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Citation: Terentyev, A.; Marusin, A.;
Evtyukov, S.; Marusin, A.; Shevtsova,
A.; Zelenov, V. Analytical Model for
Information Flow Management in
Intelligent Transport Systems.
Mathematics 2023,11, 3371. https://
doi.org/10.3390/math11153371
Academic Editors: Aleksandr
Rakhmangulov and Bahram Adrangi
Received: 5 June 2023
Revised: 9 July 2023
Accepted: 25 July 2023
Published: 1 August 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
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4.0/).
mathematics
Article
Analytical Model for Information Flow Management in
Intelligent Transport Systems
Alexey Terentyev 1,*, Alexey Marusin 2 ,3 ,* , Sergey Evtyukov 4, Aleksandr Marusin 3, Anastasia Shevtsova 5
and Vladimir Zelenov 6
1Department of Vehicles, St. Petersburg State University of Architecture and Civil Engineering,
190005 St. Petersburg, Russia
2Department of Technical Operation of Vehicles, St. Petersburg State University of Architecture and Civil
Engineering, 190005 St. Petersburg, Russia
3Department of Transportation of the Academy of Engineering, RUDN University (Peoples’ Friendship
University of Russia Named after Patrice Lumumba), 117198 Moscow, Russia; 89271333424@mail.ru
4
Department of Ground Transport and Technological Machines, St. Petersburg State University of Architecture
and Civil Engineering, 190005 St. Petersburg, Russia; s.a.evt@mail.ru
5
Department of Operation and Organization of Vehicle Traffic, Belgorod State Technological University Named
after V.G. Shukhov, 308012 Belgorod, Russia; shevcova-anastasiya@mail.ru
6Engineering Center, Plekhanov Russian University of Economics, 117997 Moscow, Russia;
vazelenov@gmail.com
*Correspondence: aleksej.terentev.67@bk.ru (A.T.); 89312555919@mail.ru (A.M.)
Abstract:
The performance of this study involves the use of the zoning method based on the principle
of the hierarchical relationship between probabilities. This paper proposes an analytical model
allowing for the design of information and analysis platforms in intelligent transport systems. The
proposed model uses a synthesis of methods for managing complex systems’ structural dynamics
and solves the problem of achieving the optimal balance between the information situations existing
for the object and the subject under analysis. A series of principles are formulated that govern the
mathematical modeling of information and analysis platforms. Specifically, these include the use of
an object-oriented approach to forming the information space of possible decisions and the division
into levels and subsystems based on the principles of technology homogeneity and information state
heterogeneity. Using the proposed approach, an information and analysis platform is developed for
sustainable transportation system management, that allows for the objective, multivariate forecasting-
based record of changes in the system’s variables over time for a particular process, and where
decision-making simulation models can be adjusted in relation to a particular process based on an
information situation existing for a particular process within a complex transport system. The study
demonstrates a mathematical model that solves the optimal balance problem in organizationally
and technically complex management systems and is based on vector optimization techniques
for the most optimal decision-making management. The analysis involves classical mathematical
functions with an unlimited number of variables including traffic volume, cargo turnover, safety
status, environmental performance, and related variables associated with the movement of objects
within a transport network. The study has produced a routing protocol prescribing the optimal
vehicle trajectories within an organizationally and technically complex system exposed to a substantial
number of external factors of uncertain nature.
Keywords:
information commutation technologies; intelligent transport systems; Pareto optimal solu-
tion; vector optimization techniques; multicriterion problem; mathematical decision-making models
MSC: 05-08; 15-11; 90-05; 90-08; 91-08; 91-10
Mathematics 2023,11, 3371. https://doi.org/10.3390/math11153371 https://www.mdpi.com/journal/mathematics
Mathematics 2023,11, 3371 2 of 16
1. Introduction
Among the features that distinguish intelligent transport systems from the conven-
tional regional and national transport systems is sustainability. System sustainability is
defined as the ability to increase (maintain) the required level of performance or provide
the most optimal performance conditions in the event of deterioration (change) in external
or internal factors. The current cargo transport systems can be characterized as a technically
and organizationally complex, unsustainable system with multiple levels. One of the
ways to make this system more sustainable is by managing its structural dynamics, which
involves a series of control actions to ensure the transition from the currently restructured
state to a synthesized multistructural macrostate.
Efforts to achieve the target level of sustainability require the introduction of the
management practice of the theory of complex systems into the transport systems—more so
because the complexity of interactions between the elements within the systems under study
is constantly increasing, requiring new methods for big data processing and analysis
[1–8]
.
In modern conditions, the practice of ITS control and management should be able to
take into account a large number of factors, including those of a cognitive nature [
9
–
13
].
Solutions to these challenges lie beyond the framework of systems theory, so the modeling
of a complex system is always about compromising between model simplicity and system
complexity. In practice, compromise decisions are often taken based on a whole range of
performance indicators, i.e., in conditions with multiple criteria [14–20].
In many cases, when a multicriterion problem is reduced to one single criterion within
a complex system, this criterion becomes the only (core) one that governs the process of
management optimization and identification of the optimal decision, in which case all
remaining performance indicators acquire upward or downward limitations, with the
deficiency in one criterion being compensated at the expense of another. In this case, the
obtained solution can be considered acceptable, yet not objective.
Our comparative analysis of the existing methods shows that they differ in their effective-
ness of determining the optimal duration required for maintaining the core quality, or group
of qualities, in conditions of dynamically evolving changes in the external environment.
According to Laplace’s principle of insufficient reason, an action can be considered
optimal if the subjectivity of its criterion results from the decision maker knowing the
possible states of the world but completely lacking the information about the plausibility of
each single state. At the same time, the principle of equal distribution builds rather on the
knowledge that some outcomes may not have a greater objective possibility of occurrence
than others, than on the unawareness of whether they have such an objective possibility or
not in comparison with others.
According to Wald’s precautionary principle (maximin criterion), the optimal action is
that which has the highest value of the effectiveness indicator for the worst-case state of
the external environment.
The precaution is the basic principle also in the Savage minimax risk criterion. Ac-
cording to the Savage criterion, one should opt for an action with the lowest worst-case
scenario risk.
The criteria proposed by A. Wald and L. Savage are subjective because they are a
priori biased towards worst-case scenarios, which makes them suitable only for idealized
decision making. However, the states of external environment exist objectively, regardless
of the choices made.
Therefore, in general, there is no reason for extreme pessimism in decision making. A
more balanced approach uses the stability criterion proposed by A. Hurwitz, in which the
evaluation function is a middle ground between the extremes posed by the optimist and
pessimist criteria. This criterion is a derivative of more traditional criteria. In the Hurwitz
rule (pessimist–optimist criterion), it is unreasonable to ignore the highest possible gain
and consider only the lower one. A certain coefficient should be introduced into decision
making. When this coefficient equals the absolute optimist criterion at 0 and the Wald’s
maximin criterion at 1, the action can be considered optimal. When this coefficient is greater
Mathematics 2023,11, 3371 3 of 16
than zero and less than one, it represents a mixture of pessimistic and extremely optimistic
outcomes of future actions.
Thus, in the case of complex systems, there can be only one decision that can guarantee
the highest possible effectiveness that would fully meet all criteria. While all existing meth-
ods do enable adequate decision making, they do not ensure their maximum effectiveness.
The method proposed herein aims to achieve the most efficient decision making within the
boundaries of the system under study.
2. Materials and Methods
When deciding on the appropriateness of a particular method for performance eval-
uation purposes or for identifying the most optimal action scenario within an intelligent
transport system, one should first determine the complexity class of the system under
analysis and which methodology best fits its complexity class. One landmark study in the
field of complex systems theory (CST) is by Sayama and states that CST incorporates a
whole range of methodologies including general systems theory (GST). CST is therefore
more general in relation to GST and classical systems analysis. The CST structure according
to [20] is shown in Figure 1.
Mathematics 2023, 11, x FOR PEER REVIEW 3 of 16
is greater than zero and less than one, it represents a mixture of pessimistic and extremely
optimistic outcomes of future actions.
Thus, in the case of complex systems, there can be only one decision that can guar-
antee the highest possible effectiveness that would fully meet all criteria. While all existing
methods do enable adequate decision making, they do not ensure their maximum effec-
tiveness. The method proposed herein aims to achieve the most efficient decision making
within the boundaries of the system under study.
2. Materials and Methods
When deciding on the appropriateness of a particular method for performance eval-
uation purposes or for identifying the most optimal action scenario within an intelligent
transport system, one should first determine the complexity class of the system under
analysis and which methodology best fits its complexity class. One landmark study in the
field of complex systems theory (CST) is by Sayama and states that CST incorporates a
whole range of methodologies including general systems theory (GST). CST is therefore
more general in relation to GST and classical systems analysis. The CST structure accord-
ing to [20] is shown in Figure 1.
Figure 1. The structure of complex systems theory [21].
As can be seen from the structure above, CST comprises elements of emergence [21]
and self-organization [22], and is closely related to game theory, collective behavior the-
ory, distributed systems (mass service systems) theory, adaptation and evolution, nonlin-
ear dynamics theory, structural modeling, and general systems theory. It is our opinion
that this structure lacks one important element—game theory of nature, where “nature”
defines the nature of the behavior factors inherent in the external environment under
study [23]. The relevance of the games theory of nature in CST is explained by the fact that
it allows for analyzing a complex system’s reliability and susceptibility to the impact of
external factors, and Hiroki Sayama himself describes his complex system model as sus-
ceptive, valid, and reliable, noting that these important characteristics are missing in the
classical GST. In this case, “susceptibility” is defined as a cognitive factor [24] and is some-
times replaced by the term “simplification”. But, simplification is more of an arbitrary
concept, as it depends on the choice of the effectiveness evaluation criteria. The term “va-
lidity” is sometimes used in reference to complex systems to denote the quality of the
consistency of the obtained information state with the expected and the actual behavior
of a complex system. “Reliability” defines a complex system’s sensitivity to external influ-
ences. In other words, a system’s decision regions must display sufficient stability when
exposed to certain spectra of external disturbances. If minor external disturbances (influ-
ences) have no effect on the decision taken toward actions that are aimed at increasing the
complex system’s efficiency, the model of such systems can be considered reliable.
While being a complex system, the ITS is classified as a dynamic system. In CST,
dynamic systems theory (DST) occupies a special place, as can be judged from its defini-
Complex systems theory
(self-organization)
Game
theory
Collective
behavior GST
Nonlinear
dynamics
Structural
modeling
Evolution and
adaptation
Distribution
systems
Theory of games with nature
Game theory of nature
Figure 1. The structure of complex systems theory [21].
As can be seen from the structure above, CST comprises elements of emergence [
21
]
and self-organization [
22
], and is closely related to game theory, collective behavior theory,
distributed systems (mass service systems) theory, adaptation and evolution, nonlinear
dynamics theory, structural modeling, and general systems theory. It is our opinion that
this structure lacks one important element—game theory of nature, where “nature” defines
the nature of the behavior factors inherent in the external environment under study [
23
].
The relevance of the games theory of nature in CST is explained by the fact that it allows
for analyzing a complex system’s reliability and susceptibility to the impact of external
factors, and Hiroki Sayama himself describes his complex system model as susceptive,
valid, and reliable, noting that these important characteristics are missing in the classical
GST. In this case, “susceptibility” is defined as a cognitive factor [
24
] and is sometimes
replaced by the term “simplification”. But, simplification is more of an arbitrary concept,
as it depends on the choice of the effectiveness evaluation criteria. The term “validity” is
sometimes used in reference to complex systems to denote the quality of the consistency
of the obtained information state with the expected and the actual behavior of a complex
system. “Reliability” defines a complex system’s sensitivity to external influences. In other
words, a system’s decision regions must display sufficient stability when exposed to certain
spectra of external disturbances. If minor external disturbances (influences) have no effect
on the decision taken toward actions that are aimed at increasing the complex system’s
efficiency, the model of such systems can be considered reliable.
While being a complex system, the ITS is classified as a dynamic system. In CST,
dynamic systems theory (DST) occupies a special place, as can be judged from its definition.
In the dynamic systems theory, a system’s state is characterized by a set of predetermined
laws intended to effect change in the system’s parameters [
25
]. While this definition
Mathematics 2023,11, 3371 4 of 16
cannot be considered comprehensive and sufficiently clear, it reflects the basic principle
of the existence and development of complex dynamic systems. It is only natural that
dynamic systems evolve not only through intended modifications, but also through their
self-organization ability. Being a property of complex systems, self-organization is missing
only in rigidly determined technical systems, a class to which ITSs can clearly not be
attributed. Figure 2shows the structure of a complex system with features inherent in
intelligent transport systems [21–29].
Mathematics 2023, 11, x FOR PEER REVIEW 4 of 16
tion. In the dynamic systems theory, a system’s state is characterized by a set of predeter-
mined laws intended to effect change in the system’s parameters [25]. While this definition
cannot be considered comprehensive and sufficiently clear, it reflects the basic principle
of the existence and development of complex dynamic systems. It is only natural that dy-
namic systems evolve not only through intended modifications, but also through their
self-organization ability. Being a property of complex systems, self-organization is miss-
ing only in rigidly determined technical systems, a class to which ITSs can clearly not be
aributed. Figure 2 shows the structure of a complex system with features inherent in
intelligent transport systems [21–29].
Figure 2. The example of an ITS-integrated complex system [21].
As can be seen from Figure 2, the ITS has all the features essential for a complex sys-
tem, one being the high number of elements and interconnections. Like sociotechnical sys-
tems [26], organizational and technological systems [27], technological systems, and com-
plex social systems [28], ITS is classified as a complex system in the definition given to it
by GST and CST. And like any complex system, the ITS must have the ability to handle
big data [29,30]. Therefore, when modeling an ITS and its optimization principles, it is
necessary to
1. Ensure that the ITS is presented simply, with all its dominant features, i.e., there
should be a balance between the description of complexity and the simplicity of mod-
eling.
2. Give theoretical form to the complexity of ITS, using as a basis the system’s infor-
mation states, which, in turn, depend on possible internal and external disturbances.
3. Identify divisibility criteria with due account of the heterogeneity of elements within
the ITS [31].
4. Design the tools for managing and optimizing ITS performance using the existing
decision-making methodologies. This is particularly important from the perspective
of the system management processes, as their level of complexity is growing steadily,
requiring new, CST-based models.
3. Theoretical Studies
The ambiguity with regard to quantifying the effectiveness of various decision-mak-
ing management models has been addressed in quite a number of studies, evidencing the
need for the further search for more objective decision-making models what would meet
the modern requirements to complex systems and their interconnections within road
transport systems [32–35]. The need for more in-depth, system-based theoretical research
into the processes of complex transportation systems management is stated in all studies
that we mentioned earlier. That said, these studies appear to pay lile aention to the
Complex
system
Complex application system
Complex technical system
Complex sociotechnical system
Complex organizational and technical system
Complex social system
Complex applied system
Figure 2. The example of an ITS-integrated complex system [21].
As can be seen from Figure 2, the ITS has all the features essential for a complex
system, one being the high number of elements and interconnections. Like sociotechnical
systems [
26
], organizational and technological systems [
27
], technological systems, and
complex social systems [
28
], ITS is classified as a complex system in the definition given to
it by GST and CST. And like any complex system, the ITS must have the ability to handle
big data [
29
,
30
]. Therefore, when modeling an ITS and its optimization principles, it is
necessary to
1.
Ensure that the ITS is presented simply, with all its dominant features, i.e., there should
be a balance between the description of complexity and the simplicity of modeling.
2.
Give theoretical form to the complexity of ITS, using as a basis the system’s informa-
tion states, which, in turn, depend on possible internal and external disturbances.
3.
Identify divisibility criteria with due account of the heterogeneity of elements within
the ITS [31].
4.
Design the tools for managing and optimizing ITS performance using the existing
decision-making methodologies. This is particularly important from the perspective
of the system management processes, as their level of complexity is growing steadily,
requiring new, CST-based models.
3. Theoretical Studies
The ambiguity with regard to quantifying the effectiveness of various decision-making
management models has been addressed in quite a number of studies, evidencing the
need for the further search for more objective decision-making models what would meet
the modern requirements to complex systems and their interconnections within road
transport systems [
32
–
35
]. The need for more in-depth, system-based theoretical research
into the processes of complex transportation systems management is stated in all studies
that we mentioned earlier. That said, these studies appear to pay little attention to the
mathematical modeling of the processes under study and to present their outcomes as
objectively verified mathematical algorithms [
36
,
37
]. The algorithmization of processes
is key to the transition to digital management models and further evolution of software
products [
38
]. This prompts a conclusion that streamlined data (databases) is a prerequisite
for implementing multi-level, hierarchical systems for evaluating the efficiency of road
Mathematics 2023,11, 3371 5 of 16
container transport, reinforcing the relevance of the systems approach and its underlying
principles. It is through the systems approach that the need for the improved coherence
of data on the system’s heterogeneous properties can be validated, with the process of
streamlining representing the “methodology” itself and determining the form and degree
of the formalization of the outcomes.
By revealing the essence of the systems approach and its use in various fields of knowl-
edge, we help ourselves to answer questions as to what this given process is about, what
algorithm it uses, and, most importantly, what element serves as the object of management.
In our case, the object of management is the multitude of road container transport systems,
as well as the processes responsible for their efficient management.
The problem of systematizing the numerous indicators of road transportation perfor-
mance is not new, and the need for specific solutions is only growing. Concluding what
has been said above, there is a need for applied mathematical methods, especially those
that use discrete mathematics and the theory of information interaction. These two fields of
mathematics offer tools for formulating answers to the questions set above, guaranteeing
the objectivity and the effectiveness of research. Let us use the theory of sets to show the
essence and the algorithm of one of the main types of systematization—ordering, which is
represented as a classification process within a particular field of knowledge.
Having noted the importance of classification in creating and improving the thesaurus,
let us formulate a mathematically precise and, consequently, exact statement of the problem
of the performance of classification. The need for this is due to the presence of multiple
objects, their characteristics, interaction processes, and, hence, the frequent use of ordering,
on the one hand, and the requirements posed to the object of research, on the other. As this
task will arise more often in the future, in each specific case, it will have its own specific
content, which poses the need for an evaluation methodology that would be suitable for
use in any information situation existing for a particular road container transport system.
For the purposes of description, let us take a set X= {X}, the elements of which can be
of diverse nature within the limits of the problem being analyzed. These elements can be
objects or properties, processes, or situations pertaining to system performance.
We will call a finite set {
x1
,
. . .
,
xi
,
. . .
,
xl
} of nonvacuous pairwise disjoint subsets as
a classification of set (X), i.e.,
xi=0, xi∈X,i=1, L(1)
The sets (x) that give the cumulation of all (X):
∪l
i=1xi=X(2)
xi∩xi+1=0, ∨i,i∈i=1, L(3)
The classification procedure is performed based on a property or attribute:
P={P1, . . . , Pi, . . ., Pl}(4)
when the elements of each distinguished subset {
x1
,
. . .
,
xi
,
. . .
,
xl
} possess one of the
varieties (L).
The characteristic of classification Pcan take the form of a set of characteristic proper-
ties
m={m1, . . . , mi, . . . , ml}(5)
or of one particular property with (L) of non-spannable intervals.
The distinguished subsets, called classes, are ordered based on a set of characteristics
(P) and form a “cluster”. The numbering of characteristics is possible through exhaustion
and formal search for each individual representative of the “cluster” of classes.
Mathematics 2023,11, 3371 6 of 16
The above description of a single-level classification does not embrace the whole
variety of cases of ordering. More often than not, the number of characteristics and their
varieties predetermines the need for multilevel structure of classification.
Let us consider a simply ordered set of characteristics:
P={P1, . . . , Pv, . . . , PL}(6)
where each Pyhas varieties Py(1), . . . , Pv(n).
Once the set of characteristics and their varieties have been brought into order, we
can “distract” from their semantic content and replace them, for classification purposes, by
ordered numerical sets of indexes, or identifiers:
i={1, . . . , v, . . . , L}(7)
or of their varieties, if the classification of characteristics is multilevel:
{(1)v, . . . , (n)v}(8)
Fixing values
i∗
1, . . . , i∗
v
as v-initial characteristics and the values of the remaining
(iv+1, . . . , iL)
as variables, we obtain a formalized, multilevel structure of classes that are
connected into a hierarchy and can be presented as the “tree of classes”.
∑L
v=1∏v
x=1nx=Sr(9)
Given the total number of characteristics, this tree will have (L) levels in the hierarchy
of the system under study:
∏v
x=1nx=sr(10)
with (n) classes at the (v) level and with nv−1classes subordinate to each particular level.
One important property of the proposed method is the possibility of determining
the set of effective plans (Pareto sets) and thus allowing the decision-making process
to remain focused solely on the most expedient scenarios and ignore less competitive
ones. [
39
,
40
]. In other words, using these methods provides an objective opportunity for
subjective decision making [
21
,
40
]. Generally, the method for determining the Pareto set in
multicriterion problems does not claim to prescribe one certain action, but rather aims to
eliminate uncertainty, identify critical parameters, and provide requirements to improved
information on such critical parameters (performance criteria) [
21
,
41
]. This reveals a
strong correlation between the determination of the Pareto set and decision making with
insufficient information, both aiming to achieve the maximum possible elimination of
uncertainty and to improve the information on the probability of states occurring within
the environment to which the method is applied [
21
,
40
,
42
]. Therefore, in order to enhance
the reliability of the decision making in multicriterion problems, it is expedient to create an
analytical tool based on vector optimization methods for obtaining the Pareto set, which
would allow for the most optimal decision making under multicriterion conditions where
the information on the state of the external environment is minimal.
The drawback of the proposed method consists in the fact that the capacity of a
system’s object can be determined based solely on empirical data. This drawback can
be eliminated with the use of analytical methods for obtaining the values forming the
Pareto set, i.e., the multicriterion problem methods where sets of possible environment
states are classified according to the principle of the hierarchical relationship between the
probabilities of their occurrence (zoning methods).
The main principles of the zoning method rely on hierarchical relationships between
the probabilities of possible environmental states (ESs) and are as follows:
1.
Since zoning represents an inverse parametric problem of linear programming, it is
expedient that zoning is performed based on the principle of maintaining a preset
Mathematics 2023,11, 3371 7 of 16
hierarchical relationship between all possible environmental states, not according to
the dominant effect principle.
2.
When dealing with “game with nature”-related problems, it is expedient to use vector
optimization techniques, and many multicriterion problems can generally be solved
using the tools of game theory of nature. When passing from a multicriterion problem
to a “game with nature”, the probabilities of nature states
pj
are coincident with
relative significance coefficients for criteria cj, i.e., pj≡cj.
3.
The procedure for zoning that uses hierarchical relationships between the proba-
bilities of possible environment states is determined by manifestations of the ESs
under analysis.
Let us form an effectiveness matrix of possible actions under various environmental
states. Any task relating to decision-making optimization is characterized by three basic
concepts—a set of candidate decisions; a set of environmental state types; and the efficiency
of a proposed decision under each environment state. Here and elsewhere, we shall use the
following designations:
•m—number of possible action scenarios;
•n—number of possible environmental states or criteria that correspond to them;
•aij —effectiveness of i-th action for j-th criterion, i= 1, m,j= 1, n.
Then, the effectiveness matrix of action scenarios under various ESs shall have
the form:
||ai j|| =
a11 a12 . . . a1n
a21 a22 . . . a2n
. . . . . . . . . . . .
am1am2. . . amn
(11)
It has been found in studies [21,41–44] that there is a relationship between the Pareto
set in multicriterion tasks and the zoning method for solving “games with nature”-related
problems under uncertainty [
45
–
48
]. Provided that the effectiveness function of the differen-
tial state vector is continuous, the matrix game with nature can be reduced to a linear vector
optimization problem [
49
–
52
]. The distribution of the effectiveness relative significance
coefficients is constrained by
0≤cj≤1, j=1, n,∑n
j=1cj=1, (12)
i.e., it is determined by a set (n−1) of independent values.
Every possible complete set of ES distributions corresponds to a distribution field
presented as a rectangular hypertetrahedron in dimensional space (
m−
1) [
21
,
53
–
55
]. In
the Cartesian coordinate system
(c1, c2. . . , cn−1)
, this hypertetrahedron is a product of the
intersection of a positive hyperoctant by a hyperplane that cuts on each coordinate axis a
segment equal to unity [56–60].
∑n−1
j=1cj=1 (13)
Let us arrange the values of the coefficients cjinto a sequence [21,46,61–67]:
c1≥c2≥. . . ≥cj≥. . . ≥cn−1≥cn(14)
The total of the sequences of this type for system allocations is determined by the
number of permutations Pn=n!:
•
With
n=
3, the distribution of the field of relative significance coefficients degenerates
into a right triangle with ordinary sides (Figure 3). The number of subsets, each having
its own relative significance ratio, equals P3=3! =6;
•
With
n=
4 (Figure 4), the number of subsets, each having its own relative significance
ratio, equals P4=4! =24.
Mathematics 2023,11, 3371 8 of 16
Mathematics 2023, 11, x FOR PEER REVIEW 8 of 16
Figure 3. The distribution field for coefficients C
, 𝑃
=3!=6.
Figure 4. The distribution field for coefficients C
, 𝑃
=4!=24.
Let us analyze Figure 3. Table 1 presents the side and median equations for triangle
ABC, each of the six subsets has been assigned with its own distribution of relative signif-
icance coefficients, as shown in Table 2 [40–42].
Figure 3. The distribution field for coefficients Cj,P3=3! =6.
Mathematics 2023, 11, x FOR PEER REVIEW 8 of 16
Figure 3. The distribution field for coefficients C
, 𝑃
=3!=6.
Figure 4. The distribution field for coefficients C
, 𝑃
=4!=24.
Let us analyze Figure 3. Table 1 presents the side and median equations for triangle
ABC, each of the six subsets has been assigned with its own distribution of relative signif-
icance coefficients, as shown in Table 2 [40–42].
Figure 4. The distribution field for coefficients Ci,P4=4! =24.
Let us analyze Figure 3. Table 1presents the side and median equations for trian-
gle ABC, each of the six subsets has been assigned with its own distribution of relative
significance coefficients, as shown in Table 2[40–42].
Table 1. Side and median equations for triangle ABC.
Triangle Segments Segment Equations
AB side c2+c3=1; c1=0
AC side c1+c3=1; c2=0
BC side c1+c2=1; c3=0
AE median c1=c2; c1+c2+c3=1
BF median c1=c3; c1+c2+c3=1
CD median c2=c3; c1+c2+c3=1
Mathematics 2023,11, 3371 9 of 16
Table 2. The geometric field of distribution of relative significance coefficients.
Subset Triangle Coefficients Ratio
I AOD c1<c2<c3
II DOB c1<c3<c2
III BOE c3<c1<c2
IV EOC c3<c2<c1
V COF c2<c3<c1
VI FOA c2<c1<c3
Triangle ABC reflects the system-described distribution of coefficients. As can be
seen from Table 2, each of the six subsets has been assigned with its own distribution of
relative significance coefficients [
68
–
73
]. For example, all possible solutions of the system
of equations and inequations
0≤cj≤1; j=1, 2, 3; c1+c2+c3=1; c3≤c2≤c1(15)
are in subset IV, i.e., within triangle EOC, and point (O) has coordinates
c1=c2=c3=1/3. (16)
4. Results
The proposed approach to zoning modeling has enabled the following algorithm for
finding the maximum possible variant of the desired solution or of system boundaries:
1.
The relative significance of indicators
Cj
, or their corresponding criteria, will be
arranged as a sequence (14);
2. For each comparable variant i, there is a linear programming problem:
(Di=∑n
j=1aij cj→max,
∑n
j=1cj=1, 0 ≤cj≤1, cj≥cj+1,j=1, n−1(17)
3. The values of the relative significance coefficients will be determined analytically:
cj=1
k, if j≤k,
0, if j>k,(18)
or to increase solution’s sensitivity to optimization parameters:
cj=
1
k, if j=k
λ
k, if j<k, where λ=n−1
n.
1−λ
n−k, if j>k
(19)
where kis defined by akj =max
jaij .
The proposed mathematical model can serve as a tool for transferring a transport
system to a sustainable state, as well as a mechanism for managing the structural dynamics
responsible for a system’s transition from a currently restructured state to a synthesized
multistructured macrostate [
40
,
42
]. Thus, a method is in place for developing the transport
efficiency management algorithms in intelligent transport systems [21,40–42].
In the present-day ITS, the management processes are characterized by highly dynamic
changes within its operating environment; the dynamic evolution of the system’s process
parameters; the high degree of uncertainty of information state; and the existence of a
large number of logic criteria and possible solutions [
40
–
42
]. Therefore, to enhance the
management efficiency in ITS, it is logical to switch from the traditional subject-oriented
(scenario development) methods to object-oriented analytical models, the latter allowing for
management automation through the use of the decision-making theory and its analytical
Mathematics 2023,11, 3371 10 of 16
techniques [
40
]. The proposed analytical solution for achieving a sustainable state through
the transport system is facilitated through the use of vector optimization techniques,
which allow an extremum problem to embrace multiple criteria and an unlimited number
of indicators [
42
]. The proposed zoning-based mechanism for managing the structural
dynamics of the objects within ITS allows for the decision making to take place under
uncertainty, specifically, when
1.
The study has no clearly defined quantitative or qualitative characteristics of its target;
2.
The object of the study has not received thorough analysis at the stage of investigating
the phenomena accompanying the system’s performance; or
3.
The external environment causes no counteraction to system parameters or the process
under analysis.
In such problems, the choice of a decision often depends on the state of the “nature”
of factors, and their mathematical models are called “games with nature”. Given this, the
desired effective decisions can be obtained using the proposed method.
The quantified effectiveness of the proposed method, as compared to those of other
methods, is presented in Table 3.
Table 3. Method effectiveness comparison table.
Decision-Making Method Solution Variant Quantified Effectiveness
Wald criterion 1 0.200
Savage criterion 2 0.660
Hurwitz criterion 3 (4) 0.366 (0.676)
Laplace criterion 4 0.4975
Fishburne sequences 4 0.5034
Proposed method 4 0.8400
The proposed solutions have a high potential for facilitating the further introduction
of information commutation and digital technologies into the modern reality of transport
systems operation, as they allow for the parallel processing of multiple “inputs” and “out-
puts” within intelligent transport systems and ensure adequate amounts, or a “database”,
of optimization parameters.
The digital mechanisms that are currently used in the transport sector have a high
potential for employing the proposed method for more effective decision making and
process optimization. Possible applications include intersection traffic safety analysis and
signaled crossing safety assurance [
74
,
75
]; hardware and software packages for measuring
driver reaction time in road accidents investigation [
76
]; and transport safety assurance with
the use of intelligent driver assistance systems [
77
]. The proposed method will prove useful
also in modeling the transport infrastructures for modern cities with growing efficiency;
measuring the effectiveness of automated road accident scene sketching based on data
from a mobile device camera; analysis of road safety, the reliability of the sustainability
criteria for urban passenger transport, and route-optimization-based mechanisms for
improving the safety of cargo transportation in urban agglomerations; forecasting the
levels of energy consumption and greenhouse gas emissions from vehicles; introducing
pedestrian early warning systems into intelligent transport system infrastructures; traffic
accident risk analysis in conditions of urban traffic demand change; and designing a man–
machine interface for self-driving vehicles with account of the time needed to take back
control [78–89].
Since the proposed model is suitable for use in problems with an unlimited number of
system inputs (indicators), it is possible, and advisable, to include in databases not only
vehicle performance (mileage, volume of transportation, cargo turnover, etc.), but also the
performance of cargo-handling facilities (terminals).
Mathematics 2023,11, 3371 11 of 16
5. Discussion
The introduction of the information commutation, or digital, technologies into the
modern reality of transport systems operation allows for the parallel processing of multiple
“inputs” and “outputs” within intelligent transport systems [
74
–
79
] and ensures adequate
amounts, or a “database”, of optimization parameters [
80
–
89
]. Using the proposed an-
alytical model, it is possible to equip the ITS management system with an information
analysis platform that builds on a synthesis of methods for managing the complex systems’
structural dynamics and allows for optimal correspondence of the information situation to
the decision making. In this model, the formation of the candidate decisions space and the
division of the information situation into levels and subsystems based on the principles
of technology homogeneity and information state heterogeneity, are achieved through an
object-oriented approach. In this way, the essence is revealed of the process of assigning
to the general ITS model the specific contents that meet all pre-set conditions, economic
feasibility requirements, and efficiency standards. It should be noted here that the proposed
ITS modeling principles have a dual nature:
1.
On the one hand, when the task is to provide the state forecast, any object or process
should be considered as an organized, dialectically developing system.
2.
On the other hand, when the task is to analyze this system for structural arrangement,
properties, and internal and external interactions with the environment, a multidi-
mensional study is required – the one that will provide an in-depth knowledge and
description of the system’s current state as a prerequisite of problem solving.
This duality is not a drawback and reinforces the relevance of the systems approach
in dealing with innovative developments. The proposed approach has been used by the
authors to create an analytical model of ITS management that allows the tracing of the
variables over time for a particular process from the perspective of multivariate forecasting.
The decision-making simulation tools used in this model can be modified in relation to a
particular process depending on the information situation existing for the given conditions
of transportation.
When developing a management routing method, the task is to provide not only the
descriptions of the objects and criteria of multicriterion systems, but also methods for
transforming attributes that are essential for deriving structures for more complex system
states and making the management process more flexible and versatile.
The proposed method offers the analytical tools allowing the decision-making process
to be effective under stochastic uncertainty, i.e., in conditions where the information needed
for the choice of the (normal, lognormal, etc.) random variable distribution law is limited.
6. Conclusions
This study has as its main outcome the method that enables the identification of
the most optimal traffic routes in the dynamically changing, multi-criterion context of
commercial operations planning. Specifically, the method allows for the following:
•
Formalizing a transport system with due account of its information states, which, in
turn, are determined by exposure to internal and external disturbances;
•
Identifying a transport system’s criteria that take into account the heterogeneity of
its elements;
•
Achieving the tools for managing and optimizing transport systems’ performance,
that build on the existing decision-making methods and allow the disadvantages
of the heuristic methods used in determining the weighted coefficients of factors to
be avoided.
The proposed analytical method employs a zoning technique that builds on the
hierarchical relationship between probabilities and allows
•Big data in transportation systems to be processed;
•
AI-based analysis of transport systems’ operating environments that involves an
unlimited number of criteria or performance attributes.
Mathematics 2023,11, 3371 12 of 16
Further, the proposed method allows the avoidance of the limitations of the subjective
methods used in the decision-making theory, namely, the methods for
•The a priori ranking of factors (methods based on expert assessments);
•The a priori distribution of probabilities;
•Ensuring guaranteed decision levels.
With complex systems, the number of decisions that can fully meet all the criteria is
always one. While the existing methods do yield competent decision making, they do not
ensure its maximum efficiency. The method proposed herein allows for the most effective
decision making within the boundaries of the system under study.
The proposed method is 23.81% more effective than Wald’s method, 78.57% than
Savage’s, 43.57% than Hurwicz’s, 59.23% than Laplace’s, and 59.93% than Fishburne’s.
The outcomes of this study have been implemented in the research and development
project entitled “Developing the Object-Oriented Management Models and Their Software
Prototypes for Transport and Logistics Systems”. The project has produced an intelligent
transport management system for a domestic logistics provider, that has been tested on a
busy interchange road in one of the districts of St. Petersburg. According to the test results,
the proposed method leads to a 15% increase in cost reduction.
The method’s high degree of reliability is assured by the use of systems analysis and
systems engineering methods in creating the digital technologies implementation concept
for road freight traffic, as well as by the use of vector optimization and linear programming
methods in creating the analytical models for the more optimized design of road freight
traffic management plans. Further, its high reliability is attested by the proprietary soft-
ware product intended for automating the newly developed, centralized transportation
management methods: “The Software for Identifying Optimal Transportation Routes in
Dynamically Changing Road Environments”, authored by Andreev A.Yu., Egorov V.D., Ter-
entyev A.V., Evtyukov S.A. Software Registration Certificate 2021667592, country: Russia,
2021, registration date: 1 November 2021.
The principal advantages of the proposed model for assuring effective decision making in
complex, big-data-based information and analysis systems and software products, consist of:
•
The absence of a formalized relationship between the weighted coefficients obtained
for individual criteria and action options in transport systems;
•
The resultant decision being the maximum possible under the initial values of perfor-
mance indicators for the criteria under consideration;
•
The resultant decision allowing not only the desired Pareto-optimal decisions to be
obtained, but also the number of required computations to be substantially reduced.
The proposed method is unique in the sense that it offers analytical tools, allowing
the decision-making process to be effective under stochastic uncertainty, i.e., in conditions
where the information needed for the choice of random variable distribution laws is limited.
Author Contributions:
Conceptualization, A.T. and A.M. (Alexey Marusin); methodology, A.T. and
A.M. (Alexey Marusin); validation, S.E. and A.M. (Aleksandr Marusin); investigation, A.M. (Alexey
Marusin) and A.M. (Aleksandr Marusin); writing—original draft preparation, A.T. and A.M. (Alexey
Marusin); writing—review and editing, A.T., A.M. (Alexey Marusin), S.E., A.M. (Aleksandr Marusin),
A.S. and V.Z.; visualization, A.M. (Aleksandr Marusin), A.S. and V.Z.; supervision, A.T., A.M. (Alexey
Marusin) and S.E. All authors have read and agreed to the published version of the manuscript.
Funding:
This research was funded by project “Development of mechanisms to improve transport
security using digital technologies”, Grant No. 202234-2-074.
Data Availability Statement:
The data that support the findings of this study are available from the
corresponding author upon reasonable request.
Acknowledgments:
This paper has been supported by the RUDN University Strategic Academic
Leadership Program.
Conflicts of Interest: The authors declare no conflict of interest.
Mathematics 2023,11, 3371 13 of 16
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