Thanks in advance to anyone can help!
Background
Say I have an initial cost of e.g. £200k which is to be distributed across four business units. Using a certain allocation rule (unimportant for problem at hand) the distribution is:
Business Unit A: £50k. Business Unit B: £100k. Business Unit C: £30k. Business Unit D: £20k.
Challenge
Now I have a £100k reduction in this cost that is to be distributed across these four business units.
It is to be allocated based on how each business unit collects invoices. For example, in a given time frame Business Unit A collects a hypothetical £800k out of £1,000k invoiced. For completeness a sample data may be represented by:
Business Unit A: £800k / £1,000k. Business Unit B: £500k / £1,000k. Business Unit C: £200k / £250k. Business Unit D: £50k / £50k.
I want to distribute the cost reduction of £100k fairly across these four business units taking into account both the amount of invoices collected and the amount invoiced i.e. A should not receive a large advantage simply because they have collected the most invoices because this would unfairly penalise Business Unit D who has collected 100% of their invoices. Likewise Business Unit D should not receive a large advantage simply because they collected 100% of their invoices - this ignores the fact they have collected the lowest absolute total amount of invoices.
So my challenge is finding a mathematical solution which will fairly distribute the cost reduction between the business units while utilising both these factors.
Thanks a lot, and let me know if you would like any clarity!