Concept Architecture
Concept
Theoretically, Integer Programming Health is a mathematical optimisation method in which some or all decision variables are constrained to take integer values. The method is grounded in operations research and optimisation theory and is used to solve discrete decision problems where fractional solutions are infeasible, such as selecting healthcare facilities, allocating finite resources, scheduling services or determining intervention portfolios. In health economics, Integer Programming supports efficient resource allocation under financial, capacity and policy constraints.
Mathematically, Integer Programming seeks to maximise or minimise an objective function subject to a set of linear equality and inequality constraints while requiring designated decision variables to be integers. Binary Integer Programming is a special case in which decision variables take values of 0 or 1, representing decisions such as whether to fund an intervention or construct a facility. Solutions are typically obtained using branch-and-bound, branch-and-cut or related optimisation algorithms.
In practice, Integer Programming models are formulated by defining decision variables, objective functions and operational constraints from healthcare data. The models are implemented using optimisation software to identify optimal allocations of budgets, workforce, facilities or technologies. Health economists apply Integer Programming in service planning, capital investment, screening programme design and health system resource allocation.
Purpose
Used to identify optimal discrete resource allocation and planning decisions in healthcare while satisfying budgetary, operational and policy constraints.
Mathematical Formulae
Primary Formula
General Integer Programming problem:
Minimise or Maximise c?x
subject to Ax � b
x? ? ?, i = 1, ?, n
where:
- x = decision variable vector
- c = objective function coefficients
- A = constraint matrix
- b = constraint vector
- ? = set of integers
Supporting Formulae
Binary Integer Programming:
x? ? {0, 1}
Budget constraint:
????� c?x? � B
where:
- c? = cost of intervention i
- B = available budget
Related Mathematical Methods
- Mixed Integer Programming
- Linear Programming
- Binary optimisation
- Branch-and-bound
- Branch-and-cut
- Constraint optimisation
- Operations research
Example
A health authority has a budget of �10 million to fund new screening programmes.
Five candidate programmes are available, each requiring a fixed investment and producing a projected health benefit. Decision variables are binary, where x? = 1 if a programme is funded and x? = 0 otherwise.
The optimisation model maximises total expected QALYs:
max ? QALY?x?
subject to:
? Cost?x? � �10,000,000
The Integer Programming solution selects the combination of programmes that delivers the greatest total health gain without exceeding the available budget.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| Solver | Set integer constraints on decision variables | Solve healthcare resource allocation problems. |
| SUMPRODUCT | =SUMPRODUCT(B2:B10,C2:C10) | Calculate total programme costs or health benefits. |
| SUM | =SUM(B2:B10) | Aggregate resource utilisation or selected interventions. |
| IF | =IF(B2=1,""Selected"",""Not Selected"") | Display whether an intervention is included in the optimal solution. |
VBA (Optional)
Automate repeated Integer Programming analyses across alternative budgets, policy constraints and resource allocation scenarios using Excel Solver.
Sources
- Winston WL. Operations Research: Applications and Algorithms.
- Hillier FS, Lieberman GJ. Introduction to Operations Research.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed.
- Crown W, Buyukkaramikli N, Sir MY, et al. Application of Constrained Optimisation Methods in Health Services Research and Health Care Decision Making. Value in Health. 2017.
Related Concepts (3)
Library
Publications
1
A Taxonomy of Model Structures for Economic Evaluation of Health Technologies — Brennan, Chick & Davies, Vol. 15, No. 12 ed., 2006 (Health Economics)
An influential paper classifying decision-analytic model structures along axes of expected value vs randomness, entity heterogeneity, and Markovian vs non-Markovian structure — providing a framework for choosing between decision trees, Markov cohort models, microsimulation, discrete event simulation and system dynamics.
Journal ArticleView source →
Frequently Asked Questions (6)
What is integer programming?
An optimisation technique for resource allocation problems where decision variables must take whole-number values, such as the number of clinics to open.
Source: Dantzig 1963
What is an example of an integer programming problem in health?
Integer programming suits allocation decisions where the answer must be a whole number because the thing being chosen cannot be split. Deciding how many clinics to open, how many ambulances to buy, or how many beds to add are examples, since half a clinic or a fractional ambulance is meaningless. The technique finds the combination of whole-number choices that best meets an objective within the constraints, such as maximising coverage within a budget. Rais and Viana (2011) review such applications in health.
Source: Rais & Viana 2011
How does integer programming differ from linear programming?
Integer programming differs from linear programming in requiring the decision variables, or some of them, to take integer values, whereas linear programming allows them to be continuous. This restriction suits decisions about indivisible quantities, but it makes the problem much harder, since the efficient methods that solve linear programs do not directly apply and the integer requirement creates a combinatorial problem. Integer programs are therefore solved by specialised methods and can be computationally demanding for large problems, unlike the relatively tractable linear program.
Source: Dantzig 1963
In integer programming, why are whole-number values required in some problems?
Whole-number values are required when the quantities being decided are indivisible, so that fractional solutions are meaningless: one cannot open half a clinic, hire a fraction of a nurse, or build part of a facility. In such cases the decision variables must be integers to represent feasible choices. Allowing fractional values, as linear programming would, could give a solution that cannot be implemented, so the integer restriction ensures the optimisation respects the discrete nature of the decision.
Source: Dantzig 1963
How is integer programming used in health?
In health, integer programming is used for resource allocation decisions involving discrete quantities, such as how many clinics, beds, or facilities to provide, where to locate services, or how to schedule staff and resources, subject to constraints like budgets and capacity. Because these decisions concern indivisible units, integer programming finds the combination that best meets the objective while respecting the constraints. It supports planning problems where the discrete nature of the choices must be captured for the solution to be feasible.
Source: Dantzig 1963
What are the limitations of integer programming?
Integer programming is computationally hard: requiring integer values turns the problem into a combinatorial one, so large problems can be very slow or impractical to solve exactly, and solution times can grow steeply with size. Formulating the problem correctly is also demanding, and the results depend on the accuracy of the objective and constraints, which may be uncertain. For large or complex problems, exact solutions may be infeasible, so approximate or heuristic methods are sometimes used instead.
Source: Dantzig 1963
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 1 Oct 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-EM-DM-046
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