Every executive 'always measures the burden of his responsibility to control the subordinates in terms of single relationship between himself and his subordinates. Graicunas feels that in any group, the relations between executive and his subordinates cannot just be calculated based on single relationship alone. According to him, there also exists cross relationships which increase in mathematical proportion. The direct single relationship always increase in the same proportion as the number of subordinates. In such a case each addition to t h ~ I, group would only create a single direct relationship. But according to Graicunas there also exists direct group and cross relationships which increase very rapidly than the 1I1 increase in the number of subordinates. This is n~a because the addition of each \ !/ individual results in many cross and drrett group relationships as there are persons already in the group. Therefore, the number of relationships increases in exponential proportion. Based upon his studies, Graicunas develops a formula to calculate the /I/ number of relationships to enable the executives to-examine the complexity of span of control.
According to these formulae, in any organisalion if there are three subordinates direct single relationships would be three, cross relationships six and direct group relationships nine. But if one more member is added there would nor be any change in the direct single relationships which would be four, but the cross relationships would increase to 12 and direct group relationship, however, rise exponentially to 28. This explains that addition of each member to the group under the control of the executive would increase the number of direct group relationships to such an extent that direct contribution becomes difficult in some cases even impossible. Graicunas also added that for subordinates it is clhite easy to grasp and remember every combination of groups. But from five on, it is not possible to remember because the relationships become more of confusion.
When the x principle enunciated by Graicunas is valid or not, whether the formulae has empirical validity or not, the problem that any increase in the number of subordinates would lead to complexity in the relationships between the individual and groups has aptly been brought out by Graicunas. It is this factor that needs to be carefully considered in any discussion on how many subordinates an executive can effectively control.
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