Applied science and technological progress - C&EN Global Enterprise

European Journal of Operational Research 174 (2006) 1491–1500 www.elsevier.com/locate/ejor

Discrete Optimization

Models of adding relations to an organization structure of a complete K-ary tree Kiyoshi Sawada a

a,*

, Richard Wilson

b

Department of Information and Management Science, University of Marketing and Distribution Sciences, 3-1 Gakuen-nishi-machi, Nishi-ku, Kobe 651-2188, Japan b Department of Mathematics, University of Queensland, Brisbane, Queensland 4072, Australia Received 27 May 2002; accepted 19 January 2005 Available online 19 May 2005

Abstract This paper proposes three models of adding relations to an organization structure which is a complete K-ary tree of height H: (i) a model of adding an edge between two nodes with the same depth N, (ii) a model of adding edges between every pair of nodes with the same depth N and (iii) a model of adding edges between every pair of siblings with the same depth N. For each of the three models, an optimal depth N* is obtained by maximizing the total shortening path length which is the sum of shortening lengths of shortest paths between every pair of all nodes. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Graph theory; Organization structure; Complete K-ary tree; Shortest path length

1. Introduction The basic type of formal organization structure is a pyramid organization [5] which is a hierarchical structure based on the principle of unity of command [2] that every unit except the top in the organization should have a single immediate superior. In the pyramid organization there exist relations only between each superior and his direct subordinates. However, it is desirable to have formed additional relations other than that between each superior and his direct subordinates in advance in case they need communication with other departments in the organization. In companies, the relations with other departments are built by

*

Corresponding author. Tel.: +81 78 796 4806; fax: +81 78 794 3054. E-mail addresses: [email protected] (K. Sawada), [email protected] (R. Wilson).

0377-2217/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.01.067

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K. Sawada, R. Wilson / European Journal of Operational Research 174 (2006) 1491–1500

meetings, group training, internal projects, and so on. Personal relations exceeding departments are also considered to be useful for the communication of information in the organization. The pyramid organization structure can be expressed as a rooted tree, if we let nodes and edges in the rooted tree correspond to units and relations between units in the organization respectively. Then the pyramid organization structure is characterized by the number of subordinates of each unit (that is, the number of children of each node) and the number of levels in the organization (that is, the height of the rooted tree), and so on [3,4]. Moreover, the path between a pair of nodes in the rooted tree is equivalent to the route of communication of information between a pair of units in the organization, and adding edges to the rooted tree is equivalent to forming additional relations other than that between each superior and his direct subordinates. The purpose of our study is to obtain an optimal set of additional relations to the pyramid organization such that the communication of information between every unit in the organization becomes the most efficient. This means that we obtain a set of additional edges to the rooted tree minimizing the sum of lengths of shortest paths between every pair of all nodes. This paper proposes three models of adding relations to an organization structure which is a complete K-ary tree of height H (H = 1, 2, . . .): (i) a model of adding an edge between two nodes with the same depth N, (ii) a model of adding edges between every pair of nodes with the same depth N and (iii) a model of adding edges between every pair of siblings with the same depth N, where siblings are nodes which have the same parent. A complete K-ary tree is a rooted tree in which all leaves have the same depth and all internal nodes have degree K (K = 2, 3, . . .) [1]. Fig. 1 shows an example of a complete K-ary tree (K = 2, H = 5). In Fig. 1 the value of N expresses the depth of each node. The above model (i) corresponds to the formation of an additional relation between two units in the same level of an organization such as a personal communication. Model (ii) is equivalent to additional relations between every pair of all units in the same level such as section chief training. Model (iii) is equivalent to additional relations between every pair of all units which have the same immediate superior such as a department meeting. If li, j(=lj,i) denotes the path length, which is the number of edges in the shortest path from P a node vi to a node vj (i, j = 1, 2, . . . , (KH+1 1)/(K 1)) in the complete K-ary tree of height H, then i 0 in Eq. (9), we have DS1,H(1) < 0 for H = 2 and DS1,H(1) > 0 for H = 3, 4, . . .

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In the case of (ii), by differentiating T1,N(x) in Eq. (8) with respect to x, we obtain T 01;N ðxÞ

¼

1 ðK 1Þ

2

2ðNK 2 þ 2K þ N Þ 4K x N þ1 . K N þ1 K

Since T 1;N ðK N þ1 Þ ¼

1 ðK 1Þ

2

ðNK 2 þ 2K þ N ÞK N þ1 2ð2K 1Þ < 0

and T 01;N ðK N þ1 Þ ¼

1 ðK 1Þ

2

2ðNK 2 þ 2K þ N Þ

4K K N þ1

< 0;

we have T1,N(x) < 0 for x P KN+1. Therefore, we have DS1,H(N) < 0 for H = N + 1, N + 2, . . . ; that is, N = 1, 2, . . . , H 1. From the above results, the optimal adding depth N* can be obtained and is given in Theorem 2. Theorem 2 (1) If K = 2, then we have the following: (a) If H = 1, 2, then the optimal adding depth is N* = 1. (b) If H = 3, 4, . . . , then N* = 2. (2) If K = 3, 4, . . . , then we have N* = 1.

Proof (1) Assume K = 2. (a) If H = 1, then N* = 1 trivially. If H = 2, then N* = 1 since DS1,H(1) < 0. (b) If H = 3, 4, . . ., then N* = 2, since DS1,H(1) > 0 and DS1,H(N) < 0, for N = 2, 3, . . . , H 1. (2) Assume K = 3, 4, . . . If H = 1, then N* = 1 trivially. If H = 2, 3, . . ., then N* = 1 since DS1,H(N) < 0, for N = 1, 2, . . . , H 1. h

3. Adding edges between every pair of nodes with the same depth This section obtains an optimal depth N* by maximizing the total shortening path length, when new edges between every pair of nodes with the same depth N (N = 1, 2, . . . , H) are added to a complete Kary tree of height H. 3.1. Formulation of total shortening path length Let S2,H(N) denote the total shortening path length, when we add edges between every pair of nodes with a depth of N. The total shortening path length S2,H(N) can be formulated by adding up the following three sums of shortening path lengths: (i) the sum of shortening path lengths between every pair of nodes whose depths


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are equal to or more than N, (ii) the sum of shortening path lengths between every pair of nodes whose depths are less than N and those whose depths are equal to or more than N and (iii) the sum of shortening path lengths between every pair of nodes whose depths are less than N. The sum of shortening path lengths between every pair of nodes whose depths are equal to or more than N is given by A2;H ðN Þ ¼ fMðH N Þg

2

N K N ðK 1Þ X ð2i 1ÞK i1 ; 2 i¼1

where M(h) is as before. The sum of shortening path lengths between every pair of nodes whose depths are less than N and those whose depths are equal to or more than N is given by B2;H ðN Þ ¼ MðH N ÞK N ðK 1Þ

N 1 X i X i¼1

ð2j 1ÞK j1

j¼1

and the sum of shortening path lengths between every pair of nodes whose depths are less than N is given by C 2;H ðN Þ ¼

N 2 X i K N ðK 1Þ X ð2j 1Þði j þ 1ÞK j1 ; 2 i¼1 j¼1

where Eqs. (2) and (3) apply. From these equations, the total shortening path length S2,H(N) is given by S 2;H ðN Þ ¼ A2;H ðN Þ þ B2;H ðN Þ þ C 2;H ðN Þ ¼ fMðH N Þg2 þ

N N 1 X i X K N ðK 1Þ X ð2i 1ÞK i1 þ MðH N ÞK N ðK 1Þ ð2j 1ÞK j1 2 i¼1 i¼1 j¼1

N 2 X i K N ðK 1Þ X ð2j 1Þði j þ 1ÞK j1 . 2 i¼1 j¼1

ð10Þ

From Eqs. (5) and (10), we have that S 2;H ðN Þ ¼

1 4ðK 1Þ

3

2ðK þ 1ÞK 2H N þ2 þ 2fð2N 1ÞK 2N 1gK 2H þ2 8K H þN þ2

þ 4ðNK 2 þ 2K N ÞK H þ1 þ N ðN 1ÞK 3 ðN 3ÞK 2 ðN þ 3ÞK þ ðN þ 1Þ K N . 3.2. An optimal adding depth In this subsection, we seek N = N* which maximizes S2,H(N) in Eq. (11). Let DS2,H(N) S2,H(N + 1) S2,H(N), so that we have 2KðK þ 1Þ 2H 4K DS 2;H ðN Þ ¼ K þ 8K N þ2 þ 4KðK þ 1Þ K H N 2 K 4ðK 1Þ 2 N 2 2 þðK 1Þ ðK 1ÞN þ ðK þ 4K 1ÞN þ 2K K ; 1

2

ð11Þ

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for N = 1, 2, . . . , H 1. Let us define a continuous variable x which depends on H as in Eq. (7), so that DS2,H(N) becomes

1 2KðK þ 1Þ 2 2 4K T 2;N ðxÞ ¼ x þ 8K N þ2 þ 4KðK þ 1Þ x N 2 K 4ðK 1Þ 2 N 2 2 þ ðK 1Þ ðK 1ÞN þ ðK þ 4K 1ÞN þ 2K K ; which is a quadratic function of x. By differentiating T2,N(x) with respect to x, we obtain

1 KðK þ 1Þ 2 N þ2 T 02;N ðxÞ ¼ 2K þ KðK þ 1Þ . x 2K KN ðK 1Þ2 Since T2,N(x) is convex downward from 4K2 2K(K + 1)/KN > 0, and 1 T 2;N ðK N þ1 Þ ¼ 2ðK 2Þð2K N þ1 K 1ÞK N þ2 þ ðK 1Þ ðK 2 1ÞN 2 2 4ðK 1Þ þ ðK 2 þ 4K 1ÞN þ 2K K N > 0 and Kð2K N þ1 K 1Þ > 0; K 1 we have T2,N(x) > 0 for x P KN+1. Hence, we have DS2,H(N) > 0 for H = N + 1, N + 2, . . . ; that is, N = 1, 2, . . . , H 1. From the above results, the optimal adding depth N* can be obtained and is given in Theorem 3. T 02;N ðK N þ1 Þ ¼

Theorem 3. The optimal adding depth is N* = H. Proof. If H = 1, then N* = 1 trivially; that is N* = H. If H = 2, 3, . . . , then N* = H since DS2,H(N) > 0, for N = 1, 2, . . . , H 1. h

4. Adding edges between every pair of siblings with the same depth This section obtains an optimal depth N* by maximizing the total shortening path length, when new edges between every pair of siblings with the same depth N(N = 1, 2, . . . , H) are added to a complete K-ary tree of height H. 4.1. Formulation of total shortening path length Let S3,H(N) denote the total shortening path length, when we add edges between every pair of siblings with a depth of N. Since addition of the new edges between every pair of siblings shortens path lengths only between pairs of descendants of the siblings on which the new edges are incident, we obtain S 3;H ðN Þ ¼ S 2;H N þ1 ð1ÞK N 1 ; where S2,H(N) is the total shortening path length in the case of adding edges between every pair of nodes with the same depth N in Eq. (10) of Subsection 3.1. Thus, we have


S 3;H ðN Þ ¼

1 ðK 2H N þ2 2K H þ1 þ K N Þ. 2ðK 1Þ

1499

ð12Þ

4.2. An optimal adding depth In this subsection, we seek N = N* which maximizes S3,H(N) in Eq. (12). Let DS3,H(N) S3,H(N + 1) S3,H(N), so that we have DS 3;H ðN Þ ¼

KN ð1 K 2H 2N þ1 Þ; 2

for N = 1, 2, . . . , H 1. Thus, we have DS3,H(N) < 0 for H = N + 1, N + 2, . . .; that is, N = 1, 2, . . . , H 1. From this result, the optimal adding depth N* can be obtained and is given in Theorem 4. Theorem 4. The optimal adding depth is N* = 1. Proof. If H = 1, then N* = 1 trivially. If H = 2, 3, . . ., then N* = 1 from DS3,H(N) < 0, for N = 1, 2, . . . , H 1. h

5. Conclusions This study considered the addition of relations to an organization structure such that the communication of information between every unit in the organization becomes the most efficient. For each of three models of adding edges between nodes of the same depth N to a complete K-ary tree of height H which can describe the basic type of a pyramid organization, we obtained an optimal adding depth N* which maximizes the total shortening path length. Theorems 1 and 2 on adding an edge between two nodes with the same depth in Section 2 show that the most efficient manner of adding a single relation between two units in the same level such as a personal communication is to add the relation between two units which doesnÕt have common superiors except the top at the optimal level which is the first level or the second level below the top depending on the number of subordinates and the number of levels in the organization structure. Theorem 3 on adding edges between every pair of nodes with the same depth in Section 3 shows that the most efficient way to add relations between every pair of all units at the same level such as section chief training is to use the lowest level, irrespective of the number of subordinates. Theorem 4 on adding edges between every pair of siblings with the same depth in Section 4 shows that the most efficient way of adding relations between all units at the same level and with the same immediate superior such as a department meeting is to do so at the level directly below the top, irrespective of the number of subordinates and the number of levels in the organization structure. These three results reveal optimal sets of additional relations to the pyramid organization which is a complete K-ary tree such that the communication of information between every unit in the organization becomes the most efficient.

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[3] S.P. Robbins, Essentials of Organizational Behavior, seventh ed., Prentice Hall, Upper Saddle River, N.J., 2003. [4] Y. Takahara, M. Mesarovic, Organization Structure: Cybernetic Systems Foundation, Kluwer Academic/Plenum Publishers, New York, 2003. [5] N. Takahashi, Sequential analysis of organization design: A model and a case of Japanese firms, European Journal of Operational Research 36 (1988) 297–310.

Applied science and technological progress - C&EN Global Enterprise

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