What is Graicuna's formula

Control span

defines how far the area of ​​control competence is for a holder of a managerial position if the effectiveness of the control is to be guaranteed. In practice, a mediocrity has to be found in the distance between superiors and employees. If the distance is too great, it is no longer possible to carry out exact controls. The control then remains anonymous and the sanctions remain ineffective. If, on the other hand, the distance is too small, the control is perceived as a disruption of the formal and informal relationship. The determination of the control range depends on a number of factors and can vary between 5 and 100 people, which has been empirically established. Examples of influencing factors for determining the control range: complexity of the tasks of the employees, management style, spatial distance between the workplaces, type of employees, number of levels of authority, wages, frequency of work repetitions, similarity of workplaces, etc.

is the number of persons subordinate to a position or its owner. Only the immediately subordinate level of the hierarchy is counted.

In economic sociology: span of control, in an organization the area of ​​those who are directly subordinate to a superior. The controversial assumption that the control margin must be small for the control to be successful is the basis of all hierarchical theories.

Number of people or work process steps that are monitored by a specific inspector (line span).

(span of control), see line span.

(Supervision range): The number of positions within the management pyramid of a company that report to the next higher level. So that is the number of employees who report directly to an instance. The control margin depends on the scope and differentiation of the tasks to be controlled as well as the scope of the managerial authority of the post holder vis-à-vis the subordinate units. The control span and the number of contacts determine the effort and intensity of the information. In addition, there is informal communication. It relates to the area that is not regulated by the formal information, e.g. ad-hoc coordination and agreements in the event of special problems, the collection of additional information or external contacts.
The number of employees who report directly to a manager must be such that the manager is busy managing this group, but not overloaded. If this number is too large, the intensity of the leadership and the cooperation between supervisors and employees suffers. The supervisor does not have enough time for the individual, the group of employees becomes too large to form groups, the supervisor no longer has his group fully in hand. Informal subdivisions are formed with informal group leaders. Formal and informal organization must be largely congruent.
If the number of employees is too small, the independence of the employees suffers. The supervisor is underutilized, takes work away from employees and creates his own areas of responsibility. This means that he is no longer a neutral superior, but a competitor of the employees. In addition, the circle becomes too stress-free to be able to work in a group educational way.
The optimal control range depends on the degree of difficulty of the task of the area, the degree of independence and quality of the employees, the difficulty of the management task, the required intensity of monitoring, the distances between the locations of the employees, the clarity of the spatial conditions and the format of the people involved. It can be greater on lower levels than on upper levels, which are more difficult tasks and stronger personalities. Favorable numbers are 3 to 7 on the upper levels and 8 to 20 employees on the lowest level. In extreme cases, other numbers are also possible. A superior with only 2 directly subordinate employees should be avoided as a matter of principle.
With these figures, it is assumed that the line manager does not take on any of his own responsibilities, if possible, in order to maintain the neutrality of his line manager function towards his employees. The supervisor should be fully utilized with the performance of his managerial tasks towards his employees and with his employee function towards his supervisor, so that he does not have the opportunity to take on his own responsibilities that should be delegated. Exceptions can be made at the lowest level in the case of small supervisory spans and group-based working methods.
The number of levels in the corporate hierarchy depends on the control margins. The smaller the number of these levels of authority, the shorter the information path between above and below. Instance paths that are too long hinder the leadership influence of the top and the flow of information in both directions. The consideration of a few levels of authority can force you to go to the upper limit with the control ranges.
The extent of the optimal control span was one of the great topics of classical organization theory. She assumed that the employees were in great need of guidance and control and therefore recommended that the control range be kept relatively small. The ranges considered optimal vary between 3 and 10.
The Frenchman V. A. Graicunas made the first attempt to optimize the control span not only on the basis of everyday experience, but to subject it to a more thorough analysis. According to this, the number of maximum possible relationships between superiors and subordinates is the decisive factor for dimensioning the control margin.
Graicunas distinguishes three types of relationships between superiors and subordinates in order to measure the scope of a managerial task:
(1) Direct relationships between the superior and each of his subordinates: The number of possible direct individual relationships corresponds to the number of subordinates (= n);

Relationships between the superior and different groups of subordinates: To determine the number of direct group relationships, Graicunas assumes that in contacts between the superior and a group of subordinates, one member of this group is always in the foreground. With two subordinates A and B, a distinction would have to be made, for example, between group AB and group BA. The total number of group relationships is calculated according to the group relationships:
(3) Mutual relationships between the subordinates: To determine the possible number of mutual relationships between the subordinates, Graicunas only considers groups of two, although one member is the initiator each, so that AB and BA represent two groups. The total number is calculated accordingly:

If all three relationship groups are combined, the total number of possible relationships (z) results from the formula:
According to this formula, an increase in the number of subordinates leads to a disproportionate increase in the possible relationships between superiors and subordinates and thus to a disproportionate increase in the scope of management; the capacity limit is reached quickly.

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