Internationalof ComputationalIntelligence
Systems
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XX(Z);
Month(2019),(Year), pp.
pp. 128–132
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DOI: 10.1080/XXXXXXXXXXXXXX; ISSN
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DOI: https://doi.org/10.2991/jrnal.k.190828.012;
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Research Article

An Efficient Structure of Organization with Complete
Group GuidanceSatoshi Ikeda1,*, Makoto Sakamoto1, Takao Ito2
Department of Computer Science and System Engineering, Miyazaki University, 1-1 Gakuen Kibanadai Nishi, Miyazaki, 889-2192, Japan
Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8526, Japan

12ARTICLE INFO
Article History
Received 23 October 2018
Accepted 23 November 2018
Keywords
Hierarchical organization
mathematical model
discrete combination optimization

ABSTRACT
Organizational structure is one of the most important factors in determining its performance. Our interest is what kind ofstructures is efficient, and what conditions are required for an efficient structure? In our previous papers, we proposed amathematical model for evaluating the hierarchical organization quantitatively and used it to investigate this problem. As a resultof that research, we proved that the shape of the hierarchical organization that maximizes the evaluation value of organization,
can be classified into three types depending on the capacity value of the members when number of the evaluation criteria is onlyone. The purpose of this paper is to extend previously obtained results to an improved mathematical model.
© 2019 The Authors. Published by Atlantis Press SARL.
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION
There are many criteria for evaluating members of an organization. For example, technical skills, management skills, etc. In Ikeda[1] and Ikeda et al. [2], we proved by using a mathematical modelthat an efficient shape of a hierarchical organization with only one
criterion is limited to three types (Figure 1).
Subsequently, the mathematical model for evaluating the shapeof the organization has been generalized more realistically [3,4].
Although the basic idea of the total output to the outside of theorganization as an evaluation value has not changed, the major
improvements in the revised mathematical model are the following
two points.
(a) Contribution of the members for a given organization can be
categorized as the internal contribution to maintain the organization and the external contribution to realize the purposeof the organization in society.
(b) For any fixed organization, the sum of the effort each person
spends is constant, but its influence is different depending on
how to take a communication style.
The ratio of one’s external contribution effort to the total of one’seffort is called the external output coefficient. In the mathematical model of Ikeda et al. [3,4] with one evaluation criterion under
the assumption that the external output coefficient takes a constanta ∈[0, 1] independent of member, previous results in Ikeda [1] andIkeda et al. [2] correspond to a = 0 and the communication stylecalled “the complete group guidance”.

Corresponding author. Email: bisu@cs.miyazaki-u.ac.jp*

When the number of evaluation criteria is one, in the paper Ikeda
et al. [3,4], we showed that the structure of an efficient organization
becomes line shape (Figure 1b), in the case that personal ability valueof organization members is high. This is an extension of the resultsof Ikeda [1] and Ikeda et al. [2] to the case of the model of Ikeda
et al. [3,4] in part. However, in general it was not known whether theresults of Ikeda [1] and Ikeda et al. [2] (Figure 1c), could be extended
to the case of the model of Ikeda et al. [3,4]. We expected that theresults could be extended in the general case, but in reality it wasnot. Yoshimura et al. [5] by simulation experiments showed that anefficient tree is generally a brush tree shape (Figure 2), when thenumber of evaluation criteria is one. After much consideration, weconcluded that the efficient tree shape as shown in Figure 1 may
depend on the communication style.
The purpose of this paper is to show that the efficient organizational shape when the evaluation criteria is only one, is limited
to the three types shown in Figure 1 under the assumption of thecommunication style called the complete group guidance.

2. MATHEMATICAL MODEL
Now we will introduce a mathematical model for evaluating thehierarchical organization quantitatively [3,4].

2.1. Notations and Tree Graph as
Hierarchical Organization
Suppose that G = (V(G), E(G)) is a graph. Throughout this paper,a graph is always finite, directed and simple, with order n = |V(G)|(n ≧ 2) and size m = |E(G)|.

S. Ikeda et al. / Journal of Robotics, Networking and Artificial Life 6(2) 128–132

cb129

idea of 1–3, the evaluation function of a given evaluation system 1(S, {ji}i ∈A) is defined as follows. For i ∈A, total output of s ∈Sa ,ithat written by f s → is the product of s′s personal ability ji (s)a ,iand s′s input that written by f s ← . That is,
f sa→,i = j i (s) f sa←,i .

aa ,i

For i ∈A, s′s total output f s → is classified into s′s external output,ia ,if sa→out
and s′s internal output f s →in (Figure 3). That is,
Figure 1 | Three types of efficient trees when the number of theevaluation criteria is one.

f sa→,i = f sa→,iout + f sa→,iin .
For the subordinate s ∈S who received instructions from his supea ,irior, it is necessary to transmit appropriate instructions f s →in as
superior to his own subordinates, while s as subordinate carries out
a ,ithe instructions f s →out (Figure 3).
We assume that the ratio of s′s external output to s′s total output isa constant independent of s ∈S. Through this paper, we denote thea ,ia ,iratio of f s →out to f s → as a ∈[0, 1] for any s ∈S. That is,Figure 2 | A brush graph.

For u ∈V(G), by N(u) = {v|{u, v} ∈E(G)}, we denote the set of vertices adjacent to u, and call deg(u) = |N(u)| the degree of u ∈V(G).
For an arbitrarily fixed rooted tree T with root r and its leaves setL(T), we denoteif s Î {r } È L(T ),ìdeg sdeg * s = ídeg1otherwise.
sîFor a rooted tree T with V(T) = S, we define natural partial ordering$(S, <)$ that so-called the tree-order associated with T. That is, theroot r ∈S of T is the least element and the leaves of T are the maximal elements in this partial order. This ordering will be consideredas the expression “depth”: if x < y, we say x lies below y in T. Notethat the root r is the least element, and that the leaves of T are its
maximal elements in this partial order.
Suppose that S = {s1, s2, …, sn} (n ≥ 2) and A (|A| ≧ 1) are finite
sets. Through this paper, A is the set of the evaluation measures. And S is interpreted as the set of members of a given organization, which consists of s1, s2, …, sn. There are many rooted
trees with S as vertices. After that, we identify each rooted tree
as its corresponding organization tree. Let T(S) be the set ofrooted trees with S as its elements.

external output f sa®,iout : internal output f sa®,iin: 1-a=afor any s ∈S. We call a the external output coefficient and call 1 - a
the internal output coefficient.

2.3. Weights’ Policy
For a given organizational structure tree T, we assume that thevalue of the input for subordinate x ∈N(s) with x > s is obtained bya ,ii
multiplying its weight w sx to f s →in (see Figure 3 again). Where{ }0 ≤ w sxi ≤ 1
for any s ∈S \ L(T), x > s of x ∈N(s) and1 ≤ ∑ x ∈N (s ), x > sw sxi ≤ deg * s
for any s ∈S \ L(T). Therefore, the total contribution of s ∈S \ L(T)for the organization can be expressed by(∑x ∈N ( s ), x > s)w sxi f sa→,iin + f sa→,iout .

2.2. External Contribution andInternal Contribution
For a given S and i ∈A, we define {ji}i∈A byj i : S ® R + º {x Î R | x > 0} for i ÎWe call ji(s) the personal ability of s ∈S with respect to i ∈A.
And we call (S, {ji}i ∈A) an evaluation system. According to theFigure 3 | Input and output for s ∈S.

S. Ikeda et al. / Journal of Robotics, Networking and Artificial Life 6(2) 128–132

130( )i
We assume that w sx x ∈N (s ), x > s for s ∈S \ L(T) is a sequence depend-ing only on deg*s and i ∈A. That is, for s, s′ ∈S \ L(T) s,s′ and( )ii ∈A, deg*s = deg*s′ implies that w s ¢x x ÎN (s ¢ ), x > s ¢ is a permutation of(w )isx x ∈N s , x > s(). Thus, for a fixed deg*s and i ∈A, the selection that

we can do is which weight to assign whom. We call the way of thedetermination a weights’ policy. For any weights’ policy, we assume
that if deg*s ≧ deg*s′ for s, s′ ∈S,w sxi1 ³ w sxi 2 ³  and w si ¢x ¢1 ³ w si ¢x ¢ 2 ³ ,( )()iithen w sx x ∈N (s ), x > s and w s ′x ′ x ′∈N (s ′ ), x ′> s ′ satisfy

Figure 4 | Conversion from T to T ′.
a ,ia ,ioutput coefficient a ∈[0, 1], f s ← and f s → denote the input and thea ,ia ,ioutput of s ∈S, respectively. And f s →out and f s ®in denote the external and the internal outputs of s ∈S for the organization, respectively. Then, we define as follows:

f sa→,i = j i ( s ) f sa←,i
if s = root,1f sa←,i =  ia ,iwfot
 p (s ) s p (s )→ in herwise.

iw sxj£ w si ¢xj

for j = 1, 2, …, deg*s′.

{ }i2.4. Communication Style
The weights’ policy is a parameter determined by the communicationstyle in the organization. Typical ones are the following two cases;

Where w xy denote the weights from x ∈S \ L(T) to y ∈N(x)with respect to i ∈A in an arbitrarily fixed T. And p(s) denotes theparent node (as superior) of x ∈S on T. Under the assumption of
external and internal output, we define
0if s ∈ L (T ) ,f sa→,iin = a ,iaf(1)otherwise,s→a ,is ∈ L (T ) ,f
iff sa→,iout =  s → a ,iotherwise.
a f s →f sa→,i = f pa(s,i)→ in + f pa(s,i)→ out ,(1) Complete Individual Guidance (CIG)åx ÎN ( s ), x > sw sxi = 1 for any s Î S  L (T ).

(2) Complete Group Guidance (CGG)∑x ∈N ( s ), x > sw sxi = deg * s for any s ∈ S  L (T ).

The case of (1) CIG is an organization like a secret society. Since asuperior gives an instruction to his/her subordinates individually,
the members of the organization do not know about each other’swork. In such organizations, when new member joins the organization, or a member of the organization leave the organization,
its efficient organization cannot be constructed unless the existingorganization is broken once.
On the other hand, the case of (2) CGG is an open organization.
The characteristic of an organization with such a communicationstyle is that it is stable in the sense that there is no need to significantly change the composition of the whole organization when newmember joins it. Roughly speaking, the reason why CGG has sucha property is that when we convert T toT ¢ = T - xs + ys
for T, T ′ ∈T(S), s, x, y ∈S, s ∈N(x), s > x, the evaluation function isaffected only by the part of the up closure of s (Figure 4).
CIG and CGG are extreme cases, the general style is intermediate
between them.

For a given evaluation system (S, {ji}i∈A) an external output coefficient a ∈[0, 1] and a given weights’ policy, let T be a rooted tree
graph with V(T) = S. Then, we will evaluate the rooted tree T asorganization model byF(a )(T ) =i)(T )å F (a ,i ) (T ), F (a ,i ) (T ) = maxå fs(®a ,out{w }i Îixyi Î{ }iHere w xy shall be taken about all the possible combinations
under the given weights’ policy. For a given (S,{ji}i∈A) and a ∈[0, 1],
we call F(a) (T) the ability value of T. Under a given weights’ policy,
(a )
Topt
is an efficient tree for a given external output coefficient

( )

(a )
a ∈[0, 1], if max F(a,i ) (T ) is attained by F (a ) Topt
. Here T(S)T ∈ ( S )denotes the set of rooted tree graph with V(T) = S.

3. RESULTS
Theorem 1. Under the setting of Complete Group Guidance, supposethat (S,{ji}i∈A) is an evaluation system such that S = {s1, s2, ..., sn} withj i ( s1 ) ≥ j i ( s2 ) ≥  ≥ j i ( sn )( )for some i ∈A. Assume that max F(a,i ) (T ) is attained by Topt,T ∈ ( S )then we have the following.
a,i(a,i )2.5. Evaluation Function(a) Assume that ji(sn) ≥ 2 / (1 – a), then we see that one of Toptis the path graph in Figure 5 (1).

To summarize so far, for a given evaluation system (S, {ji}i ∈A), arooted tree T with root r and V(T) = S, s ∈S, i ∈A and an external(b) Assume that ji(s2) ≤ 1/(1 – a), then we see that one of Topt is
the star graph in Figure 5 (2).

(a,i )

S. Ikeda et al. / Journal of Robotics, Networking and Artificial Life 6(2) 128–132

131

Figure 5 | An efficient organization structure.

Figure 7 | Three types of efficient trees.

tional shape can be classified into the three types shown in Figure 7,
whenever the organizational communication style is CGG and there
is only one organizational evaluation.

(a ,i )

Figure 6 | General form of Topt

for some i ∈A.

The typical form of an efficient tree with respect to some i ∈A is apath graph or star graph as in Theorem 1. In general, we can show
(a,i )
that the form of Topt becomes the form which seems to have put
these two figures together.
For a given (S,{ji}i ∈A) and its organizational structure tree T, let
us define}

D2 (T ) = {s ∈ S | deg * ( s ) ≥ 2Theorem 2. Under the setting of CGG, suppose that (S,{ji}i∈A) isan evaluation system. Assume that max F(a,i ) (T ) is attained byT ∈ ( S )(a,i )

Topt for some i ∈A, then we have the following.

) is equal to 0 or 1.
( )( )(b) Putting D (T ) = {x } when D (T ) = 1, then we see that(a)(( )D2 Topta ,i2

a ,iopt

2a ,iopt{s ∈S|s > x in T ( ) } = L (T ( ) ).
a,iopta,iopt(a,i )The general form of the most efficient tree Topt which Theorem 2insists on is shown in Figure 6.

4. CONCLUSION
In this paper, we introduced that the conclusions of the papers [1,2]are also valid under a new mathematical model of the organization[3,4]. The details of the proof are omitted, but they can be shown inthe same way as the papers [1,2]. Eventually, even in the improved
new mathematical model, we found that the efficient organiza-However, Yoshimura et al. have experimentally shown that there
are efficient organizational shape other than these three in the general communication style. The theoretical analysis of the generalcase remains an open problem.

CONFLICTS OF INTEREST
The authors declare they have no conflicts of interest.

ACKNOWLEDGMENTS
The authors are much indebted to M. Yoshimura for his invaluablecomments.

REFERENCES
[1]
[2]
[3]

[4]
[5]

S. Ikeda, T. Ito, M. Sakamoto, An efficient structure of organization,
International Interdisciplinary Workshop on Robotics, Ecosystem,and Management, Yamaguchi, Japan, 2009, pp. 72–78.
S. Ikeda, T. Ito, M. Sakamoto, Discovering the efficient organization structure: horizontal versus vertical. Artif. Life Robot. 15(2011), 478–481.
S. Ikeda, T. Ito, M. Sakamoto, A mathematical model of hierarchical organization. International Conference on Pure Mathematics
and Applied Mathematics (PM-AM2015), Vienna, Austria, 2015,
pp. 423–432.
S. Ikeda, T. Ito, M. Sakamoto, T. Ito, An evaluation model of internal and external contributions in hierarchical organizations, Int.
J. Math. Mod. Methods Appl. Sci. 9 (2015), 636–645.
M. Yoshimura, T. Ito, M. Sakamoto, T. Ito, Y. Shinano, S. Ikeda,
An efficient structure of organization with complete individualguidance. Proceedings of the 2018 International Conference on
Artificial Life and Robotics (ICAROB2018), Beppu, Oita, Japan,
2018, pp. 264–267.

S. Ikeda et al. / Journal of Robotics, Networking and Artificial Life 6(2) 128–132

132Authors Introduction
Dr. Satoshi Ikeda

Dr. Takao Ito
He received PhD degree from Hiroshima
University. He is an Associate Professor inthe Faculty of Engineering, University ofMiyazaki. His research interests include
graph theory, probabilistic algorithm,
fractal geometry and measure theory.

Dr. Makoto Sakamoto
He received the PhD degree in Computer
Science and Systems Engineering fromYamaguchi University in 1999. He is presently an Associate Professor in the Facultyof Engineering, University of Miyazaki.
His first interests lay in hydrodynamics
and time series analysis, especially thedirectional wave spectrum. His current
main research interests are automata theory, languages andcomputation.

He is Professor of Management ofTechnology (MOT) in Graduate Schoolof Engineering at Hiroshima University.
He is serving concurrently as Professor ofHarbin Institute of Technology (Weihai)
China. He has published numerous papers
in refereed journals and proceedings, particularly in the area of management science, and computer science. He has published more than eight
academic books including a book on Network Organizations
and Information (Japanese Edition). His current research interests include automata theory, artificial intelligence, systems control, quantitative analysis of inter-firm relationships using graphtheory, and engineering approach of organizational structuresusing complex systems theory.