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Optimisation

Definition

A 1st-Order approach treats a system as a concrete ontological entity (i.e. a collection of components and dependencies that exists “in the real world”) that can be captured by a formal model – at a single level of abstraction – that is amenable to description and analysis using research methods from the formal sciences and engineering (e.g. operations research, mathematical programming/optimisation and control theory). 1st-order approaches emphasise the regulation/control of a system’s output by feedback and error correction (as in industrial control) generally using analytically-tractable – often deterministic – and mature mathematical methods.

1st-Order approaches are suited to optimising components of larger healthcare systems where:

  • the boundaries of a given model are well-circumscribed -- for example, modelling a single outpatient department, ward or other finite healthcare resource
  • inputs, outputs and processes can be captured quantitatively, precisely (i.e. without significant unmeasurable uncertainty) -- for example, the inputs are numbers or rates of patients requiring or entering a service and similarly, outputs are the number of patients provided a service in a given period of time
  • sources of natural variation (or noise) admit well-understood formalisms -- for example, variability can be modelled as a random variable, with a convenient parametric distribution on the inputs to the model
  • and the interactions (and dependencies) between components can be described using well-understood mathematics -- for example, stochastic queueing models

## Examples Griffiths, Knight, and Komenda (2013) used queue theory to optimise bed utilisation in a critical care unit (CCU). In this example, a queueing model can be described by three components:

  • the arrival distribution: here, a Poisson model was used
  • the service distribution: modelled as two processes, one for elective and another for emergency admission with exponential distributions
  • and the number of service channels: in Griffiths et al, there are 24 beds in the CCU

The proposed parameterised queue model was fit to existing hospital data and the resulting model used to optimise the numbers of nurses required to service a critical care unit with varying demands in terms of elective and unplanned / emergency admissions.

While similar in focus, Kheiri et al (2019) used methods from scheduling (timetabling) and operations research to maximise surgical operating theatre allocations, minimising the number of cancelled surgeries while allowing for unplanned emergency surgical case demand. In this work

  • there are a finite number of surgical specialties and each has a required number of weekday sessions, in an operating theatre
  • there are a finite number of theatres and operating slots
  • a plan is one possible allocation of surgical specialties to theatre slots

In Kheiri et al's work, the space of all candidate plans is a graph and the vertices (nodes) of the graph represent a plan. Edges between vertices represent situations where plans conflict. A specialty is then represented as a partition of the graph. With this network representation, deriving the "master" schedule (for theatres and specialties over a week) can be further constrained by including bed number requirements for post-operative care. Then, algorithms for optimal graph colouring are used to find the best "master" schedule satisfying as many of the constraints as possible.

A helpful systematic review of pathway modelling can be found in Aspland et al (2021).

References and Further Reading