Bonus scheme calculation

Question

Problem Background:

We are a project based company. Employees work on project basis and each project duration can be between 3 days to 30 days. At the end of each project, the employee will get rated from a score of 1 (worst) to 3 (best) across 5 different categories, let's call it the project rating.

We have a year end bonus scheme and the project component weightage is 30%. Each employee can get up to 30% and it's not shared between all employees.

Consider the following example:

Employee A
| Project Name | Project rating (across 5 categories) |
| ------------ | ------------------------------------ |
| One  | 3,2,2,2,2 | 
| Two  | 2,2,2,2,2 |
| Three| 2,2,2,2,2 |

Employee B
| Project Name | Project rating (across 5 categories) |
| ------------ | ------------------------------------ |
| One  | 3,2,2,2,2 | 
| Two  | 2,2,2,2,2 |
| Three| 2,2,1,1,2 |
| Four | 1,1,2,3,2 |

Questions:

1. Is there a fair good way translate the project ratings into the weightage given the following limitations:
   • The number of projects completed between different employees differs (some did more while some did lesser)
   • Not all projects have the same level of difficultly, this data wasn't captured in the project rating as well. Employees dont get to choose the project either.
   • There must be a way to get full 30% weightage without compromising the fairness criteria, if possible

Answer 1

With the data provided there is no way to distinguish work on hard or long projects from work on easy or short ones. Perhaps the employees who did fewer projects did so because those were long or hard.

You ask for a "fair" bonus calculation, but "fair" is not a mathematically well defined term. I think that given the data the "fair" assumption is that each of the five ratings on each employee's projects contributes the same weight to that employee's bonus, independent of what the other employees have done.

The easiest calculation is to find the average rating for each employee. For the first one in the question that would be $$ \frac{3 + 14 \times 2}{15} = 2.07. $$ Then you could award that employee a bonus of $2.07 \times 10\% = 20.7\%$.

With this algorithm the minimum bonus is $10\%$ and the maximum is $30\%$.

You could do a little more work to count ratings of $3$ as more than three times as good as ratings of $1$ or to scale the result so that the bonus range was from $0\%$ to $30\%$. I suspect that any scheme would look "unfair" to someone. The simplest one would be the easiest to defend.