General Practitioner Funding Formula Orville D’Silva David Worthington Lancaster University

1. General Practitioner Funding Formula Orville D’Silva David Worthington Lancaster University James Crosbie Department of Health

2. Outline • Background and aims of this study • Approach • Performance of algorithms and results • Heuristics learnings

3. 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.

4. 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

5. 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.

6. 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.

7. 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.

8. 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. • If the payment falls below the minimum of the Carr-Hill and FRG during the transition period. • If there is more than a sixth drop in payments from one year to the next. (a) (b)

9. 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.

10. 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)

11. 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.

12. 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.

13. 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.

14. 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.

15. 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.

16. 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.

17. Q&A