More on unnecessary meetings and optimisation

Having written about meetings yesterday, I remembered this article, which combines the O.R. ideas of optimisation with a discussion about meetings.  As will be apparent, it is not my own work, and I am not sure about copyright ...

The article below appeared in the journal "Eureka", issue 33, October 1970.  As is clear from the introduction, it is reproduced from another source, which is unnamed.

The Optimum Size for an Establishment
by H. W. O. Petard
(The Editor would like to thank Professor M. V. Wilkes who discovered this paper quite recently and very kindly passed it on to Eureka. As older readers will know the author was famed in mathematical circles for his incredible article on the Mathematical Theory of Big Game Hunting which appeared in the American Mathematical Monthly during 1938 and later in Eureka no. 16. The article below was probably written some time during the war)
1. Introduction
It is a commonplace that the amount of work done by an Establishment of Civil Servants is not directly proportional to its size; indeed, cases may readily be quoted in which an increase in staff has impeded rather than furthered the objects for which an establishment exists. In the present paper I shall examine the problem analytically; and derive formulae by means of which the optimum size for an establishment may be determined. I shall make use for this purpose of the Kinetic Theory of Civil Servant Swarms which has been developed by Ponticelli. As this theory is not as well known as it deserves, I propose in the next section to give a brief summary of the most important results.
2. The Kinetic Theory of Civil Servant Swarms
The fundamental concept in this theory is 'pressure', which may be defined as the force that the establishment as a whole can bring to bear in order to achieve its objects, for example, to overcome the obstruction of other establishments or departments, or itself to obstruct some design originating from without. At first sight it might be thought that a Civil Service establishment, by virtue of the random motions of its members, would exert a pressure of precisely zero; a moment's consideration of the analogous case of a gas consisting of molecules moving with random velocities, will, however, show that this is not the case, and that in fact a finite pressure is always exerted. This pressure is, of course, less than the pressure of a hypothetical pseudo-establishment in which the energies of the staff are all oriented in the same direction.
I will not here enunciate all the results of the Kinetic Theory, for which reference should be made to the original sources; two results, however, are of particular importance:

(a) In order to do work an establishment must continually expand.

(b) The larger the establishment becomes the less is the pressure it exerts

3. The Optimum Size for an establishment.

We shall suppose that the establishment consists of n members; this number is only to include those members of staff in positions of responsibility, as it is not my intention in this paper to examine in detail the relations existing, for example, between an officer and his secretary. Let a fraction k of each officer's time (office hours only being considered) be devoted to the work of the establishment, and let a fraction m be devoted to productive work; the difference, namely (k-m), will then be devoted to internal liaison with other members of the establishment. We will suppose for simplicity that the quantities k and m are the same for all members. Normally, k² and higher powers may be neglected. In what follows it will be convenient to take the working day as the unit of time.
Since each officer has to liaise with each of his (n-1) colleagues, we have, assuming he spends the same amount of time with each,

k - m = a(n - 1)

where a is a constant.
The total time spent on liaison by all the officers taken together is then

a n (n - 1)

while the total time spent on useful work is

nm

It will thus be seen that as n increases, the time spent on liaison increases much more rapidly than the time spent on useful work.
If we assume that the size of the establishment is optimum when these two quantities are equal, we have

a n (n - 1) = n m

or n =1+ m/a

Since exactly half the time k is spent on useful work, we must have

m=0.5k

so that the above equation for n becomes

n =1+ k/(2a)

4. A Numerical Example
To illustrate the foregoing results in a practical case, let us suppose that the daily liaison between officers is limited to the writing of one short minute, or the making of one short telephone call, the total duration being on the average 3 minutes. If the length of the working day is five hours, we have

a = 3/(5 x 60) = 0. 01

If we now take for k the value 0.5, we get for n
n =1 + 0.5/(2 x 0. 01)= 26
Thus the optimum size for an establishment under these conditions is 26.