Concept Architecture
Concept
Theoretically, Multi-Objective Optimisation is an optimisation framework used to identify solutions that simultaneously consider two or more competing objective functions. Unlike single-objective optimisation, no single solution is generally optimal for all objectives. Instead, the method seeks efficient or Pareto-optimal solutions in which improvement in one objective can only occur at the expense of another. In health economics, it is used where decision-makers must balance competing goals such as maximising health outcomes, minimising costs, improving equity, and increasing service accessibility.
Mathematically, multi-objective optimisation is formulated as the simultaneous optimisation of multiple objective functions subject to a common set of constraints. The mathematical framework identifies the Pareto frontier, comprising all non-dominated solutions. Various solution methods transform the multiple objectives into a single optimisation problem, including weighted-sum methods, �-constraint methods, and goal programming, allowing decision-makers to explore trade-offs among competing objectives.
In practice, multi-objective optimisation is implemented by defining decision variables, specifying multiple objective functions and operational constraints, and solving the resulting optimisation problem using specialised optimisation software. In health economics it is applied to healthcare resource allocation, service planning, screening programme design, workforce planning, vaccine allocation, and health system optimisation where multiple policy objectives must be considered simultaneously.
Purpose
Used to optimise healthcare decisions involving multiple competing objectives by identifying Pareto-efficient solutions and quantifying trade-offs between health, cost, equity, capacity, and other decision criteria.
Mathematical Formulae
Primary Formula
Optimise F(x) = (f?(x), f?(x), ?, f?(x))
subject to
g?(x) � 0, i = 1, ?, m
h?(x) = 0, j = 1, ?, p
where:
- f?(x) = objective functions
- g?(x) = inequality constraints
- h?(x) = equality constraints
- x = decision variables
Supporting Formulae
Weighted-sum formulation:
max ????? w?f?(x)
subject to
????? w? = 1, w? � 0
�-constraint formulation:
max f?(x)
subject to
f?(x) � �?, r = 2, ?, k
Related Mathematical Methods
- Pareto optimisation
- Pareto frontier analysis
- Weighted-sum optimisation
- �-constraint method
- Goal programming
- Evolutionary multi-objective optimisation
- Linear programming
- Integer programming
Example
A health authority must allocate funding between preventive and acute care programmes.
Objectives:
- Maximise QALYs gained.
- Minimise programme cost.
Decision variable:
x = proportion of the budget allocated to prevention.
Suppose two feasible solutions are obtained:
| Solution | QALYs | Cost (� million) |
|---|---|---|
| A | 8,500 | 60 |
| B | 8,900 | 66 |
Neither solution dominates the other because Solution B provides greater health benefit but at higher cost. Both belong to the Pareto frontier, allowing policymakers to select the preferred balance between health gain and expenditure.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| SUMPRODUCT | =SUMPRODUCT(B2:B20,C2:C20) | Calculates weighted objective functions. |
| Solver Add-in | Maximise weighted objective subject to constraints | Solves multi-objective optimisation problems. |
| IF | =IF(B2>=Target,1,0) | Applies objective or constraint thresholds. |
| INDEX/MATCH | =INDEX(ResultRange,MATCH(MAX(ObjectiveRange),ObjectiveRange,0)) | Retrieves optimal or Pareto-efficient solutions. |
VBA (Optional)
A VBA procedure can automate repeated optimisation runs across alternative objective weights to generate an approximate Pareto frontier for healthcare decision analysis.
Sources
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
- NICE. Health Technology Evaluation Manual.
- Ehrgott M. Multicriteria Optimization. Springer.
- Miettinen K. Nonlinear Multiobjective Optimization. Springer.
- ISPOR Good Practice Reports on health economic modelling.
Related Concepts (3)
Library
Publications
1
Conceptualizing a Model: A Report of the ISPOR-SMDM Modeling Good Research Practices Task Force-2 — Roberts, Russell, Paltiel, Chambers, McEwan & Krahn, Task Force Report 2 ed., 2012 (Value in Health / Medical Decision Making)
Best-practice guidance on model conceptualisation — defining the decision problem, scoping, and choosing an appropriate model structure before implementation.
Journal ArticleView source →
Frequently Asked Questions (6)
What is multi-objective optimisation?
An optimisation approach used when a decision involves competing objectives, such as maximising benefit while minimising cost, producing a set of trade-off solutions.
Source: Deb 2001
How does multi-objective optimisation handle competing goals?
When two goals conflict, such as raising benefit while holding down cost, no single solution is best on both at once, so multi-objective optimisation does not force them into one number. Instead it identifies the set of solutions for which neither goal can be improved without sacrificing the other, leaving the final trade-off to the decision-maker. This preserves the choice between the objectives rather than hiding it inside a fixed weighting. The output is a range of efficient options, not one answer. Deb (2001) sets out the approach.
Source: Deb 2001
Why are multiple objectives difficult to optimise together?
Multiple objectives are difficult to optimise together when they conflict, so that no single solution is best on all of them: improving one, such as reducing cost, worsens another, such as benefit. There is then no unique optimum, only a set of solutions representing different trade-offs. Combining the objectives into one measure requires weighting them, which involves value judgements, so multi-objective optimisation instead identifies the efficient trade-offs, leaving the choice among them to the decision maker rather than imposing a single weighting.
Source: Deb 2001
What is a Pareto front in multi-objective optimisation?
A Pareto front is the set of trade-off solutions in multi-objective optimisation for which no objective can be improved without worsening another, so each is efficient in the sense that it is not dominated by any other. The solutions on the front represent the best achievable balances among the competing objectives, differing in how they trade one against another. Presenting the Pareto front shows the decision maker the range of efficient options, from which one must be chosen according to their priorities.
Source: Deb 2001
How does multi-objective optimisation present results?
Multi-objective optimisation presents its results as a set of efficient trade-off solutions, the Pareto front, rather than a single answer, showing how the objectives trade against one another. The decision maker can then see the range of efficient options, for instance the achievable combinations of cost and benefit, and choose according to their priorities. This makes the trade-offs explicit and leaves the value judgement about how to weigh the objectives to the decision maker, rather than embedding it in a single combined criterion.
Source: Deb 2001
How does multi-objective optimisation apply to health?
In health, multi-objective optimisation applies to decisions involving competing goals, such as maximising health benefit while minimising cost, or balancing efficiency against equity, where the objectives conflict and cannot be reduced to one measure without value judgements. It can present the trade-offs between, for example, total health gain and its distribution, or between benefit and budget, showing the efficient options. This helps decision makers see the compromises available and choose according to their values, rather than relying on a single, implicitly weighted criterion.
Source: Deb 2001
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Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 2 Oct 2025
Content version: 1.0.0
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