=Paper=
{{Paper
|id=Vol-3826/paper10
|storemode=property
|title=Mathematical models and methods for decision coordination in critical infrastructure operations
|pdfUrl=https://ceur-ws.org/Vol-3826/paper10.pdf
|volume=Vol-3826
|authors=Hryhorii Hnatiienko,Oleksii Hnatiienko,Tetiana Babenko,Larysa Myrutenko
|dblpUrl=https://dblp.org/rec/conf/cpits/HnatiienkoHBM24
}}
==Mathematical models and methods for decision coordination in critical infrastructure operations==
https://ceur-ws.org/Vol-3826/paper10.pdf
Mathematical models and methods for decision
coordination in critical infrastructure operations ⋆
Hryhorii Hnatiienko1,†, Oleksii Hnatiienko1,†, Tetiana Babenko1,2,† and Larysa Myrutenko1,*,†
1
Taras Shevchenko National University of Kyiv, 64/13 Volodymyrska str., 01601 Kyiv, Ukraine
2
International Information Technology University, 34/1 Manas str., A15M0E6 Almaty, Kazakhstan
Abstract
This paper considers the problems associated with ensuring the functioning of the critical infrastructure
network. It is proposed to consider a poorly formalized system of ensuring the functioning of critical
infrastructure facilities using expert information processing methods. The urgency of solving the problem
under consideration is confirmed by the massive attacks on critical infrastructure by the Russian troops
during Russia’s large-scale aggression against Ukraine. The paper presents a mathematical model of the
problem of maintenance of a network of critical infrastructure facilities developed by the authors. A scheme
of sequential analysis of options for solving the problem of ensuring the operation of the critical
infrastructure system is proposed. Methods for finding a valid solution to the problem, searching for a
reference solution to the problem, and algorithms for improving the reference solution in various variations
are described. The problem statement and the mathematical model of decision coordination in a three-level
hierarchical system for ensuring the operation of a network of critical infrastructure facilities are also
described. An algorithm for coordinating decisions in a three-level hierarchical system is presented.
Keywords
organizational system, functional stability, critical elements, weighting factors, layering method 1
1. Introduction in particular, it is difficult for subject matter experts to build
metric relations on a set of objects [18, 19].
The possibilities of applying mathematical models and In particular, a person cannot set reliable weighting
decision-making methods to study the problems of coefficients for the relative importance of parameters or
vulnerability, protection, and management of critical criteria [20, 21], expert competence coefficients [22, 23],
infrastructure systems are in the field of view of many elements of metricated pairwise comparison matrices [24,
researchers [1, 2]. This issue has been studied by scientists 25], or build a reasonable reliable membership function
from different countries for many years [3–5] and its using direct methods [26]. Meanwhile, such problems
relevance is not decreasing [6–8]. To date, a large number regularly arise in everyday life and require their solution.
of approaches and mathematical models have been
developed that demonstrate the authors’ attempts to ensure 2. Critical infrastructure facilities
the effective functioning and protection of critical
infrastructure [9–11]. At the same time, the problems of Critical infrastructure facilities are those that are
protecting critical infrastructure from terrorist attacks
remain extremely relevant [12, 13]. This is explained, in Particularly important for the state.
particular, by the fact that the problem of critical Capable of significantly affecting other critical
infrastructure protection is poorly structured, and the infrastructure facilities.
systems that describe the network of critical infrastructure Whose disruption causes a crisis of national
facilities are poorly formalized organizational systems [14, importance.
15]. Vital at the regional level.
In many practical decision-making situations in poorly Whose disruption or malfunction causes a
formalized organizational systems, the decision-maker is crisis of regional, local, or local significance.
forced to act in poorly structured subject areas [16, 17]. To
ensure the quality of decision support in poorly structured Critical infrastructure facilities include enterprises and
subject areas, expert knowledge is traditionally and institutions operating in the following industries:
effectively involved. In addition, building a preference
structure in a formalized form is a difficult task for humans: Energy
CPITS-II 2024: Workshop on Cybersecurity Providing in Information 0000-0002-0465-5018 (H. Hnatiienko);
and Telecommunication Systems II, October 26, 2024, Kyiv, Ukraine 0000-0001-8546-5074 (O. Hnatiienko);
∗
Corresponding author. 0000-0003-1184-9483 (T. Babenko);
†
These authors contributed equally. 0000-0003-1686-261X (L. Myrutenko)
g.gna5@ukr.net (H. Hnatiienko); © 2024 Copyright for this paper by its authors. Use permitted under
Creative Commons License Attribution 4.0 International (CC BY 4.0).
oleksii.hnatiienko@knu.ua (O. Hnatiienko);
babenkot@ua.fm (T. Babenko);
myrutenko.lara@gmail.com (L. Myrutenko)
CEUR
Workshop
ceur-ws.org
ISSN 1613-0073
105
Proceedings
Chemical 𝛽 here 𝑘 is the number of days when the execution of the𝑗
Food is request starts; 𝑚 is the number of types of resources;
Transport 𝛽 , 𝑖 = 1, … , 𝐷 are restrictions on the readiness of the OCI’s
Financial and banking command center for the EWM brigades, which means that
Information technology and telecommuni- no more than 𝑖 is requests can be serviced on a given day,
cations 𝛽 ≤ 𝐵, 𝑖 = 1, … , 𝐷. (3)
Utilities: water, heat, and gas supply It should be noted that requests for activation of the next
Healthcare, etc. I&CS teams are received directly from CI facilities, I&CS
subsystems (SS), or the governing bodies of the I&CS
According to [27], critical infrastructure assets are system.
systems that are essential for maintaining vital social Array of inconsistencies (incompatibilities) of
functions, health, safety, security, and economic or social applications
well-being of people. 𝐶= 𝐶 , 𝑠 = 1, … , 𝑆, ℎ = 1,2, …, (4)
Timely restoration of critical infrastructure facilities where 𝐶 is the number of requests that cannot be
during military operations is particularly important. As a executed simultaneously; 𝑆 is the number of array lines; ℎ
result of unprovoked aggression by Russia, Ukraine faced is the indices of incompatible requests of the 𝑠th line of the
large-scale and targeted attacks on its critical infrastructure unstacked array.
[28]. Attacks were carried out in more than a hundred cities
in Ukraine [29]. 3.2. Scheme for solving the problem of
ensuring the operation of the QMS
3. The task of maintaining a network
The solution to the problem is sought in two stages: building
of critical infrastructure facilities a reference solution and building an optimal solution. To
The maintenance task was considered and studied in [30] as describe the algorithms, we present the necessary
the task of maintaining a network of communication definitions.
elements. Subsequently, the problem statement, approaches Definition 1. A variant of the problem (1)–(4) is a vector
to its solution, and the algorithm for sequential analysis of 𝑣 = (𝑣 , … , 𝑣 ), whose elements are triples𝑣 =
options were adapted to the extremely relevant problem of 𝑣 , 𝑣 , 𝑣 , where 𝑖 is the number of the request; 𝑣 is the
timely and efficient operation of the C&I system. duration of the ith request; 𝑣 is the number of the team that
3.1. Setting the task of ensuring the executes ith the request; 𝑣 is the day the request starts ith
operation of the OCI network the request.
Definition 2. An admissible (complete) variant (solution)
Let 𝑅 = 𝑟 , 𝑖 = 1, … , 𝑁 , 𝑏 = 1, … , 𝐵, 𝑁 ≤ 𝐷, 𝑅 is the of the problem (1)–(4) is the variant 𝑣 = (𝑣 , … , 𝑣 ), that
set of working intervals in a month, 𝑁 is the number of satisfies constraints (2)–(4).
intervals (windows) for the 𝑏 is the team that ensures the Definition 3. A partial solution to problem (1)–(4) is a
operation of the OCI network; 𝐵 is the number of teams, vector, 𝑣 = (𝑣 , … , 𝑣 ), 𝑠 < 𝑛, that satisfies constraints (2)–
∩ 𝑟 , 𝑖 = 1, … , 𝑁 = (in particular, the system of (4).
intervals may coincide for all teams); 𝑟 is determined by Definition 4. A locally valid subvariant 𝑣 =
the beginning 𝜂 and duration 𝛿 , 𝑟 = 𝜂 , 𝛿 , 𝑖 = 𝑣 , 𝑣 , 𝑣 , 𝑙 ∈ {1, … , 𝑛} of problem (1)–(4) is the
1, … , 𝑁 ; 𝐷 number of working days in a month; 𝛼 , 𝑗 = placement of a request in some working interval that
1, … , 𝑛, is the set of requests for activation of the work of satisfies conditions (2)–(4).
Definition 5. An admissible subvariant of problem (2)–(4)
the CMI NMS teams, the duration of which is equal to 𝜏 =
is a locally admissible subvariant that leads to an admissible
𝜏 𝛼 , 1 ≤ 𝜏 ≤ 𝐴; 𝑛 is number of requests; 𝐴 is the variant.
maximum length of the request; 𝑓 = 𝑓 𝛼 , 𝑖 = 1, … , 𝐷, Definition 6. A reference solution to problems (1)–(4) is a
𝑗 = 1, … , 𝑛, is the efficiency of execution of the 𝑗 is request feasible solution (variant) that can, without being optimal,
if it starts to be executed on the 𝑖 th day of the month. Target make the most of resources and at the same time deliver a
function of the task: local optimum to the objective function.
𝐹(𝛼) = 𝑓 𝛼 → 𝑚𝑎𝑥 ,
∈{ ,…, }
(1) 3.3. Method for finding a valid solution to
the problem of ensuring the operation
where 𝑘 is the number of working days in the calendar of the QMS
month on which the 𝑗 is request starts to be executed. Let
The method of finding a feasible solution consists of the
𝑔 = 𝑔 𝛼 is the resource of the 𝑙 is the type required to
sequential construction of a reference solution as a union of
satisfy the 𝑗 is request if it starts executing on the 𝑖 th day, locally feasible subvariants. Therefore, it is reduced to the
𝑙 = 1, … , 𝑚, 𝑖 = 1, … , 𝐷, 𝑗 = 1, … , 𝑛. Then the resource sequential application of the following procedure.
constraints are Procedure for finding a locally admissible subvariant PS.
∑ 𝑔 𝛼 ≤ 𝐺 , 𝑙 = 1, … , 𝑚, (2) The basis of the method is the formation of a set of
admissible subvariants 𝑉 = (𝑣 ), 𝑖 = 1, … , 𝑚 + 2, 𝑙 =
106
1, … , 𝐿, from which a compromise subvariant is selected, 𝐿— There are four options for modifying the system of working
the number of admissible subvariants. The condition for intervals of brigades by changing the interval in which the
generating a sub-variant is, firstly, 𝛿 ≥ 𝛼 , 𝑖 = 1, … , 𝑁 , compromise order is placed:
𝑏 = 1, … , 𝐵, 𝑗 = 1, … , 𝑛, i.e., the application 𝛼 should not 1. The length of the interval 𝑟 , 𝑘 ∈ {1, … , 𝑁 } is equal
exceed the interval in which it is supposed to be placed, and to the length of the bid 𝜏 and the interval is completely
secondly, the compatibility of the current application with excluded from consideration. At the same time, 𝑁 = 𝑁 −
the one already accepted in the partial solution. In parallel 1, 𝑏 = 𝑡 , where 𝑘—is the index of the compromise bid.
to the set of sub-options, a set of indices corresponding to
2. 𝑟 >𝜏 and the compromise application is placed
them is formed, 𝑇 = 𝑡 , ℎ = 1,2,3, 𝑙 = 1, … , 𝐿, where
at the beginning of the interval 𝜂 > 𝑡 . In this case,
𝑡 ≤ 𝑛 is the number of the application that generates the
sub-option, 𝑡 ≤ 𝐵 is the number of the team proposed to
= + 𝑡 , 𝛿 =𝛿 −𝑡 .
execute the application, 𝑡 ≤ 𝐷 is the day the execution of 3. 𝑟 >𝜏 and the compromise bid is placed at the
the application 𝑡 by the team starts 𝑡 . end of the interval. Then = ,𝛿 =𝛿 −𝑡 .
Consider the procedure for forming a valid subvariant 4. 𝑟 ≥ 𝜏 and the placement of the application does
𝑣 = 𝑣 ,…,𝑣 , . For each request 𝛼 , 𝑗 = 𝑡 , all not correspond to any of the three cases. This generates an
possible combinations of its placement in the working additional interval with the index, i.e. 𝑑, 𝑑 = 𝑁 + 1, 𝑏 =
intervals of the teams are selected. At the same time, 𝑣 = 𝑡 , 𝛿 = 𝑡 +𝜏 , 𝛿 = 𝜂 +𝛿 −𝑡 −𝜏 −
( )
, 𝜏 ≤ 𝐴 is the length of the request 𝑡 ; 𝑣 = 𝜔 ,𝜔 = 1, and = , 𝛿 =𝑡 −𝜂 .
( ) In addition, the availability conditions are checked by
, where 𝑠 = 𝑡 , 𝐹 (𝐹 ) = max min 𝑓 ; 𝜇 >
( ) ,… comparing the number of requests accepted for execution
1 is a multiplier that plays the role of a weighting factor for on each day of the month 𝑧 , 𝑖 = 1, … , 𝐷, with the
the relative importance of the request length for finding a availability limits 𝛽 , 𝑖 = 1, … , 𝐷. If they are equal 𝑧 = 𝛽 ,
compromise sub-option; 𝑣 , = , 𝑖 = 1, … , 𝑚, where 𝑖 = 1, … , 𝐷, the interval system is adjusted on some days:
working days of teams for which 𝑧 − 𝛽 = 0, become days
𝐺 = 𝐺 − 𝛥𝐺 , 𝑠 = 1, … , 𝑛, 𝛥𝐺 is the amount of resource off, which affects the structure of intervals.
𝑖 of the type spent when including the next compromise As a result of applying the PS procedure to the original
sub-option𝑣 in a partial solution to the problem: problem, a partial solution is constructed and the problem is
𝐺 = 0, 𝑖 = 1, … , 𝑚 . modified. After 𝑛 is the application of the described
Thus, the search for a partial solution to the original procedure, three cases are possible:
problem is reduced to a discrete multicriteria optimization 1. The solution to the problem is found and one of the
model with a set of feasible solutions 𝑉 and (𝑚 + 2) criteria resources is completely exhausted:
to be minimized. If the set of admissible sub-options is not ∃𝑖: 𝛥𝐺 = 𝐺 ,
empty,𝑉 ≠∅, and not trivial, i.e. |𝑉| > 1, we will look for a and hence, 𝐺 = 0. The method of finding the reference
compromise. To find a single solution to a multicriteria solution is completed.
optimization problem, it is necessary to set the weighting 2. The solution to the problem is found, but
coefficients of the criteria [31–33]. Let’s fix the weighting for ∀𝑖 = 1, … , 𝑚, 𝛥𝐺 < 𝐺 ;
factor for the length of the application as 𝜌 —for the sake of in this case, it is necessary to reduce the total weighting of
certainty, let’s assume𝜌 = 0.5. Let’s denote by 𝜌 the resources 𝜌 by increasing the weight of the objective
weighting factor of the objective function of the initial
function 𝜌 , taking into account the following condition:
problem of the OCI MOC; the criteria that are “responsible”
𝜌 +𝜌 =1−𝜌 ; (6)
for resources are aggregated and denoted by the total
3. The task is incompatible. This, in turn, is possible
weighting factor
when:
𝜌 =∑ 𝜌,∑ 𝜌 = 1. 3.1) one or more resources have been exhausted to
We are looking for a compromise option as obtain a complete solution to the problem. Therefore, the
𝑣 = 𝑎𝑟𝑔 𝑚𝑖𝑛 𝑚𝑎𝑥 𝜌 ⋅ 𝑣 . (5)
,…, ,…, total weight of the resources 𝜌 should be increased, taking
In the case when the solution of (5) is not unique, a into account condition (6). This reduces the weight 𝜌 of the
linear convolution is applied objective function, which could also influence the
“unfavourable” placement of the suboption.
𝑣 = 𝑎𝑟𝑔 𝑚𝑖𝑛 𝜌 ⋅𝑣 , 3.2) there is no valid working interval for the next order,
∈
and therefore 𝑉 ≠∅ . Such a situation is possible if the
where 𝑈 is the set of indices of sub-variants equivalent by coefficient 𝜌 , which is “responsible” for the length of the
criterion (5). order, is not large enough. In this case, orders of short length
As a result of the search for the “best” sub-variant, we were likely prioritized and “cut” the working intervals that
complement the partial solution. This modifies the original could accommodate orders of longer length. Such a situation
problem. The number of requests is reduced, i.e. 𝑛 = 𝑛 − 1, can be managed by reducing the 𝜇 indicator. In this case,
and the number and/or length of work intervals of the teams 𝑣 , 𝑙 = 1, … , 𝐿, remain unchanged, and 𝑣 , 𝑖 = 2, … , 𝑚 + 2,
are changed. 𝑙 = 1, … , 𝐿, increase by reducing 𝜇 with unchanged
107
weighting factors 𝜌 , 𝑖 = 1, … , 𝑚 + 2, and thus “move where 𝑧—is an additional application involved in the
away” from the optima. replacement chain to generate additional solution options.
As a result of applying the described method, we obtain It is easy to see that all the more complex cases are
the reference solution 𝑣 = (𝑣 , … , 𝑣 ) or make sure that reduced to the cases a)—c) described above.
the initial problem is incompatible. In this case, the If the option remains valid, its 𝑃 is valid sub-options are
conditions of incompatibility are constructively formulated. replaced (permuted) and the process proceeds to step 0. If
If the initial problem is admissible, you can improve the the option is not valid, an attempt is made to make the
solution. To describe this method, let’s introduce a permutation valid by making concessions on the sub-
definition. options found for the permutation.
Definition 7. A P-admissible sub-variant of 𝛼 ∈ 𝛼, 𝑖 = At the same time, if 𝑓 𝛼 ( ) > 𝑓 𝛼 ( ) or
1, … , 𝑠, is one or more bids placed in the same working 𝑓 𝛼 ( ) = 𝑓 𝛼 ( ) and 𝑔 𝛼 ( ) > 𝑔 𝛼 ( ) , are used,
interval (𝑠 is number of admissible sub-variants, 𝑃— the option 𝛼 ( ) , where 𝑘—is the iteration number is
admissible placement option). Moreover, the 𝑃—valid sub- accepted.
option must be valid.
Definition 8. The length 𝑃—of a valid sub-variant will be 3.5. Algorithm for improving the reference
the distance from the start of the first order in a fixed solution by changing its P-admissible
working interval to the end of the last order in that interval.
subvariants
Definition 9. 𝑃—Valid sub-options are comparable when
the sum of the order lengths of one sub-option does not Step 1. Search for the maximum possible length of the empty
exceed the length of the working interval containing the segment in the intervals that make up 𝑃—valid subvariants.
second sub-option, and vice versa. Step 2. Applications whose length does not exceed the
Definition 10. 𝑃 is valid sub-option 𝛼 will be more value of the found segment are sorted in descending order.
promising than 𝑃 is valid sub-option 𝑦. If these sub-options Step 3. The applications of the found set are “tried on”
are comparable for a fixed interval 𝑧 and 𝑓(𝛼) > 𝑓(𝑦), or to the empty segments in which they can be placed. If
𝑓(𝛼) = 𝑓(𝑦) and (𝑔(𝛼) = 𝑔(𝑦) 𝛼 dominates 𝑦 in terms of 𝑓 𝛼 ( ) = 𝑓 𝛼 ( ) , the request is moved. If not, other
resources). applications are considered. If there is no improvement as a
𝑓(𝛼) denotes 𝑚𝑎𝑥 𝑓 (𝛼), where 𝑋—is the set of options result, the option is locally optimal.
∈
The combinatorial formulation of the problem of QMS
for placing requests on the interval 𝑧.
and the sequential algorithm for its solution is a convenient
tool for research, structuring the subject area and
3.4. Algorithm for improving the reference
“penetration” of the user into the problem and information
solution by changing its P-valid content of the QMS problem. At the same time, the
variants described heuristic algorithms are an effective apparatus for
The initial data for this algorithm are the data described in finding a locally optimal solution to the problem, since they
the problem statement, as well as the reference solution allow generating an acceptable variant of request service
𝑣 = (𝑣 , … , 𝑣 ), obtained as a result of the previous with its subsequent improvement and identifying
method and described in terms of 𝑃—admissible incompatibilities of the problem.
subvariants.
Step 0. Ordering by the quality 𝑃—of the admissible 4. The task of coordinating decisions
subvariants that make up the reference solution. If 𝑓(𝛼) = in a three-level hierarchical
𝑓(𝑦) and the vector 𝑔 (𝛼), … , 𝑔 (𝛼) are incomparable system for ensuring the operation
with the vector 𝑔 (𝑦), … , 𝑔 (𝑦) , then the subvariant of
of a network of critical
shorter length is considered more promising.
Step 1. The master selects 𝑃—the best quality admissible infrastructure facilities
sub-variant contained in the reference solution. An attempt In group decision-making and determining the properties of
is made to improve it by placing it in other working an object, there is almost always a problem reconciling
intervals or by permissible permutations in its working assessments [34, 35]. Experts’ opinions often do not coincide
interval. and must be aggregated to obtain a single conclusion [36,
Step 2. If the leading sub-option cannot be made more 37]. In some practical tasks, the definition of an aggregated
promising, the next best sub-option is considered. If no 𝑃 is (integral, resultant, etc.) solution is carried out in the form
valid sub-option has improved during the algorithm, the of intervals or a membership function of a fuzzy set [38].
algorithm ends. In [39], the authors considered and studied the problem
With improvement, such cases are possible: of coordinating decisions in a two-level hierarchical model
a) the application 𝛼 is “exchanged” by the working for choosing the mode of operation of a communication
interval of placement with the application 𝑦. system. Due to the urgency of the problem of critical
b) the application 𝛼 shall be placed in the time slot infrastructure protection at the regional and state levels,
previously occupied by the application 𝑦, and the this task was adapted to the problems of ensuring the
application 𝑦 shall be placed in the previously free time slot. operation of a hierarchical system of critical infrastructure
c) cyclical replacement 𝛼 → 𝑦 → 𝑧 → 𝛼, facilities. The technology for coordinating decisions in a
hierarchical system was improved and refined to ensure that
108
decisions are coordinated in a three-tier hierarchical system. three-level hierarchical model, considering that the modes
The interpretation of the problem area of research was also of operation of the network of elements are subsystems (SS)
naturally adapted to the issues related to the functioning of the lower level, and the capital costs for the creation and
and characteristics of the critical infrastructure network. maintenance of the system, operation of the I&C and
maintenance of crews are arguments for the quality
4.1. Formulation of the problem of decision criterion of the upper-level SS [42, 43].
coordination in a three-level Let us consider the problem of reconciling the decisions
hierarchical system of critical of a certain set of PS of a hierarchical organizational system
infrastructure with indices 𝐼 = {0, … , 𝑀}, connected by a three-level
hierarchical structure [44]. We will assume that the
Let there be given a set of alternatives (objects, options, analytical or tabular dependencies of the values of 𝑓 (𝑢), 𝑙 ∈
plans, projects, etc.) 𝑎 ∈ 𝐴, 𝑖 ∈ 𝐼 = {1, … , 𝑛}, each of which 𝐼, of the quality criteria of the PS on the attributes
is characterized by 𝑚 features (attributes, characteristics, (characteristics, attributes, etc.) are known. The
factors, etc.) relationships between the criteria (also given analytically or
𝑎 = 𝑎 ,…,𝑎 , 𝑖 ∈ 𝐼. tabularly) are denoted by
To build a model of a specific task, it is often necessary 𝑀 (𝐷), 𝑙 ∈ 𝐼,
to determine the relative weight of characteristics and their where 𝐷 is the set of permissible variants of the values of
influence on decision-making—to increase certainty and the features that affect the values of the quality criteria
increase the structure of the subject area. Since a person in 𝑓 (𝑢), 𝑙 ∈ 𝐼, 𝑢 ∈ 𝐷.
most cases is not able to adequately assign relative weights,
indirect methods are a promising direction for solving the 4.2. A mathematical model of the problem
problem of determining the weighting coefficients of of decision coordination in a three-
characteristics [40, 41].
level hierarchical system
As a rule, there is a history of preferences between
objects (alternatives, players, projects, units, etc.)—based on The three-tier hierarchical management system under
the results of measuring experts’ preferences or any other consideration consists of:
natural information. This can be a series of tournaments or
a ranking of objects, i.e. a complete preference relation. ● One top-level PS (denoted by the index 0).
We will assume that the topology of the set (network) ● 𝑛 of medium-sized substations isolated from each
of IEDs requiring regular maintenance and ensuring other with a set of indices 𝐼 = {1, … , 𝑛 }.
uninterrupted operation in the event of emergency outages ● 𝑛 subordinate systems of the set 𝐼 of isolated
or planned rolling outages is given: the number of IEDs and lower-level substations with a set of indices 𝐼 =
their geolocation coordinates on the plane. It is necessary to {𝑛 + 1, … , 𝑛 + 𝑛 }.
determine:
As a rule, the model of the hierarchical system of the
● A sufficient number of QMS centers. above structure is built as follows.
● Their location (the QIS RM center can only be The top-level PS 𝑆𝑆𝐻 coordinates the operation of the
located in an element of the QIS network). middle-level PS 𝑆𝑆𝑀 , 𝑖 = 1, … , 𝑛 , using control vectors
● Provision of the CMI centers with brigades 𝑢 , 𝑙 ∈ 𝐼 , whose values are determined when solving the
(network equipment), i.e. the optimal number of top-level optimization problem according to the top-level
brigades in each center (we will assume that all optimality criterion. The control influences of each
brigades are integrated, interchangeable, and of medium-level AC 𝑢 , 𝑙 ∈ 𝐼 , are determined when solving
the same type). optimization problems according to the medium-level
● Distribution (division) of the network into service optimality criterion, taking into account the calculated
zones, i.e. finding the best option for clustering control influences ul, 𝑙 ∈ 𝐼 , received from the upper-level
network elements with the identification of the AC. In turn, each 𝑙—and (𝑙 ∈ 𝐼 ) of the middle level
element number in which the OCI MIS center is coordinates the work of isolated lower level 𝑆𝑆𝐿 , 𝑖 = 𝑛 +
located and the number of elements served by each 1, … , 𝑛 , associated with it (the set of indices of which is
center.
denoted by 𝐼 , 𝑙 ∈ 𝐼 , ∑ 𝐼 = 𝑛 . Each 𝑙 and lower level
At the same time, it is necessary to ensure the minimum substation, taking into account the control influence of 𝑙 is
cost of building the I&CMS system and operating the the middle-level substation 𝑢 , finds its solution 𝑢
network in three modes of operation and the maximum according to its optimality criterion.
probability of maintaining operability for each mode while A diagram of the relationships between subsystems of
ensuring restrictions on the average recovery time of each different levels of the three-tier hierarchical system is
network element. We will interpret the task in terms of a shown in Fig. 1.
109
Figure 1: Schematic diagram of the three-level hierarchical structure for ensuring the sustainable operation of the critical
infrastructure network
We will assume that the choice of control influences 𝑢 in where all notations correspond to those in problems (7)–
the 𝑙 of the lower-level AC 𝑆𝑆𝐿 , 𝑖 = 𝑛 + 1, … , 𝑛 , is carried (10), 𝑢 is control influence transmitted to 𝑙 that middle-
out when solving a discrete optimization problem of the level PS from the upper-level PS 𝑆𝑆𝐻 .
form The choice of control influences 𝑢 , 𝑙 ∈ 𝐼 , in the upper-
𝑓 𝑢 → 𝑚𝑖𝑛, (7) level control system is carried out when solving the discrete
𝐻 𝑢 ≤𝐻 , ∗ (8) optimization problem of the form
(9) 𝑓 (𝑢) → 𝑚𝑖𝑛, (15)
𝑔 𝑢 ≤𝑢 ,
𝐻(𝑢) ≤ 𝐻 ∗ , (16)
𝑢 ∈𝑈 = 𝑈 , (10)
𝑢∈𝑈= 𝑈 , (17)
where 𝑢 is the solution vector, the dimension of which is
where 𝑢 = {𝑢 , 𝑙 ∈ 𝐼 }; 𝑈 are finite sets; 𝑓 is a scalar
determined by condition (10); 𝑈 is the finite sets of possible
function; 𝐻 is a vector function; 𝐻 ∗ is a vector of constants
variants 𝑗 and components of the vector 𝑢 ; 𝑓 are scalar (the requirements for 𝑓 are similar to those for problem
functions; 𝐻 , 𝑔 are vector functions of a discrete (7)–(10)).
argument of the corresponding dimension, specified The variants of the values of characteristics (factors,
analytically or in the form of tables, such that for each features, attributes, etc.) that satisfy condition (8) for a
component of 𝑢 vectors 𝑢 the values of particular model are set explicitly (taking into account
𝑎𝑟𝑔 𝑚𝑖𝑛 max () 𝐻 ,𝑔 to 𝑢 at fixed values of information about the peculiarities of the functioning of 𝑙 of
the PS obtained from experts or by solving auxiliary tasks
other variables do not depend on the value of this
of finding the quality of functioning of 𝑙 of the PS).
component (this condition is satisfied, for example, by
Conditions (9) in a particular model are set by tables of
monotonic or separable functions); 𝐻 ∗ —vector of given correspondence of variants of values of external
constants. characteristics to internal ones.
Relationship (9) characterizes the own constraints of the A “partial” mathematical model of the general model
PS, i.e. the relationship between the own parameters of the discussed above is presented.
PS (𝐻 ∗ —its resource, so condition (9) sets the law of We will assume that the set of possible values of
distribution of the own resources of the PS), and relations characteristics U is divided into the set of 𝑈 з general
(8) determine the relationship with the medium-level PS (i.e. characteristics, on which the quality of the system
the relationship between the own parameters of the PS and functioning as a whole depends, and 𝑈 , 𝑙 ∈ 𝐼 , the set of
external influences—interpreted as the “external” resource own characteristics of the PS, i.e., characteristics that affect
of the PS). This is carried out with the help of control
only the functioning of individual PS; 𝑈 𝑢 , 𝑙, 𝑖 ∈ 𝐼, 𝑢 , 𝑙 ∈
influences 𝑢 , that is determined by solving a similar
𝑖 ≠ 𝑗, characteristics that reflect the “vertical” links between
problem for each 𝑙 of the medium-level PS 𝑆𝑆𝑀 , 𝑖 =
PS of different levels.
1, … , 𝑛 ,
Let us denote by 𝑈 = 𝑈 з × ∏ ∈ 𝑈 × ∏ , ∈ 𝑈 , the set
𝑓 𝑢 → 𝑚𝑖𝑛, (11)
of possible values of characteristics in the functioning of a
𝐻 𝑢 ≤𝐻 , ∗ (12)
hierarchical system.
𝑔 𝑢 ≤𝑢 , (13) Then, the partial mathematical model of the OCI CMM
system is interpreted in terms of the general mathematical
𝑢 ∈𝑈 = 𝑈 , (14) model as follows. The values of the criterion functions (7),
(11), (15) are set as a function of the given points of the
110
corresponding hyperparallelepipeds (10), (14), (17). The where 𝐶 , 𝐶 are the average specific costs of restoring a
constraints (9) and (13) are set by the tables of network element and maintenance, respectively; 𝜏 , 𝑇
correspondence between the binding and own control are the given constants;𝑇БР is the average failure-free life,
influences. The eigenconstraints of the PS (8), (10), and (12) which is determined by the formula
are given by the table dependencies on their 𝜇 (1 − 𝑒𝑥𝑝(−(𝜇 + 𝜆) ⋅ 𝑇 ))
eigenparameters. 𝑇БР = 𝑇 ⋅ +𝜆⋅ ,
(𝜇 + 𝜆) (𝜇 + 𝜆)
A consistent solution of a hierarchical system is in which 𝜆 is a constant: 𝜇 is the failure rate, 𝜇 = 1/𝑇 , 𝑇
calculated as a “compromise” solution:
is the average recovery time 𝑇 = 𝑇 + 𝜏 ; 𝑇 is a
𝑢 = 𝑎𝑟𝑔 𝑚𝑖𝑛 𝑚𝑎𝑥 𝜌 𝜔 (𝑢), (18)
∈ ∈ constant related to the restoration of operability; 𝜏 = 𝑆/𝑣
where 𝜔 (𝑢) = 𝜔 𝑓 (𝑢) , 𝑖 ∈ 𝐼 relative deviations from the is the time spent on moving from the OCI’s emergency
optimums of the quality criteria for the functioning of 𝑖 th PS response center to the network element,𝑣 = 𝑐𝑜𝑛𝑠𝑡 is the
at the values of parameters 𝑢, 0 ≤ 𝜔 (𝑢) ≤ 1; 𝜌 is average speed of brigades’ movement, 𝑆 is the distance
weighting coefficients of the importance of PS for achieving between points.
the goals of the entire hierarchical system. Finding the 𝑆 = 𝑇 ∙ 𝐶 ∙ (1/𝑇БР + 1/𝑇 )
weighting coefficients is an independent task. We only note 𝑇Е and 𝑇 are the specified constants.
that additional restrictions may be imposed on the The values of the functionalities of the quality of
weighting factors, for example, operation of the lower-level substation are given in the form
∑∈ 𝜌 =𝜌 ,𝑖∈𝐼 , of a table in the form of a correspondence
where 𝜌 —are the coefficients of the lower-level PS, 𝜌 —are 𝑢 , 𝑢 ⇔ 𝑓 , 𝑞 ∈ 𝑄, 𝑗 ∈ 𝐼 , 𝑖 = 1,2,3,
the “weights” of the upper-level PS. where 𝑢 , 𝑞 is a string of values of “connecting” (“linking”)
𝑈 , 𝑢 ∈ 𝑈 . In the case when the solution (18) is not
characteristics, 𝑢 , 𝑗 is a string of eigenvalues of
unique, an additional criterion of the form
characteristics 𝑖 of the lower-level PS, 𝐼 is a set of indices
𝑝 𝑤 (𝑢 ) → min . of variants of values of characteristics of the lower-level PS.
∈
∈
The function of the quality of the upper-level PS functioning 4.3. Algorithm for reconciling decisions in
is calculated by the formula a three-level hierarchical system
𝑆 =𝐸 ∗∑ 𝑆 +∑ ∑ ∈ 𝑆 +
The algorithm for matching solutions of a three-level
∑ ∑ ∈ 𝑆 ,𝑖∈𝐼 , hierarchical system that models the solution of the OCI
where 𝐸 is the coefficient related to capital expenditures; WRM problem can be described by the following sequence
𝑁 is the number of QMS centers (a factor whose value of steps.
should be found); 𝑆 = 𝑆 = 𝑐𝑜𝑛𝑠𝑡 is the capital Step 0. Reading the data required for the algorithm to
expenditures for the establishment and operation of the function: variants of the values of general characteristics,
𝑆 = 𝑆 = 𝑐𝑜𝑛𝑠𝑡 center (for the sake of simplicity, we variants of the values of the characteristics of the
substations of all three levels. Calculate or explicitly enter
assume 𝑆 = 𝑆 = 𝑐𝑜𝑛𝑠𝑡 for ); ∀𝑗 = 1, … , 𝑁, 𝑅 , 𝑗 = 1, … , 𝑁
the weighting coefficients of the relative importance of the
is the number of service teams in the 𝑆 = 𝑆 = 𝑐𝑜𝑛𝑠𝑡
PS for the functioning of the system as a whole. Calculation
center (we assume that all teams are of the same type, i.e. or specification of the optimal and worst values of the
𝑆 = 𝑆 = 𝑐𝑜𝑛𝑠𝑡 for 𝑏 ∈ 𝐵 , ∀𝑗 = 1, … , 𝑁) characteristics quality criteria for the functioning of all the PSs of the
whose values are to be calculated; 𝑟 ∈ 𝑅 , ∀𝑗 = 1, … , 𝑁 is the hierarchical system.
composition of sets of indices of elements served by the 𝑆 Step 1. Formation of sets of indices of variants of values
center values of characteristics whose values are to be of characteristics of those PSs that are connected by the
calculated; 𝑆 is operating costs to maintain the operability same common characteristics. If a certain variant of the
of the 𝑟—network element served by the 𝑆 = 𝑆 = 𝑆 (𝑆), common values of characteristics is absent in a PS, then it
the center of the MIS OCI. cannot be included in the search for an agreed solution—it
The value 𝑆 = 𝑆 = 𝑆 (𝑆), where 𝑆 is the distance is concluded that it is inadmissible.
Step 2. Based on the coordinates of the network elements,
between the center of the OCI RFM with the index 𝑗 and the
distances between them are calculated to use their values in
network element to be served with the index 𝑟. The value of
calculating the quality of the upper-level substation, which is
the distance 𝑆 is calculated by the formula
set in an analytical form.
/
𝑆= 𝑥 −𝑥 + 𝑥 −𝑥 , Step 3. Set the initial values of the trade-off 𝜔 = 1,
where 𝑥 , 𝑥 , are the geolocation coordinates of the 𝑗 of the 𝜔 = 1, where 𝜔 is the initial value of the parameter 𝑘 ,
OCI network element on the plane (this network element 𝜔 is the initial value of the linear function of the PS criteria
may or may not contain an OCI center). with the weighting coefficients set in step 0.
The meaning of 𝑆 is calculated as follows: Step 4. A complete search of the sets of indices of
variants of the values of the characteristics of the PS,
𝑆 = 𝑆 +𝑆 ,
connected by the same common characteristics, is
where 𝑆 are the costs of CMM O&M and repairs, 𝑆 are organized. If the search is completed, an agreed solution is
transport costs. displayed, i.e., a variant of the values of the characteristics
𝑆 = 𝑇 ⋅ (𝐶 ⋅ (𝑇 − 𝑇БР ) + 𝐶 ⋅ 𝜏 )/𝑇 ,
111
of the PS that delivers a minimum of 𝜔 and 𝜔 . This model of decision coordination in a three-level hierarchical
completes the algorithm. system for ensuring the operation of a network of critical
Step 5. The value of the upper-level PS quality criterion infrastructure facilities are also presented. An algorithm for
is calculated on the next variant of the parameter values, the coordinating decisions in a three-level hierarchical system
values of the lower-level PS criteria are searched for in the is developed and described.
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