General Practitioner Funding Formula
Orville D’Silva
David Worthington
Lancaster University
James Crosbie
Department of Health
OR54 Conference, Edinburgh, September 2012
Outline
 Background and aims of this study
 Approach
 Performance of algorithms and results
 Heuristics learnings
OR54 Conference, Edinburgh, 2012
Background
 About half of all general medical practices (4822) are financially
supported by the Department of Health, through the Carr-Hill formula.
 The formula takes into account 6 factors that reflect health need (and
therefore costs) such as the number and demographics of patients in
each practice and the location of the practice to determine the true
workload of the practice.
 A review of the Carr-Hill formula was published in 2007 by the Formula
Review Group (FRG) which concluded that the FRG formula should
replace the Carr-Hill formula.
 The Department of Health want to investigate the impact of using the
revised formula.
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Background: The Carr-Hill and FRG formulae
 The Carr-Hill formula consists of
six indices:
 Age-sex
 Nursing and residential
homes
 List turnover
 Additional needs
 Staff market forces factor
 Rurality
 The FRG formula also consists of
six indices:
 Workload
 Consultation length and
home visits
 Staff market forces factor
 Cost of recruitment and
retention (CORR)
 Cost of unavoidable
smallness (CUS)
 Rurality
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Background: Aims of this study
 Produce an algorithm in Microsoft Excel that determines the optimal
order for replacing Carr-Hill formula components with those of the FRG
formula during a transition period.
 It was important for the algorithm to be in Microsoft Excel as the
Department of Health may need to re-run the algorithm in future years
with updated data, etc.
 Assessing the results generated from the algorithm and investigating
the impact of some of the issues that may arise during negotiations with
the BMA.
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Impact of implementing the FRG global sum formula
 The change in payments to practices will range from -19% to 84%.
 73% of practices will face between a 5% decrease and a 2.5% increase in
payments and about 13% of practices (over 600) will face over a 5%
decrease in payments which could be destabilising.
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The ideal transition path
 It was agreed that the ideal
transition path would be for
practices to have a smooth
incremental change in each of the
phases of the transition period
(agreed to be 5 years).
 Thus, the optimal transition path is the one that is as close to this
ideal path as possible and was used as the objective function of the
optimum searching algorithms.
 Measured using average absolute percentage deviation of payments to
practices from the straight-line transition path.
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Secondary objective
 Adjustments to practice payments were also considered in the following
scenarios. The sum of these adjustments would represent the overall cost
of implementing a solution.
a) If the payment falls below the minimum of the Carr-Hill and FRG during
the transition period.
b) If there is more than a sixth drop in payments from one year to the next.
(a)
(b)
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Approach
 Since there were numerous transition-path possibilities and little was
known on the properties of good solutions, it seemed sensible to use a
heuristic approach.
 A review of the literature showed that the simulated annealing and
genetic algorithm heuristics were suited to problems where properties
of ‘good’ solutions were unknown and it was possible to programme
these into Excel.
 The first and steepest descent algorithms were used as benchmarks.
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Guidance from literature
 Guidance from the literature was used to develop the framework for the
algorithms and to set initial values for the parameters.
 Simulated annealing:
 The initial value of the temperature was determined using guidance from
Aarts and Lentra (1997), which was to set it the maximum difference of
scores within a neighbourhood (obtained through a set of sampled
neighbourhoods).
 Genetic algorithms:
 An elitist strategy was adopted as suggested by De Jong (1975). This
involves carrying forward the best found solution in each generation to the
next to ensure that the best found solution is not lost through crossover.
 The size of the population was set to 30 was based on Reeves (1995)
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Developing the algorithms was an iterative process
 After developing initial versions of the algorithms using the guidance in
the literature, the results from these were analysed to identify
opportunities for further enhancement.
 For example, the graph
below shows that for the
simulated annealing
algorithm, the first phase
was more of a random
search and so could be
omitted to increase
efficiency.
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The performance of algorithms
 Four algorithms were tried: First Descent, Steepest Descent, Simulated
Annealing and Genetic Algorithms. All four algorithms are capable of
finding the ‘minimum’ solution.
 Boxplots show that more
than 50% of the solutions
from the Simulated
Annealing algorithm lie
below 120 which is superior
to the other algorithms. It’s
run time was also superior
to the Genetic algorithm.
Thus, it was decided to use
this algorithm within the
final Excel model.
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Assessing results: analysis of five ‘good’ orders
 The 5 best solutions with solution scores less than 120 and
implementation costs less than £1 million were analysed further.
 Solution 1 has the lowest solution score but Solution 4 has the lowest
found cost of implementation of £160,153.
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Assessing results: Analysis of orders by number of GPs
 Solutions 4 and 5 offer the lowest average cost per practice for each of
the categories of GPs. A similar trend was seen when analysing these
solutions by list size. Thus, solution 4 is the most desirable.
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Common characteristics of solutions with the lowest
scores
 The algorithm was run 175 times and about 50 unique solutions were
found. Of the unique solutions that were returned in the 115-120 range,
common characteristics were:
 The Carr-Hill staff MFF adjustment was phased-out in the same phase as the
FRG staff MFF adjustment was phased-in.
 The difference in these solutions was the phasing out of the Carr-Hill
nursing & residential home patients adjustment and the phasing in of the
FRG CUS adjustment.
 All other components were phased in/out in phase 3.
 Solutions with these characteristics have an average cost of £670,565
and a maximum cost of £1,136,240, which is the worst result given these
characteristics.
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Heuristics learnings
 Heuristics are suited to projects with a relatively short time-scale and can
be programmed in Excel.
 There are good reviews on the implementation of the algorithms in the
literature and these can be used to understand how to tailor the
algorithms to the problem.
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Q&A
OR54 Conference, Edinburgh, September 2012