Concept Architecture
Concept
Theoretically, Linear Programming Health is a mathematical optimisation method used to determine the optimal allocation of limited healthcare resources subject to a set of linear constraints. It is founded on operations research and mathematical programming, where decision variables represent healthcare activities or resource allocations, the objective function represents a quantity to be maximised or minimised, and constraints represent resource, policy, or operational limitations. The concept exists to identify resource allocations that achieve the greatest health benefit or lowest cost while respecting finite healthcare resources.
Mathematically, linear programming is represented as a constrained optimisation problem in which a linear objective function is optimised subject to a system of linear equality and inequality constraints. The mathematical framework estimates the optimal values of decision variables that maximise health outcomes, minimise costs, or achieve another specified objective while satisfying all constraints. Solutions are typically obtained using the Simplex algorithm or interior-point optimisation methods.
In practice, linear programming is implemented by defining decision variables, specifying objective coefficients and constraint matrices, and solving the optimisation problem using dedicated optimisation software or spreadsheet solvers. In health economics it is applied to healthcare resource allocation, workforce planning, hospital capacity management, pharmaceutical production planning, vaccination programme design, and budget allocation where multiple competing activities must be optimally selected within fixed resource limits.
Purpose
Used to determine the optimal allocation of scarce healthcare resources by maximising health outcomes or minimising costs while satisfying operational, financial, and policy constraints.
Mathematical Formulae
Primary Formula
Maximise (or Minimise)
Z = ????� c?x?
subject to
????� a??x? � b?, i = 1, ?, m
x? � 0, j = 1, ?, n
where:
- Z = objective function
- x? = decision variables
- c? = objective coefficients
- a?? = resource consumption coefficients
- b? = available resources
Supporting Formulae
Dual problem:
Minimise W = ????? b?y?
subject to
????? a??y? � c?
Related Mathematical Methods
- Simplex algorithm
- Interior-point methods
- Duality theory
- Sensitivity analysis
- Shadow price analysis
Example
A regional health authority must allocate a �10 million prevention budget between two programmes.
Programme A:
- Cost per unit = �2 million
- Produces 450 QALYs
Programme B:
- Cost per unit = �1 million
- Produces 180 QALYs
Decision variables:
- x? = number of Programme A units
- x? = number of Programme B units
Objective:
Maximise 450x? + 180x?
Subject to
2x? + x? � 10
x?, x? � 0
The optimal solution is:
x? = 5, x? = 0
Total health gain:
450 ? 5 = 2,250 QALYs
Thus, the available budget is allocated entirely to Programme A because it produces the greatest health gain per constrained optimisation solution.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUMPRODUCT | =SUMPRODUCT(B2:B6,C2:C6) | Calculates the objective function value. |
| SUMPRODUCT | =SUMPRODUCT(B2:B6,D2:D6) | Calculates total resource use for a constraint. |
| Solver Add-in | Objective: Maximise SUMPRODUCT() subject to constraints | Solves healthcare resource allocation problems. |
| MIN | =MIN(range) | Identifies limiting resource values during model construction. |
VBA (Optional)
A VBA procedure can automate repeated optimisation analyses by updating resource constraints, running Excel Solver, and exporting optimal healthcare allocation results for multiple scenarios.
Sources
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- NICE. Health Technology Evaluation Manual.
- ISPOR Task Force Reports on resource allocation and optimisation methods.
- Dantzig GB. Linear Programming and Extensions. Princeton University Press.
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 linear programming?
An optimisation technique identifying the resource allocation that maximises or minimises a linear objective subject to linear constraints, such as a fixed budget.
Source: Dantzig 1963
Who developed linear programming?
Linear programming was developed in the 1940s, most notably by George Dantzig, who devised the simplex method for solving it. The technique finds the values of decision variables that make a linear objective as large or small as possible while satisfying a set of linear constraints. It arose from planning problems involving the best use of limited resources, and it remains a standard tool wherever an allocation must be optimised under constraints such as a budget. Dantzig (1963) set out the method.
Source: Dantzig 1963
What are the elements of a linear program?
A linear program has a linear objective function to be maximised or minimised, such as total benefit or cost; decision variables representing the choices, such as how much of each activity to undertake; and linear constraints limiting the choices, such as a budget, capacity, or requirements. All relationships are linear, meaning proportional and additive. The solution is the set of variable values that optimises the objective while satisfying every constraint, representing the best feasible allocation of resources under the stated limits.
Source: Dantzig 1963
How is a linear program solved?
A linear program is solved by finding the point in the feasible region, defined by the constraints, that optimises the objective. Because the objective and constraints are linear, the optimum lies at a vertex of the feasible region, and the simplex method, developed by Dantzig, moves efficiently between vertices to find it. Other methods also exist. These techniques reliably solve even large linear programs, which is part of why linear programming is so widely used for allocation problems.
Source: Dantzig 1963
How is linear programming used in health?
In health, linear programming is used for resource allocation and planning problems where a linear objective is optimised under linear constraints, such as allocating a budget across programmes to maximise health, planning capacity, scheduling, or the diet and blending problems of nutrition. It finds the allocation that best meets the objective within the constraints, such as budget and capacity limits. Where the problem can be represented linearly and variables can be continuous, linear programming provides an efficient and reliable optimisation method.
Source: Dantzig 1963
What are the limitations of linear programming in health?
Linear programming assumes the objective and constraints are linear, which may not hold where effects diminish, interact, or involve thresholds, and it allows continuous variables, so it cannot directly handle indivisible choices, which need integer programming. Its results depend on the accuracy of the objective and constraints, which in health may be uncertain or involve equity as well as efficiency. Where these assumptions fail, linear programming may give solutions that do not fit the real problem, so its applicability is judged case by case.
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-050
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