=Paper= {{Paper |id=Vol-3909/Paper_30.pdf |storemode=property |title=Structuring Management Tasks in the Telecommunication Network Management System |pdfUrl=https://ceur-ws.org/Vol-3909/Paper_30.pdf |volume=Vol-3909 |authors=Yuri Samokhvalov,Eduard Bovda,Sergey Liubarskyi |dblpUrl=https://dblp.org/rec/conf/iti2/SamokhvalovBL24 }} ==Structuring Management Tasks in the Telecommunication Network Management System== https://ceur-ws.org/Vol-3909/Paper_30.pdf

Structuring management tasks in the telecommunication
network management system
Yuriy Samokhvalov1,2,*, , Eduard Bovda2, , Sergey Liubarskyi2,
1
Taras Shevchenko National University, Volodymyrska Street 64/13, Kyiv, 01601, Ukraine
2
Military Institute of Telecommunications and Informatization named after Heroes of Kruty, Knyaziv Ostrozkyh Street 45/1,
Kyiv, 01011, Ukraine

Abstract
One of the possible approaches to the distribution of tasks between the management levels of the
telecommunication network management system has been proposed. The issues of analysis of management
tasks on the basis of ordering information flows and management goals are considered. It is shown that the
construction of organizational structures of telecommunication network management systems should be
based on a general methodological basis, representing management activities that take place in time and
space. The procedure for detecting strongly related tasks, which corresponds to the task of constructing
rational spheres of activity. Algorithms for vertical and horizontal structuring of network management
tasks are proposed on the basis of analysis of a set of features that describe the features of the initial data
and interaction of officials of the network management body. This makes it possible to solve the issue of
the distribution of tasks in the management system of the telecommunication network, taking into account
their interconnections.

Keywords
Telecommunication network management system, management tasks, task structuring, vertical
(hierarchical) structuring, horizontal (group) structuring, organizational structure.

1. Introduction
When creating a telecommunication network management system (TNMS), the question arises of
choosing a rational version of its structure. Its solution is based on the use of the principles of
structuring and consistency of tasks and the structure of the management system.
Problem analysis is carried out in two main ways. Initially, the information flows of the
management system are streamlined on the basis of rational aggregation of management procedures
with their simultaneous inclusion in the list of works performed by individual departments or
officials of the management system. Then management processes are formed based on the
construction of the structure of the goals of the management body of the system. The structure of
goals is directly and directly related to the operational aspects of management and has the form of a
tree.
Structuring operational tasks creates the best conditions for achieving the goals of system
management. It is carried out on the basis of minimizing the amount of duplicate information
received for processing by each official or individual structural unit.
The issues of mutual coordination of the tasks solved by the management system and its
organizational structure have been considered in many studies. Thus, in the works [1, 2] it is
shown that if there is a management function, then there must be a corresponding unit that

Information Technology and Implementation (IT&I-2024), November 20-21, 2024, Kyiv, Ukraine
Corresponding author.
These authors contributed equally.
yu1953@ukr.net (Y. Samokhvalov); edepig8305@ukr.net (E. Bovda); lubarsky550@gmail.com (S. Liubarskyi)
0000-0001-5123-1288 (Y. Samokhvalov); 0000-0002-8267-2120 (E. Bovda); 0000-0001-8068-1106 (S. Liubarskyi)
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).

CEUR
Workshop
ceur-ws.org 375
ISSN 1613-0073
Proceedings
performs it. In most cases, the functions of management systems are defined through the
organizational structure, and the organizational structure is defined through functions.
The paper [3] argues that management processes are "stratified" by time levels and concentrated
"horizontally" (by time intervals), and not "vertically" (by functions). Therefore, the time level can be
a sign of specialization of the operational unit of the governing body. In [4, 5] it is shown that when
determining the lower level of aggregation of management tasks solved in organizational structures,
it is necessary to proceed from a common goal: the tasks that are constituent parts of ensuring the
implementation of the general goal should become the basis for the distribution of the organizational
structure's activities between its individual structural units. Consequently, a necessary property of
these problems is the relative closure of their connections within a set of problems, provided that
their external connections are of much less importance. On the other hand, the complexity of solving
these tasks should correspond to the average capabilities of specialists of certain professional and
qualification groups officials of the TNMS management body (MB TNMS). And in the paper [6] it
is shown that in management systems the allocation of possible signs of specialization of units or
individual officials should be based on a general methodological basis, representing purposeful
managerial activity that takes place in time and space. Based on this, the analysis of interrelations of
tasks should be carried out on the basis of the totality and mutual influence of functional, temporal
and spatial features, taking into account the features of the tasks solved in the MB TNMS.
Currently, there is no comprehensive approach to the mutual coordination of tasks solved by the
management system and its organizational structure. The absence of such mechanisms is explained,
firstly, by the lack of empirical experience in classifying various kinds of operational tasks and their
distribution according to the levels of complexity of the decisions made; secondly, insufficient
development of approaches to the rational distribution of decision-making efforts by management
bodies; thirdly, the emerging complexity of the synthesis of the hierarchy of management tasks and
the micro-level structure of management bodies due to the lack of an unambiguous correspondence
between the hierarchies of goals of tasks and the organizational structure [5, 6, 7]. In addition, as a
rule, the hierarchy of tasks is more complex than the hierarchy of management bodies. Moreover,
the hierarchy of tasks (goals) of the management body can be built both from the bottom up and
from the top down by specifying the tasks of a higher level.
The article considers one of the possible approaches to solving the issues of distribution of tasks
in the TNMS management system, taking into account the links between them.

2. Structuring of management tasks
Based on the general methodology of designing organizational structures of management systems,
the structuring of tasks should meet two criteria: 1) the set of tasks should reflect the nature of their
representation in the set of cause-and-effect relationships and correspond to the form of the
hierarchical organization of the management system; 2) tasks should not violate the integrity of its
organizational structure and should provide the necessary degree of purposefulness of its
functioning.
According to the first criterion, structuring procedures should ensure their decomposition in such
a way that they would be presented in the form of a hierarchical multi-level structure. Reflecting the
nature of the top-down relationships between groups of tasks, such a structure can be considered as
a hierarchical structure of subdivisions of different levels of the hierarchy [8].
With regard to the second criterion, it can be noted that within the framework of structuring
tasks, tasks are grouped at one horizontal level between two or more subjects. The groupings of tasks
obtained in this way will determine the potential areas of activity of the MB units that are at the
same level of the hierarchy. Figure 1 shows a variant of the decomposition of tasks X between the
levels of the hierarchy of the control system.

376
XN XN-1 Level 1
Potential Scope of Level 2
Officials

XN-2 XN-3 XN-4 Level 2
R(XN-3,XN-4)

Possible options for horizontal
structuring of tasks

R(X1,X2) R(X3,X4)

X1 X2 X6 Level 3
X3 X4 X5

R(X4,X5)

Figure 1: Example of vertical and horizontal structuring of tasks
In this figure, the 1st level is the highest, the 2nd level is the middle level of management, and the
3rd level of management is the lower level of management. Structural functions R(X1, X2), R(X3, X4),
R(X4, X5) determine the degree of mutual proximity of tasks X1 and X2, X3 and X4, X4 and X5, at the
lower level of control, and structural function R(XN-3, XN-4), which determines the degree of mutual
proximity of tasks XN-3 and XN-4 in the middle management. Tasks X6, XN-2, XN-1, XN do not have a
degree of mutual proximity to other tasks at their levels of control, so they do not have a structural
function. The dotted line shows the scope of activity of level 2 officials, and the solid line shows the
options for grouping close tasks for the subsystem (subgroup, subdivision).
The presence of intersecting connections between tasks (for example, in Fig. 1 these are the links
between XN-2 and XN, as well as between XN-3 and XN, XN-2 and XN-1), which form the basis of the
elements of the MB activity, often leads to the appearance of cycles, which excludes their vertical
structuring. The presence of cycles indicates the existence of a set of strongly connected tasks that
are resolved in the course of cyclical exchange of information (in the process of internal dialogue).
Such tasks cannot be separated within a given level of decomposition of the group's activities. They
are considered as one task (as one complex element of activity), and if it is necessary to study a more
subtle structure of activity, their analysis is carried out separately.
Identification of strongly related tasks can be done by the following procedure:
On the set of problems 𝑋, a certain structural function 𝑅(𝑥𝑖 , 𝑥𝑗 ) is calculated, which determines
the degree of their mutual proximity. The function is defined in the range from 0 to 1.
The step of varying the parameters of structuring tasks 𝛥𝑅 is selected. It determines the
discreteness of the choice of structural parameters of tasks, as well as options for organizing the
organizational structure.
The concept of "structuring step" is introduced. 𝑍 (𝑍 = 1, 2, 3, . .. ).
1, 𝐿; 𝐿 = |𝑋|;) − the number of types of control
On the set of values of the function 𝑅(𝑥𝑖 , 𝑥𝑗 ) (𝑖, 𝑗 = ̅̅̅̅̅
problems), a discrete function is formed:𝑅𝑍 (𝑥𝑖 , 𝑥𝑗 )

𝑅(𝑥𝑖 , 𝑥𝑗 ), 𝑅(𝑥𝑖 , 𝑥𝑗 ) ≥ 𝑍𝛥𝑅
𝑅𝑍 (𝑥𝑖 , 𝑥𝑗 ) = {
0, ,
and a graph 𝐺𝑍 (𝑋, 𝑌𝑍 ) of interrelations of the vertex of the graph is constructed (corrected),
X − and the arcs of the graph 𝑌𝑍 determine the directional connections of the problems equal to:

𝑌(𝑥𝑖 , 𝑥𝑗 ), 𝑅(𝑥𝑖 , 𝑥𝑗 ) ≥ 1 − 𝑍𝛥𝑅;
𝑌𝑍 (𝑥𝑖 , 𝑥𝑗 ) = { .
0,

377
When the parameter 𝑍 is changed, the structure of the graph 𝐺𝑍 (𝑋, 𝑌𝑍 ) of mutual relations of
tasks will also change at the 𝑍th step of structuring. By changing 𝑍 from 𝑍𝑚𝑖𝑛 to 𝑍𝑚𝑎𝑥 it is possible
to generate all the options for structuring them that are acceptable for a given set of tasks on the
basis of such a procedure.
Thus, on the set of vectors ordered in this way, for each variant of the definition of the discrete
coupling function, the problems 𝑅𝑍 (𝑥𝑖 , 𝑥𝑗 ) of vertical and horizontal structuring are sequentially
weighed. Algorithms of vertical and then horizontal structuring are applied to the resulting
graph 𝐺𝑍 (𝑋, 𝑌𝑍 ).

3. Algorithm of vertical structuring of management tasks
As a result of decomposition and elimination of intersecting relationships between problems, the
structure of control problems will be represented by a directed graph, in which strongly connected
subgraphs are represented by separate vertices. Therefore, an important property of the
graph 𝐺𝑍 (𝑋, 𝑌𝑍 ) is its acyclicity, which actually indicates the absence of contradictions between
control problems. It is also true for the graph 𝐺𝑍 (𝑋, 𝑌𝑍 ):

1, 𝑅𝑍 (𝑥𝑖 , 𝑥𝑗 ) ≥ (1 − 𝑍𝛥𝑅);
𝑌𝑍 (𝑥𝑖 , 𝑥𝑗 ) = {
0, 𝑅𝑍 (𝑥𝑖 , 𝑥𝑗 ) < (1 − 𝑍𝛥𝑅).
In the graph obtained in this way 𝐺𝑍 (𝑋, 𝑌𝑍 ), the internal hierarchy of tasks is presented implicitly.
The complexity of the hierarchical representation of such a graph lies in the fact that the presence
of cross-connections between tasks leads to the fact that its structure can be represented in the form
of a graph only with a predominantly hierarchical order, and not as "pure trees".
The paper [6] proposes a heuristic method for transforming a graph 𝐺𝑍 (𝑋, 𝑌𝑍 ) into its
corresponding tree. Its essence is as follows. It is necessary to select in the graph the vertex that
corresponds to the global goal (for a given task or group of tasks) and place it at the first (upper)
level of the hierarchy; then select the sub-goals and place them on the next second level of the
hierarchy; then, for each sub-goal of the second level, select those of them, which is a prerequisite
for achieving the sub-goals of the second level and placing them on the third level, and so on, until
those sub-goals that are kind of primary remain at the lower levels of the hierarchy. The resulting
tree is redundant.
The disadvantages of this graph transformation method include the complexity of its automated
implementation. In addition, this method gives an excessive structure of the hierarchical distribution
of tasks, in this respect it is not optimal, does not take into account the multidimensionality of the
target guidelines of the organizational structure and the functional differences in the tasks of the
activity of a group of operators in the process of system management. With this in mind, a different
approach to the analysis of the hierarchy of tasks solved by the organizational structure in
management systems is proposed.
Namely, the tasks of the management system structure, which are directly related to the
operational aspects of management, express the goals of management at different levels of the
hierarchy of its organization. The need for their successful solution in a dynamically changing
external environment is different. The level of this need actually reflects the degree of influence of a
particular sub-goal on the achievement of the global goal of the unit. Thus, in general terms, each
task should be associated with a certain quantitative value that determines the usefulness of its error-
free and timely solution in the process of activity. The inverse value of this parameter determines
the disorder in achieving the necessary criterion for the activity of an official (or a group of persons)
at a given level of the hierarchy of the management system structure 𝑥𝑖 𝑣𝑖 .
Defining the heterogeneity in the achievement of the goal, which is expressed by the results of
solving management problems, as a measure of divergence of any selected parameter 𝑣𝑖 in relation
to the standard of order 𝑣𝑖𝑒 , it should be noted that the goal of the activity of any official of the
management system, if it is not specifically motivated, is to achieve an equilibrium state of
378
information flows coming from the outside (i.e., operational tasks) and the flow of solved (in the
sense of accuracy and timeliness) tasks. Hence, it can be assumed that in the absence of motivational
differentiation of tasks, their weight, felt subjectively by any official of the management group, will
be determined by the necessary intensity of their solution. In this case, the disorder of the official's
activity in achieving his partial goal can be assessed by the extent of the tasks not performed by him,
due to his functional duties.
Let the structure of the problems be represented by a graph 𝐺𝑍 (𝑋, 𝑌𝑍 ) and a discrete weight
function 𝑣𝑖 (𝑥𝑖 ) is given on the set 𝑋, which characterizes, in the general case, the complexity and
importance of the timely and error-free solution of i th control problem. Then the problem of
determining the rational hierarchy of the system of control problems can be represented as the
problem of determining the ordinal function of an acyclic graph.
To do this, we define the subsets 𝑋1 , 𝑋2 , . . . , 𝑋𝑠 :

𝑋1 = {𝑥𝑖 |𝑥𝑖 ∈ 𝑋, 𝑧 −1 𝑥𝑖 = ∅};
𝑋2 = {𝑥𝑖 |𝑥𝑖 ∈ 𝑋 − 𝑋1 , 𝑧 −1 𝑥𝑖 ∈ 𝑋1 };
𝑋3 = {𝑥𝑖 |𝑥𝑖 ∈ 𝑋 − (𝑋1 ∪ 𝑋2 ), 𝑧 −1 𝑥𝑖 ∈ 𝑋1 ∪ 𝑋2 }; (1)
𝑠−1 −1 𝑠−1
𝑋𝑠 = {𝑥𝑖 𝑖 ∈ 𝑋 − ⋃𝑘=0 𝑋𝑘 , 𝑧 𝑥𝑖 ∈ ⋃𝑘=0 𝑋𝑘 } ,
|𝑥
where s is the smallest number of levels in the hierarchy that 𝑧 −1 𝑥𝑠 = ∅;
𝑧 −1 𝑥𝑖 means crossing out the vertices 𝑥𝑖 ∈ 𝑋𝑘 of the graph 𝐺𝑍 (𝑋, 𝑌𝑍 ) .
The subsets (𝑋𝑘 𝑘 = 1, 𝑠) form a partition of the graph 𝐺𝑍 (𝑋, 𝑌𝑍 ), which is ordered by the relation:
𝑋𝑘 − 𝑋𝑘 ′ <=> 𝑘 < 𝑘 ′ .
The set of vertices of a graph 𝐺𝑍 (𝑋, 𝑌𝑍 ) is divided into non-intersecting subsets, which are ordered
in such a way that if the vertex of the graph belongs to a subset with number 𝑘, then the vertex
following it is included in the subset with a number greater than 𝑘.
The subsets 𝑋𝑘 form hierarchical levels of tasks that are solved in the organizational structure.
Each 𝑘 th task of the th level (𝑘 = 1, 𝑠) is associated with a corresponding tree of its subtasks, the
results of which are considered as initial data. Obviously, the higher the level of the hierarchy of
tasks of the subset 𝑋𝑘 , the higher the level of training of officials who solve them. Therefore, each
subset of tasks 𝑋 is associated with certain requirements for the qualification characteristics of
officials, and, consequently, the cost characteristics of their professional selection, training and
maintenance.
Thus, the sequence of determining the ordinal function of a graph is as follows:
1. The adjacency matrix 𝐴 = ‖𝑎𝑖𝑗 ‖ of the graph 𝐺𝑍 (𝑋, 𝑌𝑍 ),

1, 𝑖 𝑗
𝑎𝑖𝑗 = {
0,
1
2. The first line of the matrix of weights is calculated v

𝑣 1 = {𝑣𝑗1 : 𝑗 = 1, 𝐿}, 𝑣𝑗1 = ∑𝐿𝑖=1(𝑣𝑖 )𝑎𝑖𝑗 , 𝑖 = 1, 𝐿, (2)
where 𝑣𝑗1 − is the weight of the 𝑗-task, taking into account the sum of the weights of the "subordinate"
tasks.
3. The set of tasks of the 1st level of the hierarchy is determined

𝑋𝑖 = {𝑥𝑗 |𝑣𝑗1 ≤ 𝑣э }, 𝑥𝑗 ∈ 𝑋, (3)
x
That is, the 1st level of the hierarchy includes tasks j whose weight is less than the established
threshold 𝑣е .
4. Vertices are crossed 𝑥𝑗 ∈ 𝑋𝑖 out from the graph 𝐺𝑍 (𝑋, 𝑌𝑍 ) and the adjacency matrix is corrected
(columns and rows corresponding to the vertex 𝑥𝑗 are crossed out).
379
5. Paragraphs 2 4 are performed for the following levels of the hierarchy
𝑘 = 2, . . . , 𝑠 𝐺𝑍 (𝑋, 𝑌𝑍 ), taking into account the correction of the graph and adjacency matrix:
𝑣 𝑘 = {𝑣𝑗𝑘 }, 𝑋𝑘 = {𝑥𝑗 |𝑣𝑗𝑘 ≤ 𝑣е𝑘 }, 𝑥𝑗 ∈ 𝑋 − ⋃𝑘 𝑋𝑘 .
6. The breakdown of the original set 𝑋 ends if all its elements are distributed at the appropriate
levels of the hierarchy, i.e.
𝒗𝒌 = ∅ .
The belonging of a task 𝑥𝑗 to the hierarchical level is determined either by its own weight and the
corresponding requirements for the qualifications of the official, or by the total weight of
"subordinate" tasks solved at the lower levels of the hierarchy of the organizational structure.
Let's consider an example of vertical structuring of tasks. According to the above sequence of
determining the ordinal function of a graph at the first step of structuring, we determine the matrix
of adjacency of problems and connections when an arc goes from the i-th vertex to the j-th vertex
(Fig. 2).

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11
- S1
x1 1 X10
x2 1 1
x3 1
x4 1 1 1 1 X9 - S2
x5 1 1 1 1 1
x6 1 1 1 1 1
x7 1 1 1 1 1 - S3
X8
x8 1 1 1 1
x9 1 1 1 1 1 1 1 1 1
x10 1 1 1 1 1 1 1 1
x11 X7 - S4

Figure 2: Boolean matrix of mutual relations of problems (Z=0)
X6 - S5
L0 6 5 6 4 4 3 3 2 1 0 10
L1 6 5 5 3 3 2 2 1 0 x 9
L2 5 4 4 2 2 1 1 0 x 8 8
L3 4 3 4 2 2 1 0 x x x 7 X5 - S6

L4 4 3 3 1 1 0 x x x x 6
L5 3 2 2 1 0 x x x x x 5
L6 2 1 1 0 x x x x x x 4 X4 - S7
L7 1 0 0 x x x x x x x 3
L8 0 x x x x x x x x x 1 X3 X2 - S8
L9 x x x x x x x x x x 0

X1 - S9
Figure 3: Sequence of calculation of the matrix of weights (a) b)
and vertical structure of interrelations of problems (b)

Next, we carry out the calculation according to (2), the above sequence of determining the ordinal
function of the graph, the first line of the matrix of weights 𝑣 1 (Fig. 3a). Determine according to (3)
the set of tasks of the 1st level of the hierarchy. From the graph, 𝐺𝑍 (𝑋, 𝑌𝑍 ) which is built as a result
of vertical structuring of problems, we cross out the vertices 𝑥𝑗 ∈ 𝑋𝑖 and adjust the adjacency matrix
(columns and rows corresponding to the vertex 𝑥𝑗 are crossed out). Steps 2 4 are performed for the
next levels of the hierarchy 𝑘 = 2, . . . , 𝑠, taking into account the correction of the graph
𝐺𝑍 (𝑋, 𝑌𝑍 ) and matrix adjacency. The breakdown of the original set 𝑋 ends if all its elements are
distributed at the appropriate levels of the hierarchy. As a result, we obtain a vertical structure of
interrelations of tasks (Fig. 3b).
380
In the presence of a significant number of tasks, cycles may appear in their execution. The
presence of cycles indicates a violation of the correctness of the choice of elements of activity, and
the presence of strongly related tasks solved on the basis of mutual exchange of information.
Strongly related tasks are taken into account separately they are considered as one task and special
algorithms are used to identify them [6-9].

4. Algorithm of horizontal structuring of management tasks
We will assume that as a result of the vertical structuring of tasks, the levels of the hierarchy 𝑘 =
1, . . . , 𝑠 are allocated. Consider a finite set of problems 𝑋𝑘 ∈ 𝑋. On this set, a real function
(𝑅𝑍 (𝑥𝑖 , 𝑥𝑗 )𝑥𝑖 , 𝑥𝑗 ∈ 𝑋𝑘 ) is given with the properties:
𝑅𝑍 (𝑥𝑖 , 𝑥𝑗 ) ≥ 0, 𝑅𝑍 (𝑥𝑖 , 𝑥𝑗 ) = 𝑅𝑍 (𝑥𝑗 , 𝑥𝑖 ), ∀𝑥𝑖 , 𝑥𝑗 ∈ 𝑋 .
Horizontal structuring of management tasks involves the division of the set (𝑋𝑘 ) of tasks of each 𝑘
level of the hierarchy into a given number of groups of tasks with maximum internal connections.
The groups of tasks found in this way will determine the rational spheres of activity of individual
officials at each 𝑘 level of the hierarchy of the organizational structure of the group.
The considered problem of horizontal structuring of elements of activity belongs to the class of
tasks of automatic classification. Automatic classification algorithms can be represented by three
large groups [6, 7, 10, 15]: heuristic algorithms; variational algorithms; algorithms related to the
problem of mixture separation (statistical algorithms). Selection This or that algorithm for
structuring tasks is carried out on the basis of an analysis of a set of features that describe the features
of the initial data and the features necessary for the organization of interaction of officials, which
characterize the properties of the final breakdown of tasks [6, 10].
According to a set of features that characterize the features of the initial data (the number of tasks
to be classified; the dimension of the set of problem relations; the type of restrictions on the number
of problems in the class, etc.). Taking this into account, the choice of such an algorithm can
effectively be made, for example, by one of the methods [16, 17], which are modifications of the
method of hierarchy analysis in cases of single or group selection. The result of the analysis shows
that the classification algorithm described in the works [6, 10] takes into account these features to
the greatest extent [6, 10].
For its formal description, consider the following definitions.
Definition 1.
A set 𝐵 (𝐵 = 𝑋𝑘 ) containing more than one element is called a group if for any elements 𝑥𝑖 and
𝑥𝑗 that set there exists a sequence 𝑐1 , 𝑐2 , . . . , 𝑐𝐿𝑘 , where 𝑐𝑖 ∈ 𝐵, 𝑐1 = 𝑥𝑖 , 𝑐𝐿𝑘 = 𝑥𝑗 (𝐿𝑘 = |𝑋|) such that

𝑚𝑖𝑛𝑅(𝑐𝑖 , 𝑐𝑖+1 ) > 𝑚𝑎𝑥 (𝑑, 𝑙) (4)
𝑐𝑖 ∈𝐵 𝑑∈𝐵
𝑙∈𝑋𝑘 \𝐵

Definition 2. Let 𝑎, 𝑏 ∈ 𝑋𝑘 . Let's 𝑏 call it 𝑎 neighbor if

𝑚𝑎𝑥 𝑅(𝑎, 𝑐𝑖 ) > 𝑅(𝑎, 𝑏) , ie.𝑎 → 𝑏 (5)
𝑐𝑖 ∈𝑋𝑘 \𝑎
Definition 2 implies an obvious property of a group: any element is included in a group along with
its neighbors.
Definition 3. A set 𝐷 (𝐷 = Хк ) is called a pseudogroup if any element of that set is included in it
along with all its neighbors.
Let us denote 𝐹𝑘 by the set of all possible partitions 𝑋𝑘 into pairwise non-intersecting
pseudogroups. On a set 𝐹𝑘 , partial ordering can naturally be introduced. Let 𝐹1 , 𝐹2 ∈ 𝐹𝑘 . According
to definition 3, is 𝐹1 preceded by 𝐹2 , if any element 𝐹1 can be represented as a union of elements
with 𝐹2 . In this set of breakdowns, there is a minimum that holds only one element 𝑋𝑘 − the plural
itself. It is proved [12, 14] that the maximum division of a set 𝐹𝑘 is unique, that is, there is a single

381
division of this set into pairwise non-intersecting sets that have the following property: if a and b
belong to one of these subsets, then they can be connected by a chain of the form
𝑎 → 𝑐1 → 𝑐2 →. . . → 𝑐𝑚 → 𝑏, and if not, then such a sequence cannot be constructed. This
breakdown is a breakdown into pseudogroups.
The maximum element of the set 𝐹𝑘 is called the base partition. Let us denote the set of elements
of the basic breakdown by 𝑋𝑘1 and define the function 𝑅1 (𝐷, С) on it as follows:
𝑅(𝑎, 𝑏), ≠
𝑎∈ 𝑏∈𝐶
𝑅1 ( )={
𝑅(𝑎, 𝑏), =
𝑎∈ 𝑏∈𝐷\𝑎
Definition 4.
Let 𝐷 ∈ 𝑋𝑘1 . We will call an element С a neighbor of 𝐷 if

𝑚𝑎𝑥 𝑅(𝐷, Е) = 𝑅(𝐷, С) , ie.𝐷 → С (6)
Е∈𝑋𝑘1
Next, you can enter the definition of the pseudogroup and show that the definitions formulated
earlier will be valid.
Let us denote 𝑄𝑘 in terms of the set 𝑋𝑘 all possible divisions of the set into pairwise non-
intersecting groups. It is true that 𝑄𝑘 in order for a set 𝑋𝑘 to be divided into groups, it must
correspond to the division of the set 𝑋𝑘1 into pseudogroups. The proof of the statement is given in
[13, 15].
It follows from the statement that any division of a set 𝑋𝑘 into groups can be represented as a
union of elements with 𝐹1 (𝐹1 − the maximum division of the set 𝑋𝑘1 into pseudogroups). If we denote
𝑋𝑘2 = 𝐹2 and introduce on the set 𝑋𝑘2 a measure of proximity 𝑅2 (𝐿, 𝑀) = 𝑚𝑎𝑥 𝑅1 (𝐷, С), then with
respect to the set 𝑋𝑘2 all the statements that for 𝑋𝑘1 . Similar conclusions can be drawn with respect
to any intermediate set 𝑋𝑘𝑍∗ .
This process ends when each of the following two solutions is present:
- at some step 𝑍 ∗, a set with a single element 𝑋𝑘𝑍∗ containing all the elements of the set 𝑋𝑘 is
obtained, i.e. the breakdown of tasks of a given level of the hierarchy is impossible by formal methods
and it is necessary either to use heuristic procedures or to change the structure of the initial data:
𝑋𝑘𝑍∗ = 𝑋𝑘 ; (7)
- with some 𝑍 ∗ > 2 𝑋𝑘𝑍∗ = 𝑋𝑘𝑍 ∗ −1 𝑋𝑘𝑍∗ and does not consist of any element. In this case, the
elements of the set 𝑋𝑘𝑍∗ −1 are pseudogroups, and the set 𝑋𝑘𝑍∗ −1 corresponds to the division of the
original set 𝑋𝑘 into groups:
𝑋𝑘𝑍∗ = 𝑋𝑘𝑍 ∗ −1. (8)
Thus, the algorithm for horizontal grouping of management tasks can be represented as follows:
1. The original set 𝑋𝑘 and the function 𝑅𝑍 (𝑥𝑖 , 𝑥𝑗 ) of the interconnection of the problems𝑥𝑖 , 𝑥𝑗 ∈
𝑋𝑘 are written. We get the graph 𝐺𝑍 (𝑋, 𝑌𝑍 ).
2. According to expression (4), for each element of the set 𝑋𝑘 , neighbor elements are determined
and pseudogroups of the first (𝑍 ∗ = 1) of the basic breakdown 𝐷𝑙 ∈ 𝑋𝑘1 are written.
3. Condition (7) is checked. If the elements of the set 𝑋𝑘 make up one pseudogroup, then
partitioning by formal methods is not possible.
4. On the set 𝑋𝑘1 , according to expression (5), a function of the degree of mutual proximity
between pseudogroups, is formed 𝑅(𝐷𝑖 , 𝐷𝑗 )𝐷𝑖 , 𝐷𝑗 ∈ 𝑋𝑘1 .
5. According to expressions (6) and (7), for each element 𝐷𝑙 of the set 𝑋𝑘1, neighbor elements are
determined and pseudogroups of the second (𝑍 ∗ = 2) of the basic division are written.
6. Condition (8) is checked. If it is not fulfilled, then paragraphs 4 - 8 are repeated for the third
∗
(𝑍 = 3) and so on breakdowns.
7. If condition (8) is satisfied, then the set 𝑋𝑘𝑍∗ corresponds to the maximum breakdown and it is
necessary to proceed to the horizontal grouping of the remaining tasks of the hierarchy levels 𝑋𝑘 ,
𝑘 = 1, 𝑠. With each step 𝑍 ∗ , the breakdown of the initial set of tasks at a given level of the

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organizational structure hierarchy is associated with checking the feasibility of including a group of
officials in the organizational structure 𝑚𝑘𝑍∗ = |𝑋𝑘𝑍∗ | at this level.
Let's look at an example of horizontal structuring of tasks.
Let 𝑋 = (𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 , 𝑥5 , 𝑥6 , 𝑥7 , 𝑥8 , 𝑥9 , 𝑥10 , 𝑥11 ) be the set of problems that are solved in the
network management body. Figure 4 shows a graph 𝐺𝑍 (𝑋, 𝑌𝑍 ) of mutual relations of problems, and
in Figure 5 − matrix of measures of their proximity.

X11

X10
X1

X9
X2

X8
X3

X5
X6

Figure 4: Graph of mutual relations of problems

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11
x1 1 1/7 1/14 1/13 1/13 1/8 1/10 1
x2 1 1/14 1/6 1/5 4/5 1/4 1
x3 1 1/2 1/14 1/16 1/3 1/13 1/2 1
x4 1 1/13 1/2 1/13 1/3 1
x5 1 4/5 1/13 1/7 1/2 1
x6 1 1/14 1/3 1
x7 1 1/9 1/11 1/2 1
x8 1 1/4 1/11 1
x9 1 1/10 1
x10 1 1
x11 1

( )
Figure 5: Matrix R x i , x j of measures of proximity of tasks
In accordance with the algorithm of horizontal grouping of management tasks, we will perform
the procedure of horizontal structuring of tasks. To do this, we will use the algorithm for calculating
the ordinal function of the graph. Let 𝑅 = 0.2. By sequentially changing the structuring step 𝑍 =
0 … 5, the following structures of graphs of mutual relations of problems are obtained (Table 1).
The table shows the six steps of structuring tasks (𝑍 = 0 … 5). At each of these steps, according
to the above algorithms, the options for vertical and horizontal structuring are shown. The range of
changes in the values of the hierarchy obtained on this structure 𝑆 = 2 ÷ 10.

383
Table 1.
Structure of graphs of mutual relations of problems
Step Structures Generated by Vertical Options for horizontal structuring of
Number Structuring of Tasks tasks
x1
𝑆 = 10 𝑚1 = 𝑚2 = 𝑚3 = 𝑚4 = 𝑚5 = 𝑚6 = 𝑚7 = 𝑚9
𝑆=9 = 1; 𝑚8 = 1.2
x3 x2 𝑆=8
𝑆=7
x4
𝑆=6
𝑆=5
𝑍=0 x5 x6 𝑆=4
𝑆=3
x8 x7 𝑆=2
𝑆=1
x9 x10 x11

x1 x6
𝑚1 = {1,3}; ;𝐷11 = {𝑥1 , 𝑥9 , 𝑥10 }
x2
𝑍=1 x9 x7 x4 x3 x11 𝑚2 = {1,3}; ;𝐷12 = {𝑥6 , 𝑥7 , 𝑥8 }
x10 x8 x5 𝑚3 = {1,2,3}; ;𝐷13 = {𝑥2 , 𝑥4 , 𝑥5 }
𝑆=5 𝐷23 = {{𝑥2 , 𝑥5 }, 𝑥4 }
𝑚1 = {5,3}; ;𝐷11 = {𝑥1 , 𝑥6 , 𝑥8 , 𝑥9 , 𝑥10 }
x1
x6
x2 ;𝐷21 = {{𝑥1 , 𝑥8 , 𝑥9 }, 𝑥6 , 𝑥10 }
𝑍=2 x5 x4 x3 x11
x8
x7
𝑚2 = {3,2}; ;𝐷12 = {𝑥2 , 𝑥5 , 𝑥7 }
x9
x10
𝐷22 = {{𝑥2 , 𝑥5 }, 𝑥7 }
𝑆=5

x1 x3 x4 x6 x7 x8 x9 x10
𝑚1 = {8,3,2};
𝐷11 = {𝑥1 , 𝑥3 , 𝑥4 , 𝑥6 , 𝑥7 , 𝑥8 , 𝑥9 , 𝑥10 };
𝑍=3 x2 x5 𝐷31 = {{𝑥1 , 𝑥8 }, {𝑥9 , 𝑥4 , 𝑥7 , 𝑥10 }, {𝑥9 , 𝑥6 }};
𝑆=3 𝑚2 = {2,1};
x11
𝐷 = {{𝑥1 , 𝑥6 , 𝑥8 , 𝑥9 }, {𝑥3 , 𝑥4 , 𝑥7 , 𝑥10 }}
x1 x3 x4 x6 x7 x8 x9 x10 𝑚1 = {8,3,2};
𝐷11 = {𝑥1 , 𝑥3 , 𝑥4 , 𝑥6 , 𝑥7 , 𝑥8 , 𝑥9 , 𝑥10 };
𝑍=4 x2 x5 𝐷31 = {{𝑥1 , 𝑥8 }, {𝑥9 , 𝑥4 , 𝑥7 , 𝑥10 }, {𝑥9 , 𝑥6 }};
𝑆=3 𝑚2 = {2,1};
x11
𝐷 = {{𝑥1 , 𝑥6 , 𝑥8 , 𝑥9 }, {𝑥3 , 𝑥4 , 𝑥7 , 𝑥10 }}
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
𝑚1 = {10,3,2};
𝑍=5 𝐷11 = {𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 , 𝑥5 , 𝑥6 , 𝑥7 , 𝑥8 , 𝑥9 , 𝑥10 };
𝑆=2 𝐷21 = {{𝑥1 , 𝑥2 , 𝑥8 }, {𝑥10 , 𝑥3 , 𝑥7 , 𝑥4 }, {𝑥9 , 𝑥5 , 𝑥6 }};
x11
𝐷31 = {{𝑥1 , 𝑥2 , 𝑥5 , 𝑥6 , 𝑥8 , 𝑥9 }, {𝑥3 , 𝑥4 , 𝑥7 , 𝑥10 }}

A variant of horizontal structuring of tasks has been obtained. Next, you need to define the
neighboring elements of the set and write down the pseudogroups of problems of the first basic
breakdown𝑋𝑘 . To do this, we solve the problem of horizontal structuring of the 5th variant of the
breakdown, the conditions of which are fulfilled at 𝑍 = 5. The hierarchy of vertical structuring of
the spheres of activity of officials in this variant has two levels 𝑆 = 2.
As can be seen from Table 1, all the tasks that support the group's activities are focused on the
first level. The matrix of interrelations of the tasks of the first level is presented in Fig. 6.
Using the algorithm of horizontal structuring of control problems, we obtain the first basic
breakdown of the original set N1 :

Areas of activity of Level 1 officials:
a) 𝑥11 = {𝑥1 , 𝑥2 , 𝑥8 };
b) 𝑥21 = {𝑥10 , 𝑥3 , 𝑥7 , 𝑥4 };
c) 𝑥31 = {𝑥9 , 𝑥5 , 𝑥6 }.
𝑁11 = {𝑥11 , 𝑥21 , 𝑥31 }

In particular, starting with the problem 𝑥1 , we successively obtain the set of "neighbor" problems
formed by it {𝑥1 → 𝑥2 ↔ 𝑥8 } = 𝐷1. Continuing the process, we select the sets 𝐷2 = {𝑥10 , 𝑥3 , 𝑥7 , 𝑥4 }
and 𝐷3 = {𝑥9 , 𝑥5 , 𝑥6 }. 𝐷1, 𝐷2 and 𝐷3 form pseudo-groups of problems of the first basic breakdown
of the first stage. Each pseudo-group characterizes the sphere of activity of one official at the 𝑘 = 1
level of the hierarchy of the organizational structure. Thus, for 𝑍 ∗ = 1 𝑚1 = |𝑁11 | = 3.
384
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
x1 1 1/7 1/14 1/13 1/13 1/8 1/10
x2 1 1/14 1/6 1/5 4/5 1/4
x3 1 1/2 1/14 1/16 1/3 1/13 1/2
x4 1 1/13 1/2 1/13 1/3
x5 1 4/5 1/13 1/7 1/2
x6 1 1/14 1/3
x7 1 1/9 1/11 1/2
x8 1 1/4 1/11
x9 1 1/10
x10 1

Figure 6: Matrix of measures of proximity of tasks of the first level of the hierarchy

Pseudogroups 𝐷1, 𝐷2, 𝐷3 are elements of the set 𝑁11 . A function can also be built on this set
𝑅(𝑥𝑖1 , 𝑥𝑗1 ).
Areas of activity of Level 1 officials:
a) 𝑥12 = {𝑥1 , 𝑥2 , 𝑥5 , 𝑥6 , 𝑥8 , 𝑥9 };
b) 𝑥22 = {𝑥3 , 𝑥4 , 𝑥7 , 𝑥10 }.
𝑁11 = {𝑥12 , 𝑥22 }

Pseudogroups of the basic division of the set 𝑁1 look like this: 𝐷11 = {𝑥11 , 𝑥31 }, 𝐷21 = 𝑥21 . Pseudo-
groups 𝐷11 and 𝐷21 will form spheres of activity for two officials of the 1st level of the group hierarchy.
On the set 𝑥12 = 𝐷11, 𝑥22 = 𝐷21, let us construct the function 𝑅2 (𝑥12 , 𝑥22 ). As a result of the analysis
of the obtained set 𝑁12 = {𝐷11 , 𝐷21 }, we find the following pseudogroups of the basic division of the
third stage: 𝑁13 = {𝑎12 , 𝑎22 }, i.e. 𝑁13 = 𝑁12 . From this it follows that individual problems of a set of
problems 𝑁1 will form two groups:
𝐷12 = {𝑥1 , 𝑥2 , 𝑥5 , 𝑥6 , 𝑥8 , 𝑥9 }. 𝐷22 = {𝑥3 , 𝑥4 , 𝑥7 , 𝑥10 }
Similar calculations are performed for all variants of the previous stage of vertical structuring of
tasks.

Conclusions
An approach to the distribution of tasks between the levels of management of a telecommunication
network is considered. An algorithm for vertical structuring of tasks in TNMS control systems is
presented. This algorithm makes it possible to effectively distribute tasks by management levels
based on the analysis of information flows and the hierarchy of goals, as well as to synthesize the
organizational structure of the TNMS as a whole.
An algorithm for horizontal structuring of tasks is also provided, which allows automating the
process of forming various options for the spheres of activity of officials in the organizational
structure at each level of the hierarchy. At the same time, the nature of the mutual relations of tasks
is taken into account, which reflect the target orientation of the organizational structure. The
variants of structuring tasks obtained as a result of generation serve as a further oriented basis for
the selection of appropriate structures for the organization of activities of the structural elements of
the TNMS and allow to exclude from consideration the variants of organizational structures that do
not correspond to the structure of tasks that are solved in the management system.

Declaration on Generative AI
The authors have not employed any Generative AI tools.

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