Formula 1 :
T(N=N) = ( S(N+1) -1 ) / ( S - 1)Every manager manages S employees (S = Span of Control)
Consider a company with one managementlevel: N=1
T(N=1) = 1 manager + S employees = 1 + S
Now go on to a company with managementlevel: N=2
The first manager (on top of the organisation) manages S sub-managers.
Each of sub-managers manages also S employees.
T(N=2) = 1 manager + S sub-managers + S x S = 1 + S + S2
Another way of seeing this is:
You have S times a company T(N=1) plus one manager to manage the S managers of the T(N=1)-companies.
T(N=2) = 1 + S x T(N=1) = 1 + S x (1+S) = 1 + S + S2
Now every new managementlevel introduces one new boss-manager and S times the amount of people of a company with one managementlevel less.
So: T(N=M) = 1 + S x T(N=[M-1]). This leads to:
T(N=N) = 1 + S + S2+ S3+ S4+ …. +SN
So:
|
S x T(N=N |
= |
S x ( 1 + S + S2+ S3+ S4+ …. +SN ) |
|
|
S x T(N=N) |
= |
S + S2+ S3+ S4+ …. +SN +S(N+1) |
+ |
|
T(N=N) |
= |
1 + S + S2+ S3+ S4+ …. +SN |
- |
|
-------------- |
------------------------------------------- |
||
|
(S-1) x T(N=N) |
= |
-1 + 0 + 0 + 0 + 0 + ….. +0 + S(N+1) |
= S(N+1) -1 |
So:
T(N=N) = ( S(N+1) -1 ) / ( S - 1)
Formula 2 : M
(N=N) = ( SN -1 ) / ( S - 1)From the construction of formula 1 you can easily see that in a company with N managementlevels can be split up in a sub-company of N-1 managementlevels that consists of only managers and the employees at the bottom of the organisation.
So:
M(N=N) = T(N=[N-1]) = ( S([N-1]+1) -1 ) / ( S - 1) = ( SN -1 ) / ( S - 1)
This leads to some more or less other interesting relations:
Management-ratio: MR = M/T = ( SN -1 ) / ( S(N+1) -1 )
Span of control: S = (T-1) / M