Concept Architecture
Concept
Theoretically, Particle Swarm Optimisation is a population-based stochastic optimisation algorithm inspired by the collective behaviour of biological swarms, such as flocks of birds and schools of fish. It belongs to the class of metaheuristic optimisation methods and searches for optimal solutions by allowing a population of candidate solutions, known as particles, to move through the solution space according to their own experience and that of the swarm. The method exists to solve complex optimisation problems in which objective functions are nonlinear, non-convex, discontinuous, or lack analytical derivatives, making conventional optimisation methods difficult to apply.
Mathematically, particle swarm optimisation represents each candidate solution as a particle characterised by a position vector and a velocity vector. During each iteration, particles update their velocities according to their own best previous position and the best position identified by the swarm, before moving to a new location in the search space. The optimisation process continues until convergence criteria are satisfied or a predefined number of iterations has been completed.
In practice, particle swarm optimisation is implemented by defining an objective function, selecting swarm size and algorithm parameters, initialising particle positions randomly, and iteratively updating particle velocities and positions. In health economics it is applied to calibrating decision models, estimating model parameters, fitting disease progression models, optimising resource allocation problems, and solving complex nonlinear optimisation problems where derivative-based methods perform poorly.
Purpose
Used to identify optimal model parameters or resource allocation decisions by iteratively searching complex solution spaces using a population of interacting candidate solutions.
Mathematical Formulae
Primary Formula
Velocity update:
v???? = ?v?? + c?r?(p? ? x??) + c?r?(g ? x??)
Position update:
x???? = x?? + v????
where:
- x? = particle position
- v? = particle velocity
- p? = personal best position
- g = global best position
- ? = inertia weight
- c?, c? = cognitive and social learning coefficients
- r?, r? ? U(0,1) = random numbers
Supporting Formulae
Objective function:
min? f(x)
or
max? f(x)
depending on the optimisation problem.
Related Mathematical Methods
- Metaheuristic optimisation
- Stochastic optimisation
- Swarm intelligence
- Genetic algorithms
- Simulated annealing
- Nelder-Mead method
- Numerical optimisation
- Model calibration
Example
A health economist calibrates a Markov model by minimising the sum of squared differences between observed and predicted survival probabilities.
The objective function is
min ?????� (S????????? ? S??????)�.
A swarm of 40 particles explores the parameter space of transition probabilities. After 300 iterations, the objective function decreases from 15.82 to 0.37, indicating that the calibrated model closely reproduces the observed survival data.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| RAND | =RAND() | Generates random numbers used in particle movement. |
| SUMXMY2 | =SUMXMY2(B2:B13,C2:C13) | Calculates the optimisation objective as the sum of squared errors. |
| SUMPRODUCT | =SUMPRODUCT(B2:B13,C2:C13) | Computes weighted objective functions during calibration. |
| Solver Add-in | Optimises simplified objective functions for comparison | Compares optimisation results with deterministic methods. |
VBA (Optional)
A VBA procedure can automate particle swarm optimisation by updating particle positions and velocities iteratively until convergence criteria are satisfied.
Sources
- Kennedy J, Eberhart R. Particle Swarm Optimization. Proceedings of the IEEE International Conference on Neural Networks. 1995.
- Clerc M. Particle Swarm Optimization. ISTE.
- 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.
- ISPOR Good Practice Reports on model calibration and decision-analytic modelling.
Related Concepts (2)
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 particle swarm optimisation?
A population-based optimisation technique inspired by flocking behaviour, in which candidate solutions adjust position based on their own and the group's best results.
Source: Kennedy & Eberhart 1995
What inspired particle swarm optimisation?
The technique takes its idea from the coordinated movement of flocks and shoals, where each member adjusts its course using its own experience and what the group has found. It keeps a set of candidate solutions, each moving through the space of possibilities and steering toward both the best point it has personally found and the best any of them has found. Over many steps this shared search converges on good solutions. The flocking analogy gives the method its name and its rules. Kennedy and Eberhart (1995) introduced it.
Source: Kennedy & Eberhart 1995
How does particle swarm optimisation work?
Particle swarm optimisation works by initialising a population of particles at random positions and velocities in the search space, evaluating the objective at each, and then updating each particle's velocity and position at each step. The velocity is adjusted toward the particle's own best position found so far and toward the best position found by the whole swarm, so particles are drawn to promising regions while retaining momentum. Over iterations, the swarm converges toward good solutions, guided by both individual and shared experience.
Source: Kennedy & Eberhart 1995
What guides the particles in particle swarm optimisation?
Each particle in particle swarm optimisation is guided by two influences: its own best position found so far, its personal best, and the best position found by the whole swarm, the global best. At each step, the particle's movement is adjusted toward both, combined with its current velocity, so it explores while being drawn toward promising regions. This balance of individual memory and social information lets the swarm search broadly yet converge on good solutions, mimicking how a flock coordinates its movement.
Source: Kennedy & Eberhart 1995
When is particle swarm optimisation useful?
Particle swarm optimisation is useful for optimisation problems where the objective is complex, non-linear, or has many local optima, and where derivatives are unavailable, since it uses only function values and searches broadly with a population. It suits problems that gradient-based methods struggle with, offering a way to explore difficult landscapes and find good solutions. It is relatively simple to implement and has few parameters, making it a convenient choice among population-based, derivative-free global optimisation methods for hard problems.
Source: Kennedy & Eberhart 1995
What are the limitations of particle swarm optimisation?
Particle swarm optimisation does not guarantee finding the global optimum, only good solutions, and it can converge prematurely on a suboptimal region if the swarm loses diversity. Its performance depends on parameter settings, such as the weights on individual and group influence, which may need tuning, and it can be computationally expensive, requiring many function evaluations. Where a more direct method applies, it is often preferred, so particle swarm optimisation is used for problems that resist gradient-based or exact approaches.
Source: Kennedy & Eberhart 1995
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Verified by Dr Darrin Baines
British health economist
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Verification date: 2 Oct 2025
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