<Dl> <Dd> n (2 n / 2 − 1) (\ displaystyle n (2 ^ (n) / 2 - 1)) </Dd> </Dl> <Dd> n (2 n / 2 − 1) (\ displaystyle n (2 ^ (n) / 2 - 1)) </Dd> <P> The sum of these three types of interactions is the number of potential relationships of a supervisor . Graicunas showed with these formulas, that each additional subordinate increases the number of potential interactions significantly . It appears natural, that no organization can afford to maintain a control structure of a dimension being required for implementing a scalar chain under the unity of command condition . Therefore, other mechanisms had to be found for dealing with the dilemma of maintaining managerial control, while keeping cost and time at a reasonable level, thus making the span of control a critical figure for the organization . Consequently, for a long time, finding the optimum span of control has been a major challenge to organization design . As Mackenzie (1978, p 121) describes it: </P> <P>" One could argue that with larger spans, the costs of supervision would tend to be reduced, because a smaller percentage of the members of the organization are supervisors . On the other hand, if the span of control is too large, the supervisor may not have the capacity to supervise effectively such large numbers of immediate subordinates . Thus, there is a possible trade - off to be made in an attempt to balance these possibly opposing tendencies ." </P>

The optimum number of subordinates a manager can supervise is referred to as the