also be included as constraints (for example at least x percent of transplants should
go to patients of a certain blood type).
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In other clinically-oriented problems, consider how radiation treatment planning might be framed
as an optimization problem (Shepard, Ferris et al. 1999). In this setting, a cancerous tumor within a
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patient’s anatomy is targeted with several beams of radiation passing through the tumor from different
directions (a single beam of radiation strong enough to control the growth of the tumor would do
unacceptable damage to healthy tissue in its path). A typical objective function in this setting might be
to minimize the damage to healthy tissue while a constraint might be ensuring that the tumor receives a
dose of radiation sufficient to prevent further tumor growth.
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The decision variables might be the angles at which the beams are positioned (Craft 2007) or the
intensity of the subcomponents of beams, referred to as beamlets (Romeijn, Ahuja et al.
2006). Although the basic problem sounds straightforward in principle, the complexities of the
underlying physics and biology, the existence of conflicting treatment goals, uncertainties caused by
daily setup procedures and organ motion, as well as ensuring that information garnered in the course of
treatment is efficiently integrated into the treatment plan means that accurately solving this problem
presents substantial conceptual and computational challenges (Bortfeld 2006, Sir, Epelman et al. 2012,
Akartunalı, Mak-Hau et al. 2015), hence requiring constrained optimization approaches.
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Other clinical problems where optimization can be applied relate to problems of disease management
such as the timing of the initiation of treatment. The promise of health gain from treatment must be
balanced against reasons for holding off treatment which may include cost, undesirable side-effects,
emergent drug resistance, or the possibility that the presenting symptom may be not as threatening as it
appears, e.g., maybe a tumor has been detected, but it is benign. It may be that there is an optimal stage
in the disease progression or point in the patient’s life cycle where the balance shifts from favoring nonintervention to favoring treatment. The modelling framework for identifying such critical points is the
Markov Decision Process framework (Puterman 2014). This framework has been used to analyze timing
decisions in diseases as diverse as HIV, diabetes and breast cancer (Shechter, Bailey et al. 2008, Denton,
Kurt et al. 2009, Chhatwal, Alagoz et al. 2010).
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Workforce planning in healthcare involves several important tactical and operational decisions that
can be grouped under three broad stages, which progress from organizational budgeting, to optimizing
shift templates, and then finally to assigning individual healthcare professionals to shifts. In the
budgeting stage, each department/unit is allocated a certain amount of full-time equivalent (FTE) of
different roles (e.g., physician, nurse, clinical assistants, etc.) based on historical patient demand
patterns, financial considerations, and other managerial constraints (Trivedi 1981, Kwak and Lee 1997,
Mincsovics and Dellaert 2010).
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As a tactical decision, shift templates should be periodically adjusted so that staffing levels meet patient
demand at different times of the day and days of the week. This stage involves several constraints related
to human factors (e.g., certain shift start times cannot be not allowed), mandatory staff-to-patient ratios
(Lin et al., 2013b; Sir et al., 2015; Das et al., 2016), coordination of shifts of a certain role with other roles
and hospital units (e.g., the nurse and physician shift in an emergency department should be aligned)
and budget (Warner and Prawda 1972, Green, Soares et al. 2006, Sinreich and Jabali 2007). Once the
shift templates are determined, each individual healthcare professional must be assigned to various
shifts within a certain planning period (e.g., a month) (Arthur and Ravindran 1981, Cohn, Root et al.
2009, Defraeye and Van Nieuwenhuyse 2016).
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Many models developed for this stage propose different mechanisms to incorporate individual
preferences while ensuring some degree of fairness among individuals and safety guidelines (e.g., a
resident cannot be assigned to two shifts unless they are apart from each other for a certain amount of
hours) (Azaiez and Al Sharif 2005, Lin, Sir et al. 2013).
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A related operational problem to workforce planning is appointment scheduling, which, in broad
terms, involves determining to which appointment slot associated with a specific provider or diagnostic
resource a patient is assigned based on medical urgency, patient priority, provider/organizational
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preferences, etc. (Gupta and Denton 2008). Due to dynamically changing provider and resource
calendars, appointment scheduling problems are often modeled using a dynamic programming
framework (Patrick, Puterman et al. 2008) with possible objectives of minimizing service level (also
referred to as access target) violations for different priority group and minimizing wasted capacity.
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A highly related strategic problem is to determine how to allocate limited capacity for different patient
priority groups. Several exogenous factors further complicate the appointment scheduling problem
including late cancelations or no shows, patient/provider tardiness, delays in appointment duration, and
emergency add-ons to the calendars (Cayirli, Yang et al. 2012). One special case of appointment
scheduling is surgery scheduling, which received a great amount of attention due to surgical services
being a major revenue source for hospital systems (Gupta 2007). More specifically, an operational
problem that is solved daily by surgery schedulers involves determining the operating room assignment
and start time of a set of scheduled surgeries. The most challenging aspect of this problem is the high
uncertainty associated with surgery durations (Batun, Denton et al. 2011).
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Facility location is a classical optimization problem where a decision maker determines where to
locate/build facilities among a set of candidate locations and how to assign different demand locations to
these facilities. The most common objective is to minimize cost that has two main components: the cost
of building facilities and transportation cost for serving clients at different demand locations from the
built facilities (Simchi-Levi, Chen et al. 2014). Facility location has many applications to health care
(Daskin and Dean 2005). One interesting application is optimizing hospital network planning
considering uncertainties in patient demand (Mestre, Oliveira et al. 2015). Another application is
determining optimal location of emergency medical service stations in case of a large-scale disaster (Jia,
Ordóñez et al. 2007, Lee, Chen et al. 2009).
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6.
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Table 2. Steps in an optimization process
Steps in a Constrained Optimization Process
Comprehensive descriptions of the steps involved in optimization have been published (e.g.
http://www3.nd.edu/~jstiver/FIN360/Constrained%20Optimization.pdf). An overview of the main
steps involved is presented in Table 2. It is important to emphasize that the process of optimization is
iterative, rather than comprising a strictly sequential set of steps. However, in contrast to alternative
methods such as dynamic simulation modelling, constrained optimization goes beyond the elaboration
of outcomes under different scenarios. The goal of constrained optimization is to identify the optimal
solution to a particular objective subject to existing constraints. In addition, there is software available
for supporting optimization.
Step
Problem structuring
Mathematical formulation
Model development
Select optimization method
Perform optimization
Perform validation
Report results
Description
Specify the objective and constraints, identify decision variables and
constant parameters and list and appraise model assumptions
Present the objective function and constraints in mathematical
notation using decision variables and constant parameters
Develop the model to estimate the objective function and constraints
using decision variables and constants
Choose an appropriate optimization method and algorithm based on
the characteristics of the model
Use the optimization algorithm to search for the optimal solution
Examine performance of model for reasonable values of parameters
and decision variables
Report the results of optimal solution (i.e. values of decision
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Decision making
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variables, constraints and objective function) and sensitivity analysis
Interpret the optimal solution and use it for decision making
a) Problem structuring
This involves specifying the objective, i.e. goal, identifying the decision variables, constant parameters
and the constraints involved. These can be specified using words, ideally in non-technical language so
that the optimization problem is easily understood. Table 1 presents the objective, variables, constraints
and the relevant decision makers associated with different exemplar health care optimization problems.
This step needs to be performed in collaboration with all the relevant stakeholders, including decision
makers, to ensure all aspects of the optimization problem are captured. As with any modelling technique
it is also crucial to surface key modelling assumptions and appraise them for plausibility and materiality.
b) Mathematical formulation
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After the optimization problem is specified in words, it needs to be converted into mathematical
notation. The standard mathematical notation for any optimization problem involves specifying the
objective function and constraint(s) using decision variables and constant parameters. This also involves
specifying whether the goal is to maximize or minimize the objective function. The standard notation for
any optimization problem, assuming the goal is to maximize the objective, is as shown below:
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Maximize z=f(x1, x2, …. xn)
subject to
cj(x1, x2, …. xn)≤Cj
for j=1,2,..m
where, x1, x2, …. xn are the decision variables, f(x1, x2, …. xn) is the objective function; and cj(x1, x2, …. xn,
p1, p2, …. pk)≤Cj represent the constraints. Specification of the optimization problem in this
mathematical notation allows clear identification of the type (and number) of decision variables,
constant parameters and the constraints. It should be noted that the objective function and the
constraints also include constant parameters p1, p2, …. pk, which are fixed (i.e. their values do not change
during the optimization problem).
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c) Model development
The next step after mathematical formulation is model development. The model should estimate the
objective function and the left hand side (LHS) values of the constraints, using the decision variables and
constant parameters as inputs. The complexity of the model can vary depending on the decision
problem, from simple linear equations to sophisticated health economic/health care delivery simulation
models. Similar to other types of modelling, the choice of the model depends on the outputs required
and the level of detail included in the model depends on the accuracy required, e.g., in the estimation of
the objective function and the LHS of constraints).
d) Select optimization method
This step involves choosing the appropriate optimization method. Optimization problems can be
classified depending upon the nature of the objective functions and the constraints. The broad
classifications include linear vs non-linear, deterministic vs stochastic, continuous vs discrete, single vs
multi-objective optimization. For instance, if the objective function and constraints consist of linear
functions only, the corresponding problem can be identified as linear optimization. Similarly, in
deterministic optimization, it is assumed that the constant parameters used in the optimization problem
are fixed while in stochastic optimization, uncertainty is incorporated. The optimization problems can be
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continuous (i.e. fractional values are allowed) or discrete (for example a hospital ward may be either
open or closed; the number of CT scanners which a hospital buys must be a whole number). Most
optimization problems have a single objective function, however when optimization problems have
multiple conflicting objective functions, they are considered as multi-objective optimization problems.
e) Perform optimization
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Optimization involves running the model for different sets of decision variables to find a combination of
decision variables that achieve the objective, using specific algorithms. Broadly speaking, optimization
methods use two types of optimization algorithms – iterative methods and heuristic methods. Iterative
methods might reach the optimal solution within a pre-specified maximum amount of steps or might
converge on the optimal solution. Examples of these include simplex methods for linear programming
and the Newton method for non-linear programming (e.g., Minoux, 1986; Kelly, 1999).
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Heuristic methods provide approximate solutions to optimization problems and are typically used when
iterative methods are unavailable or computationally expensive. Examples of these techniques include
relaxation approaches, evolutionary algorithms (such as genetic algorithms), simulated annealing,
swarm optimization, ant colony optimization, and tabu search. Besides these two methods, other
methods are also available to tackle large-scale problems, as well (i.e. decomposition of the large
problems to smaller sub-problems).
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There are software programs that help with optimization, interested readers are referred to the website
of INFORMS (www.informs.org) for a list of optimization software. It should be noted that the users
need to specify, and more importantly understand, the parameters for these optimization algorithms,
e.g., the termination criteria such as the level of convergence required or the number of iterations).
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f) Perform validation
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Once the optimization algorithm has finished running, the results need to be interpreted. First, the
results should be checked to see if there is actually a feasible solution to the optimization problem, i.e.
whether the optimal solution satisfies all the constraints. If not, then the optimization problem needs to
be adjusted, e.g., changing the objective, or relaxing some constraints or adding other decision variables.
If a feasible optimal solution has been found, the results need to be understood – this involves
interpretation of the results to check whether the optimal solution, i.e., values of decision variables,
constraints and objective function makes sense.
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This may also involve running sensitivity analyses, e.g. running the optimization problem using different
values for using additional decision variables and constraints, in order to verify the robustness of the
optimization results. Sensitivity analysis is an important part of building confidence in and sensechecking an optimization model, ensuring that it is a good representation of the problem at hand.
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g) Report results
The final optimal solution and if applicable, the results of the sensitivity analyses should be reported.
This will include the results of the optimum ‘objective function’ achieved and the set of ‘decision
variables’ at which the optimal solution, i.e., the optimum objective function, is found. Both the
numerical values (i.e. the mathematical solution) and the physical interpretation, i.e., the non-technical
text describing the meaning of numerical values, should be presented. The optimal solution identified
can be contextualized in terms of how much ‘better’ it is compared to the current state. For example, the
results can be presented as improvement in benefits such as QALYs or reduction in costs.
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It would also be useful to report the optimization method used and the results of the ‘performance’ of the
optimization algorithm, e.g., number of iterations to the solution, computational time, convergence level,
etc. Dashboards can be useful to visualize these benefits and communicate the insights gained from the
optimal solution and sensitivity analyses.
h) Decision making
The final optimal solution and its implications for policy/service reconfiguration should be presented to
all the relevant stakeholders. This typically involves a plan for amending the ‘decision variables’, e.g.,
shift patterns, screening frequency, (see Table 1 for a complete list) to those identified in the optimal
solution. This needs getting the ‘buy-in’ from the decision makers and all the stakeholders, e.g., frontline
staff such as nurses, hospital managers, etc., to ensure that the numerical ‘optimal’ solution found can be
operationalized in a ‘real’ clinical setting. After the decision is made, data should still be collected to
assess the efficiency and demonstrate the benefits of the implementation of the optimal solution.
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7.
Summary and Conclusions
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In this report, we introduce readers to the vocabulary of constrained optimization models and outline
the broad types of models available to analysts for a range of different health care problems. We outline
the relationship of constrained optimization methods to traditional health economic modelling and
simulation models. We illustrate the formulation of a straightforward linear program to make pizza and
solve the problem graphically. Although simple, this example illustrates many of the key features of
constrained optimization problems that would be commonly encountered in health care.
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In the second task force report, we describe several case studies that illustrate the formulation,
estimation, evaluation, and use of constrained optimization models. The purpose is to illustrate actual
applications of constrained optimization problems in health care that are more complex than the pizza
example described in the current paper and make recommendations on emerging good practices for the
use of optimization methods in health care research.
This is the first report of the ISPOR Constrained Optimization Methods Task Force. It introduces
readers to the application of constrained optimization methods to health care systems and patient
outcomes research problems. Such methods provide a means of identifying the best policy choice or
clinical intervention given a specific goal and in the face of a specified set of constraints. Constrained
optimization methods are already widely used in health care in traditional areas such as choosing the
optimal location for new facilities, making the most efficient use of operating room capacity, etc.
However, they have been less widely used for decision making about clinical interventions for patients.
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Appendix Material
793
Pizza problem
Health Care
Terminology
Options available
Regular or large pizzas
pharma, bundled episodic
payment models, ortho,
hip/knee, etc
Decision variables
Constraints
Total cost < £15
Budget constraint
Constraints
Aim
Maximise number of people
to feed
Maximise health care benefits
Objective function
Evidence base
Cost of each pizza, how many
people it can feed and the
time taken to cook
Costs of each intervention,
health benefits, and any other
relevant data
Model (to determine th
objective function and
Constraints)
Complexity
One-off, deterministic,
static problem
Repeated,
stochastic,
dynamic problem
Optimisation method
794
Complexity
Pizza problem
Health Care
Static vs Dynamic
Static (i.e. one-off) problem.
If the pizza problem was solved for multiple
time periods, then it will become dynamic
problem
Dynamic problem.
Health care is constantly evolving – ch
budgets, new policies, new interventio
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ISPOR Optimization Task Force Report
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Deterministic vs
stochastic
All the information is assumed to be certain
(e.g. costs of the pizza, how many it can feed,
how long it will take to cook)
Know that the information is uncertain
uncertainty in the costs and benefits of
interventions)
Linear vs Non-linear
Linear (i.e. each additional pizza costs the
same and feeds the same number of people)
Non-linear (e.g. Quality/outcomes ma
linear, also interactions between the in
etc)
795
796
797
23