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# SOC Quantitative Approach

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Statistical Approach to Defining and Elaborating the numbers behind span of control and quantifying the relationships a People leader manages. It also shows how to calculate the same and has embedded Excel to auto-calculate the Relationships given the number of subordinates one manages.

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### SOC Quantitative Approach

1. 1. Span of Control –Quantitative Approach to Relationships Presented to: Relevant to any People Leader Date: 09/Nov/2011 Location: NA
2. 2. Agenda – Span of Control, Quantitative Approach Definition - SOC Definition - Relationships Calculation - Relationships Application - Relationships Definition of:  Span of control  Supervisor  Subordinate The scope of activities and inter-relation between a Supervisor and the Subordinates A FieryAir Venture Slide:2
3. 3. Agenda – Span of Control, Quantitative Approach Definition - SOC Definition - Relationships Calculation - Relationships Application - Relationships Definition of: Definition of three kinds of Relationships managed by a  Span of control Supervisor:  Supervisor  (DSR) Direct Single Relationships  Subordinate  (CIR) Cross Individual The scope of activities Relationships and inter-relation between a Supervisor and the  (DGR) Direct Group Subordinates Relationships A FieryAir Venture Slide:3
4. 4. Agenda – Span of Control, Quantitative Approach Definition - SOC Definition - Relationships Calculation - Relationships Application - Relationships Definition of: Definition of three kinds of The Formulas to calculate the Relationships managed by a three kinds of Relationships  Span of control Supervisor: that a Manager handles.  Supervisor  (DSR) Direct Single The basic inputs being the Relationships number of Subordinates a  Subordinate Supervisor controls.  (CIR) Cross Individual The scope of activities Relationships It has been devised by Graicunas, and inter-relation between a a Management Consultant in Supervisor and the  (DGR) Direct Group 1933. We would discuss him at end. Subordinates Relationships A FieryAir Venture Slide:4
5. 5. Agenda – Span of Control, Quantitative Approach Definition - SOC Definition - Relationships Calculation - Relationships Application - Relationships Definition of: Definition of three kinds of The Formulas to calculate the There are various ways that Relationships managed by a three kinds of Relationships the concept of SOC can be  Span of control Supervisor: that a Manager handles. put to use for enhancing Management role and  Supervisor  (DSR) Direct Single The basic inputs being the comparing current and future Relationships number of Subordinates a workload of Supervisors.  Subordinate Supervisor controls.  (CIR) Cross Individual We would try discussing the The scope of activities Relationships It has been devised by Graicunas, same as case studies. and inter-relation between a a Management Consultant in Supervisor and the  (DGR) Direct Group 1933. We would discuss him at end. Subordinates Relationships A FieryAir Venture Slide:5
6. 6. Span of Control – Definition Span of Control is defined as the Ratio of Subordinates to one Superior. In the following sections we would discuss the relationship between the following elements:  Span of Control  Relationship Management for a Supervisor  Potential Relationships a Supervisor has to manage due to the Span s/he is handling  Quantitative approach to the Relationship Management by a Supervisor The term ‘Supervisor’ would mean an individual who:  Leads a group of individuals for a certain purpose (that may change over a period)  Gets reported ‘directly’ by this group of individuals  Has definite goals for each individual reporting to her/him and for the group as an entity The term ‘Subordinate’ would mean an individual who:  Reports to a particular superior for a particular purpose  Follows defined timelines set by the superior  Works towards achieving set goals individually and contributes to the team goals A FieryAir Venture Slide:6
7. 7. Relationships a Supervisor handles Direct Single Relationships (DSR) The relationship between a Supervisor and individual Subordinates Sup Definition (DSR) A Supervisor has to manage her/his direct relation with each individual in her/his team. Thus, the Supervisor “Sup” in our example is managing three relations with his subordinates, “Sub1”, “Sub2” & “Sub3”. The three relations managed by “Sup” here are:  DSR1 DSR2  DSR2  DSR3 Sub1 Sub2 Sub3 A FieryAir Venture Slide:7
8. 8. Relationships a Supervisor handles contd… Cross Individual Relationships (CIR) The relationship between each individual subordinates, in both ways. Sup Definition (CIR) A Supervisor has to manage the inter-relations between each individual in her/his team. Thus, the Supervisor “Sup” in our example is managing six inter-relations within his subordinates, “Sub1”, “Sub2” & “Sub3”. The six inter-relations managed by “Sup” here are:  CIR1  CIR2  CIR3 Sub2  CIR4  CIR5  CIR6 CIR2 CIR5 Sub1 Sub3 A FieryAir Venture Slide:8
9. 9. Relationships a Supervisor handles contd… Direct Group Relationships (DGR) The relationship between Superior and combination of subordinates. Definition (DGR) A Supervisor has to manage her/his group-relations between each group Sub2 Sub1 of individuals in her/his team. The number of groups would be a result Sub1 Sub2 of each possible combination that can occur with the subordinates. Sub2 Sub3 DGR9 Thus, the Supervisor “Sup” in our example is managing nine group- Sup relations within the group of subordinates, Sub1, Sub2 & Sub3. The nine group-relations managed by “Sup” here are: Sub1 Sub3 Sub3 Sub1 Sub3 Sub2 Sub3 Sub1 Sub2 Sub1 Sub2 Sub3 Sub2 Sub3 Sub1 A FieryAir Venture Slide:9
10. 10. Next step after the Theory… DSR CIR DGR Calculation & Application After defining the theory of Relationships a Manager handles, we would discuss the practical aspect of the same. There are two aspects to Practicality of this Theory:  Calculating the number of relationships of each kind a Manager handles  Applying the Theory and the Calculation to Practical life in office A FieryAir Venture Slide:10
11. 11. Calculating DSR Calculating Direct Single Relationships DSR is the easiest to calculate. It is equal to the number of Subordinates a Supervisor has. Assumption: Number of Subordinates = n Formula: DSR = n A FieryAir Venture Slide:11
12. 12. Calculating CIR Calculating Cross Individual Relationships CIR calculation is not tricky, and the formula finds out the exact number of cross relation a manager handles in her/his team. Assumption: Number of Subordinates = n Formula: CIR = n x (n-1) A FieryAir Venture Slide:12
13. 13. Calculating DGR Calculating Direct Group Relationships DGR calculation is little tricky, and the formula finds out the exact number of group relation a manager handles in her/his team. Assumption: Number of Subordinates = n Formula: DGR = n x (2n/2-1) A FieryAir Venture Slide:13
14. 14. Sum of all Relationships Total number of relationships The total number of relationships handled by a Superior would then be a sum of DSR, CIR & DGR. However, there is another formula to calculate the total number of relationships without calculating DSR, CIR & DGR separately, if required: The formula is: n*(2n/2+ n -1) Calculating the figures using an example… A chart in the embedded Excel File based on the table demonstrates that, as the number of subordinates increases past four, the complexity of the relationships increases exponentially. This owes primarily to an increase in the number of DGR created by adding a member to an existing group. For example, adding a fifth subordinate roughly doubles complexity, increasing the total direct plus cross relationships from 44 to 100. Adding a sixth subordinate more than doubles complexity again, increasing the number of relationships from about 100 to 222. For 12 subordinates, the total number of relationships that might demand a superiors attention is an astounding 24,708! To view the live example of above theory, please open the embedded Excel file at right to have the formulas and calculations already set up for you in an Excel File… A FieryAir Venture Slide:14
15. 15. Practicality & Applicability “There is nothing which rots morale more quickly and more completely than poor communication and indecisiveness -- the feeling that those in authority do not know their own minds. And there is no condition which more quickly produces a sense of indecision among subordinates or more effectively hampers communication than being responsible to a superior who has too wide a span of control” - Lyndall F. Urwick (Pg 43 of article, "The Managers Span of Control" published in May-June 1956 issue of Harvard Business Review) Practicality of quantifying SOC  SOC affects the cost A wider SOC, as in the above statement, at times creates unrest among the subordinates. However, a wider SOC ensures lower cost of management. Though, the significance lies is comparing the cost of poor quality of output of the team against cost of maintaining thinner SOC. Because, as a thumb rule, thinner SOC ensures lesser number of relations that a superior manages and therefore, efficient output.  SOC affects subordinate morale As discussed above, a wider SOC at times creates unrest in subordinates as the superior has a visible span of large number of subordinates and it exposes her/his ability to do People Management. Even a Super Efficient supervisor would have a hard time to handle 10 subordinates where s/he is actually handling a total of 5210 relationships (DSR+CIR+DGR).  SOC behaves typically for different industries Affording a wider or a thinner SOC depends on the kind of industry as well. In an industry where the output is co-dependent on machines (for e.g. manufacturing), a wider span of control can be afforded. However, in service industry as ours, thinner SOC results better as in service industries, the output is dependent on the human resource only. Therefore, managing human relations and inter-relations are more significant  SOC varies with hierarchy For any kind of industry, the general approach should be thinning the SOC towards the bottom of hierarchy. The justification being, at the top of hierarchy, the superior has to manage fewer number of ancillary tasks and major number of human relations. Thus, the people management becomes easier. However, at the lower levels, the transaction management also becomes as important as human resource management. Thus, managing the total number of relationships (DSR+CIR+DGR) becomes difficult. SOC is highly dependent on the culture of the kind of organization and industry. It can be used judiciously to control the cost and output of the organization and define the optimum mix so that there is a maximization in earning. A FieryAir Venture Slide:15
16. 16. Graicunas. The Graicunas formula is named after V.A. Graicunas, who gave a mathematical formula to explain the complexity of span of control if more subordinates are added to the executive. 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 formula, in any organization 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. www.nickols.us/graicunas.pdf http://en.wikipedia.org/wiki/Graicunas_formula http://en.wikipedia.org/wiki/Span_of_control www.jstor.org/stable/254987 www.bus.lsu.edu/bedeian/articles/VGracunias-AMJ1974.pdf A FieryAir Venture Slide:16
17. 17. K. S. Alok Ranjan+91 9818665496 | +91 124 4118792 A FieryAir Venture Slide:17