This is an abstracted version of the problem I'm facing and I have to tell you first, my question might not be precise and or even correct, so I hope you understand and in that case can improve the point. I'm striking a problem field that I'm not very familiar with. But case is this.

There is an organization that has lets say 7 departments. Departments are getting a share of the money, say 1000000. There are several variables that affects the final share of the sum 1M per department. Variables are divided to sections and meters:

Section (S) 1, 2, 3 and 4 which all has meters (m), lets say:

S1: m1, m2
S2: m1, m2, m3
S3: m1, m2
S4: m1, m2, m3, m4

Total score is calculated by summing up all meters on all sections. And total score is used to calculate portion of the share. There is some other math involved at this point, but I'm simplifying the problem now for this question.

Departments also have some variables, like size by employers, their salary, activities performed and such, which defines initial sum they get. But total of all departments is 1M.

Now after certain time period, say 4 months, variables are checked and new portion of the share is calculated based on sections and their and meters.

This raises a problem, which might not be apparent on simple example given above, but what is the actual question and problem of mine. Departments will get certain percentage of the share, so bigger departments will get bigger and smaller will get smaller. It looks like this is not rightful and equal in this system, because more the bigger departments get, slice from the share is bigger and it affects increasingly on smaller departments. At the moment share is somewhat percentage (linear) based, but I was thinking if share could be functionally (logarithm) defined, or something else. Does anyone has insight to this kind of problem?

  • $\begingroup$ do I understand you correctly that you are basically facing an optimization problem ? You would like to distribute a fixed amount of money amongt serveral departments and under certain restriction. (e.g. department one should get at least $c_1$ amount but not more than department two etc. Thus this restrictions could become arbitrarily complex) - correct ? $\endgroup$ – Probilitator Feb 22 '14 at 9:52
  • $\begingroup$ Yes @Probilitator it's optimization problem, but there is no minimum sum every department should get, to a the moment. Maybe there should be... But I'm after an algorithm which prevents smaller departments losing too much of the total share when bigger departments gets their increased percentual share. $\endgroup$ – MarkokraM Feb 23 '14 at 9:22
  • $\begingroup$ do you know how genetic algorithms work - this is precisly the type of task they excell at. You would first however need to set the restrictions. Do you have trouble formulating those restrictions mathematically ? So are you looking for a mathematical formulation of the "smaller departments losing too much of the total share when bigger departments gets their increased percentual share" ? $\endgroup$ – Probilitator Feb 23 '14 at 10:21

Your solution is an optimization, but your symptom is game theory. Managers will maximize their share through gaming by placing emphasis on growing the variables that you use for your allocation.

For example if you give department A 100,000 because there are 10 employees out of 100 total, the manager will hire another employee at a rate below the marginal increase that they expect to receive at the expense of a cost that you are not including as a variable. If they are the only deparment that increases in employee number, they will receive a 11/100 next quarter leaving 89/100 for the other 6 to share.

You need to define an allocation basis focusing on outputs rather than inputs. This way a more productive department will receive a larger secondary share and therefore your more productive departments will grow and the less productive departments will shrink. That is fundamental to optimizing ROI for your 1,000,000. If these are all sales or production departments then you can use total sales or contribution margin of products. If you use a total products produced method you need to be sure to monitor inventory fluctuations and penalize for over production to inventory that doesn't sell.

If you have some internal service departments then you need to determine a fixed portion to feed to the service departments and share out the remainder to the sales/production departments.


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