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.
The possibilities of applying mathematical models and
decision-making methods to study the problems of
vulnerability, protection, and management of critical
infrastructure systems are in the field of view of many
researchers [1, 2]. This issue has been studied by scientists
from different countries for many years [3–5] and its
relevance is not decreasing [6–8]. To date, a large number
of approaches and mathematical models have been
developed that demonstrate the authors’ attempts to ensure
the effective functioning and protection of critical
infrastructure [9–11]. At the same time, the problems of
protecting critical infrastructure from terrorist attacks
remain extremely relevant [
In many practical decision-making situations in poorly
formalized organizational systems, the decision-maker is
forced to act in poorly structured subject areas [
In particular, a person cannot set reliable weighting
coefficients for the relative importance of parameters or
criteria [
Particularly important for the state.
Capable of significantly affecting other critical infrastructure facilities.
Whose disruption causes a crisis of national importance.
Vital at the regional level.
Whose disruption or malfunction causes a crisis of regional, local, or local significance.
Critical infrastructure facilities include enterprises and institutions operating in the following industries:
Energy
0000-0002-0465-5018 (H. Hnatiienko);
0000-0001-8546-5074 (O. Hnatiienko);
0000-0003-1184-9483 (T. Babenko);
0000-0003-1686-261X (L. Myrutenko)
© 2024 Copyright for this paper by its authors. Use permitted under
Creative Commons License Attribution 4.0 International (CC BY 4.0).
[
, 1 ≤ ≤ ; is number of requests; is the maximum length of the request; = , = 1, … , , = 1, … , , is the efficiency of execution of the is request if it starts to be executed on the th day of the month. Target function of the task: where is the number of working days in the calendar month on which the is request starts to be executed. Let =
is the resource of the is the type required to satisfy the is request if it starts executing on the th day, here is the number of days when the execution of the is request starts;
is the number of types of resources; , = 1, … , are restrictions on the readiness of the OCI’s command center for the EWM brigades, which means that no more than is requests can be serviced on a given day,
It should be noted that requests for activation of the next I&CS teams are received directly from CI facilities, I&CS subsystems (SS), or the governing bodies of the I&CS (3) (4) system. applications
Array of inconsistencies (incompatibilities) of =
, = 1, … , , ℎ = 1,2, …, where
is the number of requests that cannot be executed simultaneously; is the number of array lines; ℎ is the indices of incompatible requests of the th line of the unstacked array. describe definitions. 3.2. Scheme for solving the problem of ensuring the operation of the QMS The solution to the problem is sought in two stages: building a reference solution and building an optimal solution. To the algorithms, we present the necessary Definition 1. A variant of the problem (1)–(4) is a vector = ( , … , ), whose elements are triples = , , , where is the number of the request; ∈{ ,…, } , (1) 3.3. Method for finding a valid solution to the problem of ensuring the operation of the QMS The method of finding a feasible solution consists of the sequential construction of a reference solution as a union of locally feasible subvariants. Therefore, it is reduced to the sequential application of the following procedure.
Procedure for finding a locally admissible subvariant PS. The basis of the method is the formation of a set of admissible subvariants = ( ), = 1, … , + 2, = ( ( ( ) ) ) 1, … , , from which a compromise subvariant is selected, — the number of admissible subvariants. The condition for generating a sub-variant is, firstly, ≥ , = 1, … , , = 1, … , , = 1, … , , i.e., the application should not exceed the interval in which it is supposed to be placed, and secondly, the compatibility of the current application with the one already accepted in the partial solution. In parallel to the set of sub-options, a set of indices corresponding to them is formed, =
, ℎ = 1,2,3, = 1, … , , where ≤ is the number of the application that generates the sub-option,
≤ is the number of the team proposed to execute the application,
≤ is the day the execution of the application
by the team starts .
Consider the procedure for forming a valid subvariant = , … ,
, . For each request , = , all possible combinations of its placement in the working intervals of the teams are selected. At the same time, , ≤ is the length of the request ; = , , where = , ( ) = max
min ,… 1 is a multiplier that plays the role of a weighting factor for the relative importance of the request length for finding a compromise sub-option; , = , = 1, … , , where = − , = 1, … , , is the amount of resource of the type spent when including the next compromise sub-option in a partial solution to the problem:
= 0, = 1, … , .
Thus, the search for a partial solution to the original
problem is reduced to a discrete multicriteria optimization
model with a set of feasible solutions and (
+ 2) criteria
to be minimized. If the set of admissible sub-options is not
empty, ≠∅, and not trivial, i.e. | | > 1, we will look for a
compromise. To find a single solution to a multicriteria
optimization problem, it is necessary to set the weighting
coefficients of the criteria [
In the case when the solution of (5) is not unique, a (5) linear convolution is applied = = ; > ⋅ , where is the set of indices of sub-variants equivalent by criterion (5).
As a result of the search for the “best” sub-variant, we complement the partial solution. This modifies the original problem. The number of requests is reduced, i.e. = − 1, and the number and/or length of work intervals of the teams are changed. 2. 3. and the placement of the application does There are four options for modifying the system of working intervals of brigades by changing the interval in which the compromise order is placed: 1. The length of the interval , ∈ {1, … , } is equal to the length of the bid
and the interval is completely excluded from consideration. At the same time, = − 1, = , where —is the index of the compromise bid.
and the compromise application is placed not correspond to any of the three cases. This generates an additional interval with the index, i.e. , = + 1, = − − − , 1, and = +
, = , = = − + .
In addition, the availability conditions are checked by comparing the number of requests accepted for execution on each day of the month , = 1, … , , with the availability limits , = 1, … , . If they are equal = , = 1, … , , the interval system is adjusted on some days: working days of teams for which − = 0, become days off, which affects the structure of intervals.
As a result of applying the PS procedure to the original problem, a partial solution is constructed and the problem is modified. After
is the application of the described procedure, three cases are possible:
1. The solution to the problem is found and one of the resources is completely exhausted: ∃ : = , and hence,
= 0. The method of finding the reference solution is completed.
2. The solution to the problem is found, but for ∀ = 1, … , , < ; in this case, it is necessary to reduce the total weighting of resources
by increasing the weight of the objective function , taking into account the following condition: + = 1 − ; (6) 3. The task is incompatible. This, in turn, is possible when:
3.1) one or more resources have been exhausted to obtain a complete solution to the problem. Therefore, the total weight of the resources should be increased, taking into account condition (6). This reduces the weight of the objective function, which could also influence the “unfavourable” placement of the suboption.
3.2) there is no valid working interval for the next order, and therefore ≠∅ . Such a situation is possible if the coefficient , which is “responsible” for the length of the order, is not large enough. In this case, orders of short length were likely prioritized and “cut” the working intervals that could accommodate orders of longer length. Such a situation can be managed by reducing the indicator. In this case, , = 1, … , , remain unchanged, and , = 2, … , + 2, = 1, … , , increase by reducing with unchanged weighting factors , = 1, … , + 2, and thus “move where —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 = ( , … , ) or make sure that the initial problem is incompatible. In this case, the conditions of incompatibility are constructively formulated.
If the initial problem is admissible, you can improve the solution. To describe this method, let’s introduce a definition.
Definition 7. A P-admissible sub-variant of ∈ , = 1, … , , is one or more bids placed in the same working interval ( is number of admissible sub-variants, — admissible placement option). Moreover, the —valid suboption must be valid.
Definition 8. The length —of a valid sub-variant will be the distance from the start of the first order in a fixed working interval to the end of the last order in that interval.
Definition 9. —Valid sub-options are comparable when the sum of the order lengths of one sub-option does not exceed the length of the working interval containing the second sub-option, and vice versa.
Definition 10. is valid sub-option will be more promising than is valid sub-option . If these sub-options are comparable for a fixed interval and ( ) > ( ), or ( ) = ( ) and ( ( ) = ( ) dominates in terms of resources).
( ) denotes
( ), where —is the set of options ∈ for placing requests on the interval . 3.4. Algorithm for improving the reference solution by changing its P-valid variants The initial data for this algorithm are the data described in the problem statement, as well as the reference solution ), obtained as a result of the previous described in terms of —admissible
= ( method subvariants.
, … , and
Step 0. Ordering by the quality —of the admissible subvariants that make up the reference solution. If ( ) = ( ) and the vector ( ), … , ( ) are incomparable with the vector ( ), … , ( ) , then the subvariant of shorter length is considered more promising.
Step 1. The master selects —the best quality admissible sub-variant contained in the reference solution. An attempt is made to improve it by placing it in other working intervals or by permissible permutations in its working interval.
Step 2. If the leading sub-option cannot be made more promising, the next best sub-option is considered. If no is valid sub-option has improved during the algorithm, the algorithm ends.
With improvement, such cases are possible: a) the application is “exchanged” by the working interval of placement with the application .
b) the application shall be placed in the time slot previously occupied by the application , and the application shall be placed in the previously free time slot. c) cyclical replacement → → → , the option ( accepted. reduced to the cases a)—c) described above.
If the option remains valid, its is valid sub-options are replaced (permuted) and the process proceeds to step 0. If the option is not valid, an attempt is made to make the permutation valid by making concessions on the suboptions found for the permutation.
At the same time, if ( ) > ( ) or ( ) = ( ) and (
) > ( ) , are used, ), where —is the iteration number is 3.5. Algorithm for improving the reference solution by changing its P-admissible subvariants Step 1. Search for the maximum possible length of the empty segment in the intervals that make up —valid subvariants.
Step 2. Applications whose length does not exceed the value of the found segment are sorted in descending order.
Step 3. The applications of the found set are “tried on” to the empty segments in which they can be placed. If (
) = ( ) , the request is moved. If not, other applications are considered. If there is no improvement as a result, the option is locally optimal.
The combinatorial formulation of the problem of QMS
and the sequential algorithm for its solution is a convenient
tool for research, structuring the subject area
and
“penetration” of the user into the problem and information
content of the QMS problem. At the same time, the
described heuristic algorithms are an effective apparatus for
finding a locally optimal solution to the problem, since they
allow generating an acceptable variant of request service
with
its
subsequent
improvement
and
identifying
incompatibilities of the problem.
4. The task of coordinating decisions
in a three-level hierarchical
system for ensuring the operation
of a network of critical
infrastructure facilities
In group decision-making and determining the properties of
an object, there is almost always a problem reconciling
assessments [
In [
To build a model of a specific task, it is often necessary to determine the relative weight of characteristics and their influence on decision-making—to increase certainty and increase the structure of the subject area. Since a person in most cases is not able to adequately assign relative weights, indirect methods are a promising direction for solving the problem
of determining the weighting coefficients of
characteristics [
As a rule, there is a history of preferences between objects (alternatives, players, projects, units, etc.)—based on the results of measuring experts’ preferences or any other natural information. This can be a series of tournaments or a ranking of objects, i.e. a complete preference relation.
We will assume that the topology of the set (network) of IEDs requiring regular maintenance and ensuring uninterrupted operation in the event of emergency outages or planned rolling outages is given: the number of IEDs and their geolocation coordinates on the plane. It is necessary to determine: ● ● ● ●
A sufficient number of QMS centers.
Their location (the QIS RM center can only be located in an element of the QIS network).
Provision of the CMI centers with brigades (network equipment), i.e. the optimal number of brigades in each center (we will assume that all brigades are integrated, interchangeable, and of the same type).
Distribution (division) of the network into service zones, i.e. finding the best option for clustering network elements with the identification of the element number in which the OCI MIS center is located and the number of elements served by each center.
At the same time, it is necessary to ensure the minimum
cost of building the I&CMS system and operating the
network in three modes of operation and the maximum
probability of maintaining operability for each mode while
ensuring restrictions on the average recovery time of each
network element. We will interpret the task in terms of a
maintenance of the system, operation of the I&C and
maintenance of crews are arguments for the quality
criterion of the upper-level SS [
Let us consider the problem of reconciling the decisions
of a certain set of PS of a hierarchical organizational system
with indices = {0, … , }, connected by a three-level
hierarchical structure [
The relationships between the criteria (also given analytically or tabularly) are denoted by
( ), ∈ , where is the set of permissible variants of the values of the features that affect the values of the quality criteria ( ), ∈ , ∈ . 4.2. A mathematical model of the problem of decision coordination in a threelevel hierarchical system The three-tier hierarchical management system under consideration consists of: ● ● ● + 1, … ,
+ }. { One top-level PS (denoted by the index 0).
of medium-sized substations isolated from each other with a set of indices
= {1, … , }. subordinate systems of the set of isolated lower-level substations with a set of indices = As a rule, the model of the hierarchical system of the above structure is built as follows.
The top-level PS
coordinates the operation of the middle-level PS
, = 1, … , , using control vectors , ∈ , whose values are determined when solving the top-level optimization problem according to the top-level optimality criterion. The control influences of each medium-level AC , ∈ , are determined when solving optimization problems according to the medium-level optimality criterion, taking into account the calculated control influences ul, ∈ , received from the upper-level AC. In turn, each —and ( ∈ ) of the middle level coordinates the work of isolated lower level infrastructure network We will assume that the choice of control influences in the of the lower-level AC corresponding hyperparallelepipeds (10), (14), (17). The constraints (9) and (13) are set by the tables of correspondence between the binding and own control influences. The eigenconstraints of the PS (8), (10), and (12) are given by the where ( ) =
( ) , ∈ relative deviations from the optimums of the quality criteria for the functioning of th PS at the values of parameters , 0 ≤ ( ) ≤ 1; is weighting coefficients of the importance of PS for achieving the goals of the entire hierarchical system. Finding the weighting coefficients is an independent task. We only note that additional restrictions may be imposed on the weighting factors, for example, (18) ∑ ∈
= , ∈ , where —are the coefficients of the lower-level PS, —are the “weights” of the upper-level PS.
,
∈ . In the case when the solution (18) is not unique, an additional criterion of the form ∈ ( ) → min .
∈ The function of the quality of the upper-level PS functioning is calculated by the formula = ∗ ∑
+ ∑ ∑ ∑ ∈
∑ ∈ , ∈ , + where
is the coefficient related to capital expenditures; is the number of QMS centers (a factor whose value should be found); = = where ,
are the average specific costs of restoring a network element and maintenance, respectively; , are the given constants; БР is the average failure-free life, which is determined by the formula БР =
( + ) + ⋅ (1 − (−( + ) ⋅ ( + ) )) , in which is a constant: is the failure rate, = 1/ , is the average recovery time = + ; is a constant related to the restoration of operability; = / is the time spent on moving from the OCI’s emergency response center to the network element, = average speed of brigades’ movement, is the distance between points.
Е and = ∙ ∙ (1/ БР + 1/ ) are the specified constants.
The values of the functionalities of the quality of operation of the lower-level substation are given in the form of a table in the form of a correspondence , ⇔
, ∈ , ∈ , = 1,2,3, where , is a string of values of “connecting” (“linking”) characteristics, , is a string of eigenvalues of characteristics of the lower-level PS, is a set of indices of variants of values of characteristics of the lower-level PS. 4.3. Algorithm for reconciling decisions in a three-level hierarchical system The algorithm for matching solutions of a three-level hierarchical system that models the solution of the OCI WRM problem can be described by the following sequence of steps.
Step 0. Reading the data required for the algorithm to function: variants of the values of general characteristics, variants of the values of the characteristics of the substations of all three levels. Calculate or explicitly enter the weighting coefficients of the relative importance of the PS for the functioning of the system as a whole. Calculation or specification of the optimal and worst values of the quality criteria for the functioning of all the PSs of the is calculated on the next variant of the parameter values, the values of the lower-level PS criteria are searched for in the correspondence tables and these values are converted to a dimensionless form. The next and values of are also calculated, respectively. If > dimensionless form. The next and values of , are also calculated, respectively. If > , , and or to a , or and = > , proceed to step 4. Otherwise, the values are = , = . Go to step 4. 5. Prospects for further research In further research, it is advisable to consider and improve the solution of the problem described in this paper for other classes of problems. In particular, it is promising to formalize the described problem in other classes of mathematical problems:
Determination of the collective resultant ranking
of all applications [
Determination of the relative importance of
individual applications [
Building a functionally stable QMS system, i.e.
ensuring redundancy in the QMS system and its
reasonable use [
Introduction of the concept of criticality categories
of CMI [
making decisions on the reliable and
uninterrupted operation of CMI’s CMM [
Thus, the paper considers the problems of ensuring the functioning of the critical infrastructure system. The relevance of the study is due, in particular, to the massive attacks on Ukraine's critical infrastructure during Russia’s large-scale aggression since February 2024. The paper presents a mathematical model of the problem of maintenance of a system of critical infrastructure facilities developed by the authors. The authors propose a scheme for solving the problem of ensuring the operation of a system of critical infrastructure facilities. The procedures 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 presented. An algorithm for coordinating decisions in a three-level hierarchical system is developed and described.
Protecting
Europe’s
Critical Infrastructures:
Problems and
Prospects,
J.
Contingencies Crisis Manag. 15(1) (2007) 30–41. L. Labaka, J. Hernantes,
M. Sarriegi,
A
Holistic Framework for
Building
Critical Infrastructure Resilience,
Technological Forecasting &
Social Change, 103 (2016) 21–33.
P. D. Wright, M. J. Liberatore, R. L. Nydick, A Survey of Operations Research Models and Applications in Homeland Security, Interfaces, 36(6) (2006) 514–529. S. O. Johnsen,
M. Veen,
Risk
Assessment and Resilience of Critical Communication Infrastructure in Railways, Cognition Technology & Work, 15 (2013) 95–107.
330–350. [5]
L. F. Gay, S. K. Sinha, Resilience of Civil Infrastructure Systems: Literature
Review for Improved asset
Management, Int. J. Critical Infrastructures, 9(4) (2013) [6]
V. Hrechko, H. Hnatienko, T. Babenko, An Intelligent Model to Assess Information Systems Security level, Fifth World Conference on Smart Trends in Systems Security and Sustainability (WorldS4) (2021) 128–133. [7]
A. Fekete,
K. Tzavella,
R. Baumhauer,
Spatial Exposure Aspects Contributing to Vulnerability and Resilience
Assessments of
Urban
Critical Infrastructure in a Flood and Blackout Context, Natural
Hazards, 86 (2017) 151–176.
doi: [10] J. W. Herrmann, Using Operations Research Methods for
Homeland
Security
Problems. Handbook Operations
Research for
Homeland
Security, International Series in
Operations Research of &
Management Science, 183 (2013) 1–24. [11] D. Řehák, et al., Strengthening Resilience in the