Splitting costs of a costcenter

Question

This is probably stupid-simple for you guys, but I'm not mathy and I'm having difficulty figuring this out.

In a fictional company C there are two costcenters and three employees.

Costcenters:

• Facilities has one employee (Let's call him Frank)
• Culture has one employee (Let's call her Cassie)

A third employee exists that is in neither cost center. Let's call her Naomi.

Costs

• To run Facilities for a month costs 10000
• To run Culture for a month costs 1000

Looking at the cost of the company we'd see something like:

• Facilities: -10000
• Culture: -1000
• (other): 0

However if we were to tax out the cost of facilities to all employees that are not in facilities we'd see:

• Facilities: 0
• Culture: -6000
• (other): -5000

And similarly if I instead taxed out culture to all employees that are not in culture:

• Facilities: -10500
• Culture: 0
• (other): -500

But I can't figure out how to apply both taxes such that:

• Everyone pays a tax for all taxes that they are not a part of (facilities pays culture, culture pays facilities and everyone else pays both).
• All cost centers end up with a total of zero (so people who pay facilities also need to take into account facilities new payment to culture).

My question is: can this be done? and how can this be accomplished?

EDIT:

Allow me to rephrase my question.

First to clarify how the taxing works the base cost of Facilities in my examples is 10000. I then take the whole population who is not in Facilities and they share in that cost (in my example there are only two other entities to share that cost, so they each pay 5000 to Facilities). The end result in this case is that Facilities nets out to zero, Culture comes out to -6000 (including it's existing costs) and my one other employee comes out to -5000.

If we were to imagine this as a transfers table as such:

+--------+--------+------------+
|  From  | Amount |     To     |
+--------+--------+------------+
| Frank  |  10000 | (external) |
| Cassie |   1000 | (external) |
+--------+--------+------------+

I would like to come up with additional transfer entries that will zero out these employees costs (to their cost-centers).

For example to zero out Facilities costs (which all come from Frank), I would modify the table as such:

+--------+--------+------------+
|  From  | Amount |     To     |
+--------+--------+------------+
| Frank  |  10000 | (external) |
| Cassie |   1000 | (external) |
| Cassie |   5000 | Facilities |
| Naomi  |   5000 | Facilities |
+--------+--------+------------+

Alternatively to zero out Culture:

+--------+--------+------------+
|  From  | Amount |     To     |
+--------+--------+------------+
| Frank  |  10000 | (external) |
| Cassie |   1000 | (external) |
| Frank  |    500 | Culture    |
| Naomi  |    500 | Culture    |
+--------+--------+------------+

I'd like to come up with a series of transfers based on the first table above that will zero out BOTH Facilities AND Culture. I imagine it will look something like this:

+--------+--------+------------+
|  From  | Amount |     To     |
+--------+--------+------------+
| Frank  |  10000 | (external) |
| Cassie |   1000 | (external) |
| Cassie |        | Facilities |
| Frank  |        | Culture    |
| Naomi  |        | Facilities |
| Naomi  |        | Culture    |
+--------+--------+------------+

My only problem is I have no idea what values make sense here to distribute the cost fairly, and to have the new expenditures added somehow net out to zero. (Frank paying Culture for instance increases the cost of Facilities, which in turn needs to be paid back by Culture and Naomi).

I'm sorry that my question is so convoluted. Please ask questions, and I will clarify as best I can.

EDIT:

In my examples above Frank is an employee of facilities. When he transfers money externally (out of the company) that is what I am calling a cost. Cassie is an employee of culture. Naomi is an employee of neither and thus benefits from both and foots both bills. Frank only needs to pay the Culture bill and Cassie only needs to pay facilities.

Answer 1

Ok so I finally figured it out I think.

+--------+--------+------------+
|  From  | Amount |     To     |
+--------+--------+------------+
| Frank  |  10000 | (external) |
| Cassie |   1000 | (external) |
| Cassie |   5000 | Facilities |
| Frank  |    500 | Culture    |
| Naomi  |   5500 | Facilities |
| Naomi  |   5500 | Culture    |
+--------+--------+------------+

So if I did my math right, all I have to do is take Facility's expense (10000) split it by the population that's paying it (Naomi and Cassie). Then do the same for Culture's expense. After that I take the new additional costs of both and add that back into Naomi's payments as she needs to cover both.

I apologize for the convoluted problem and explanation. This has been swimming around in my head for a while and I have been trying many different solutions so I'm sure I didn't come across very clearly, but just putting up here has helped me think through the issue more thoroughly, so thanks for the assist in that regard.