CC BY
0
БЕЗОПАСНОСТЬ В ЧРЕЗВЫЧАЙНЫХ СИТУАЦИЯХ
SAFETY IN EMERGENCY SITUATIONS
УДК 004.05, 006.015.8, 519.718, 519.876.2 doi: 10.21685/2307-4205-2024-1-14
AN INDEX-BASED METHOD FOR INTEGRAL ESTIMATION OF REGIONAL CRITICAL INFRASTRUCTURE RESILIENCE USING FUZZY CALCULATIONS (PART 1. PROBLEM STATEMENT AND METHOD GENERIC STRUCTURE)
A.V. Masloboev
Putilov Institute for Informatics and Mathematical Modeling of the Federal Research Centre "Kola Science Centre of the Russian Academy of Sciences", Apatity, Russia;
Nanomaterials Research Centre of the Federal Research Centre "Kola Science Centre of the Russian Academy of Sciences", Apatity, Russia
masloboev@iimm.ru
Abstract. Background. The study is aimed at developing well-known and designing novel models and methods for decision support in the field of security and resilient operation management of critical infrastructures and socioeconomic facilities in the Arctic region of Russian Federation. This urgent problem is especially relevant at the regional level in terms of the need to protectability heightening of critical facilities/infrastructures, cascading effects restricting of the multiple threats of various nature on higher-level systems and favorable conditions providing for mitigation of the negative consequences of influencing factors on the performance of the elements of these systems. Materials and methods. For easy understanding, the work structurally is decomposed in two parts. In the first part, a formal problem statement is given. The substantiation of mathematical apparatus for problem-solving is carried out. The generic framework of the developed method for assessment and analysis of the regional critical infrastructures resilience based on a fuzzy-set approach and expert judgements is proposed. In the second part, the efficiency Q-function computational models of the organizational and technical systems resilience, such as anticipation ability, responsiveness, recoverability and adaptability, which are the central elements of the optimization model of critical infrastructures resilience integral index, are examined. Results and conclusions. An index-based method for the integral estimation and analysis of the regional critical infrastructures resilience, based on fuzzy calculations of the level and ratio of aggregated reliability, security and robustness indices, has been developed. The method allows on basis of incomplete data to quantify systemic risks affecting the critical infrastructure resilience, performances, savings and possible losses under sampling and implementing the anti-crisis measures at all stages of the resilience management life-cycle. A distinctive feature of the method is its universality, i.e., applicability to all types of critical infrastructures. The method can be practically used by operators and analysts of regional situational centers to train and generate design decisions for counteracting the actual threats and local failures in the operation of regional critical infrastructures under uncertainty.
Keywords: system analysis, resilience, security, critical infrastructure, integral performance index, expert judgement, fuzzy calculations
Financing: the work was carried out within the framework of the State Research Program of the Putilov Institute for Informatics and Mathematical Modeling KSC RAS (project No. FMEZ-2022-0023).
For citation: Masloboev A.V. An index-based method for integral estimation of regional critical infrastructure resilience using fuzzy calculations (Part 1. Problem statement and method generic structure). Nadezhnost' i kachestvo slozhnykh sistem = Reliability and quality of complex systems. 2024;(1):124-141. (In Russ.). doi: 10.21685/2307-4205-2024-1-14
© Masloboev A.V., 2024. Контент доступен по лицензии Creative Commons Attribution 4.0 License / This work is licensed under a Creative Commons Attribution 4.0 License.
ИНДИКАТОРНЫЙ МЕТОД ИНТЕГРАЛЬНОЙ ОЦЕНКИ ЖИЗНЕСПОСОБНОСТИ РЕГИОНАЛЬНЫХ КРИТИЧЕСКИХ ИНФРАСТРУКТУР НА ОСНОВЕ НЕЧЕТКИХ ВЫЧИСЛЕНИЙ (ЧАСТЬ 1. ПОСТАНОВКА ЗАДАЧИ И ОБЩАЯ СТРУКТУРА МЕТОДА)
А. В. Маслобоев
Институт информатики и математического моделирования имени В. А. Путилова Федерального исследовательского центра «Кольский научный центр Российской академии наук», Апатиты, Россия; Центр наноматериаловедения Федерального исследовательского центра «Кольский научный центр Российской академии наук», Апатиты, Россия
masloboev@iimm.ru
Аннотация. Актуальность и цели. Исследование направлено на развитие известных и разработку новых моделей и методов поддержки принятия решений в области управления безопасностью и устойчивым функционированием критических инфраструктур и социально-экономических объектов Арктической зоны Российской Федерации. Эта задача особенно актуальна на региональном уровне с точки зрения необходимости повышения защищенности критически важных объектов/инфраструктур, сдерживания каскадных эффектов воздействия множественных угроз различной природы на системы более высокого уровня и обеспечения благоприятных условий для смягчения негативных последствий влияющих факторов на состояние работоспособности элементов этих систем. Материалы и методы. Работа состоит из двух частей. В первой части дана формальная постановка задачи, приводится обоснование математического аппарата для ее решения и представлена общая структура разработанного метода оценки и анализа жизнеспособности региональных критических инфраструктур на основе нечетко-множественного подхода и экспертных оценок. Во второй части исследуются вычислительные модели целевых функций качества устойчивости организационных и технических систем, таких как упреждае-мость, реактивность, восстанавливаемость и адаптируемость, являющихся центральными компонентами оптимизационной модели интегрального показателя жизнеспособности критических инфраструктур. Результаты и выводы. Разработан индикаторный метод интегральной оценки и анализа жизнеспособности региональных критических инфраструктур, основанный на нечетких вычислениях уровня и соотношения агрегированных показателей надежности, безопасности и устойчивости. Метод позволяет на основе неполных данных количественно оценить системные риски, влияющие на жизнеспособность критической инфраструктуры, полезные эффекты и возможные потери при выборе и реализации антикризисных мер на всех стадиях жизненного цикла управления устойчивостью. Отличительной особенностью метода является его универсальность, т.е. применимость ко всем типам критических инфраструктур. Метод может быть использован операторами и аналитиками региональных ситуационных центров для подготовки проектных решений по противодействию актуальным угрозам и локальным сбоям в работе критических инфраструктур региона в условиях неопределенности.
Ключевые слова: системный анализ, жизнеспособность, безопасность, критическая инфраструктура, интегральный показатель, экспертная оценка, нечеткие вычисления
Финансирование: работа выполнена в рамках государственных заданий ИИММ КНЦ РАН (НИР № FMEZ-2022-0023).
Для цитирования: Masloboev A. V. An index-based method for integral estimation of regional critical infrastructure resilience using fuzzy calculations (Part 1. Problem statement and method generic structure) // Надежность и качество сложных систем. 2024. № 1. С. 124-141. doi: 10.21685/2307-4205-2024-1-14
Introduction
For several decades, the main emphasis and efforts of developers and experts in the field of risk management of critical infrastructures of various types and levels have been performed on the engineering of security systems and means for comprehensive protection of these essential elements of socio-economic and natural-industrial systems. Real experience shows that in practice ensuring absolute security is an unattainable goal and unjustified in terms of the resources spent. Therefore, at present, the focus of attention of Russian and foreign researchers is gradually shifting towards studying the issues of developing and analyzing resilient systems that can withstand multiple threats of various natures. Resilience is a fundamentally new concept that significantly expands the modern theory of the safety of complex systems in terms of the development of approaches to managing the processes of timely recovery and adaptation of critical functions and operating characteristics of dynamic systems under the influence of negative factors. The application of such approaches allows, along with traditional problem monitoring and preventive risk management, to mitigate the
consequences of multiple threats implementation within critical infrastructures and facilities, as well as to design flexible self-organizing security systems in order to effectively counter these threats. Meanwhile, it is significant to note that due to the large variety of diverse aspects of resilience, measuring and maintaining this inherent performance characteristic of complex dynamic systems is a nontrivial interdisciplinary problem.
In modern scientific literature for different research disciplines and classes of systems, accordingly, one can find many different, but similar in meaning, definitions of the "resilience" concept, which historically arose in the study of the environmental sustainability of natural systems, and then became the objective of analysis in closely-related fields. For the general case in this work, the resilience of critical infrastructure is understood as such a property of self-preservation of the system, in which elements of the critical infrastructure and the infrastructure (system) as a whole, subjected to the influence of negative factors, are able to anticipate and absorb the consequences of the multiple threats intra-implementation, and timely restore their functionality (critically important functions) and adapt to changing operating conditions, thereby effectively resisting the destructive impacts and dangers emanating from objects of the external and internal environment.
In a broad sense, critical infrastructure is an important part of socio-economic systems and is a set of interdependent significant objects distributed over a certain territory that perform critical functions to ensure the normal operation of the social sphere, the economy and the state as a whole. Violation or loss of functionality of at least one of these objects can lead to a significant decrease in the level of safety and quality of life of the population, including public health and well-being, as well as have a destructive impact on other objects of this system, destabilizing the entire system and obstructing the preservation of essential functions necessary for its sustainable development. Thus, critical infrastructure, in essence, is a life-supporting subsystem for higher-level systems: local, regional, federal and international. Any critical infrastructure is exposed to external threats and has its own life-cycle of responding to crisis situations and adverse events. It is customary to distinguish between "hard" (physical and virtual objects and networks, production assets, technical systems) and "soft" (socio-economic objects and systems) critical infrastructures.
To ensure effective management of the soft critical infrastructures resilience, it is necessary to conduct an integrated assessment and audit of their stability, reliability and safety on a regular basis. Meanwhile, such an assessment is hampered by the lack of retrospective data on typical critical situations and the incompleteness/limitation of information on current threats and the ways of their impact on elements of critical infrastructure. At once, most monitoring systems used in regional situational centers are just not designed for modeling and assessing factors affecting the resilience of critical infrastructures. Additional difficulties for preventive analytics are arisen by the human factor, which is not always measurable. In addition, the integral resilience assessment uses various quantitative and qualitative indicators, which must be measurable, realistic, attainable, time-bound and agreed upon, as well as reflect the specific context and class of resilience (technological, organizational, personal, societal, etc.). However, not all reliability, safety and stability indices, as well as the factors influencing them, can be taken into account in the assessment, since they cannot be measured. This reduces the adequacy of known resilience models and the validity of resilience total estimates derived on their basis. This is especially typical for the resilience of "soft" critical infrastructures, in which critical situations are unique each time, that is, rarely repeated, and the uncertainty, fuzziness or insufficiency of initial data for analyzing the system behavior before, during and after the occurrence of adverse events can only be compensated by involving a team of experts in the field corresponding to the problem context and resilience domain.
Thus, an integral assessment of the critical infrastructures resilience should be carried out based on the use of formal models of aggregated indicators of reliability, safety and stability of dynamic systems in combination with expert judgements and fuzzy calculation methods. This will ensure the accounting of data uncertainty and expert subjectivity when assessing the resilience of critical infrastructures.
Subject to the considered features, this study is aimed at developing tools for information and analytical support of the situational control of multi-level distributed systems resilience, in particular, novel methods and models for assessing and analyzing the resilience of regional critical infrastructures, taking into account both organizational and technical factors influencing operating performance characteristics of this class of systems. The work is divided into two parts. The first part consists of three sections. The first section discusses the methodological foundations of the study and provides a rationale for the choice of mathematical apparatus for solving the stated problem. The second section provides a formal problem statement. The third section is key and encloses a description of the general structure of the developed index-based method for integral estimation of the regional critical infrastructures resilience, based on fuzzy calculations of the level and ratio
of aggregated reliability, safety and stability indices. Herein, the corresponding formalism and mathematical manipulations are expressed. The summary outlines the highlights, strengths and limitations of the proposed fuzzy approach to analyzing and measuring the critical infrastructures resilience. Conclusions are drawn on its applicability for the class of systems under examination.
Background and methodology
To date, the academic literature presents various approaches and methods of assessment and analysis of the system resilience of various types and scales, including critical infrastructures. The well-known methodologies can be conditionally classified as follows: empirical methods based on observations of the operating conditions of an object or facility, systematic problem monitoring of its condition, data acquisition and statistical analysis for the purpose of running experiments and testing hypotheses; heuristic methods based on practical experience and intuition of risk managers, security experts, analysts and operators of critical facilities and infrastructures; simulation methods based on mathematical and computer modeling; conceptual approaches (frameworks) based on conceptual analysis and ontological modeling; index-based methods based on the metrics (a system of state indicators) design of an object/process and analysis of the dynamics of these indicators.
Empirical methods are used to quantify the resilience of a system. When using empirical methods, based on the analysis of retrospective data on the system behavior over time, curves for the number of failures and recovery of the system after a malfunction are constructed. At the same time, empirical methods are divided into deterministic and probabilistic. Deterministic methods typically assume that the system's performance is in a perfect condition before the failure, and that measures to restore the system's functionality are taken immediately after the failure. However, de facto organizational and technical systems degrade over time, and the process of their restoration is characterized by delays in results due to the influence of various factors, including force majeure circumstances, e.g., financial restrictions, logistics availability or a long-term process of making managerial decisions. To effectively assess the resilience of a system, it is necessary to first identify these influencing factors and then quantify their impact. However, most studies based on empirical approaches often ignore these types of influencing factors and, in addition, rely substantially on the homogeneity, availability and validity of historical data on the manner of system behavior. In reality, such assumptions are in many cases not always justified and acceptable. Despite this, empirical methods allow one to obtain more accurate and reliable (valid) results compared to heuristic approaches, although they are more resource-intensive. Heuristic methods do not always guarantee an optimal strategy for enhancing the resilience of a system, but they can ensure prompt decision-making when the system does not have step-by-step instructions (a clear algorithm) on how to operate in previously unknown critical situations.
On the other hand, methods based on computer modeling simulate the system behavior and dynamically model its performance characteristics over time. These methods are quantitative and allow one to examine how the system structure and components affect the resilience of the system as a whole. They can be applied to various classes of systems (industrial, transport, natural, socio-economic, technical). To quantify resilience, these methods typically use fuzzy models, Bayesian networks, Monte Carlo simulations, optimization methods, etc. Bayesian networks are usually used when there is uncertainty and incompleteness of the initial data on the system status and behavior. State-of-the-art studies are based mainly on the assumption that systems are of a static nature, that is, they do not take into account the time dimension. This is done for the convenience of studying a specific object. However, most real systems still have a dynamic nature and Bayesian networks are widely used today to analyze the characteristics ofjust such systems. For example, to estimate the resilience of energy critical infrastructure, an approach based on a combination of dynamic models of Bayesian networks and Markov chains is used. When modeling the resilience of systems, optimization methods are traditionally used to determine the optimal operating modes of the system with given operating characteristics and target indicators of reliability, fault tolerance and security of the system, as well as with minimal resource costs for the process of restoring the system after a failure of one or more of its elements. Developing adequate simulation models to assess the resilience of large-scale systems is a complex, expensive process that requires substantial time outlay and the involvement of highly qualified specialists from various fields of knowledge. At once, these models may require extensive operational data, as well as a variety of multi-faceted technical and management information. Thus, the development and application of simulation-based methods and models for assessing resilience is limited only by the capabilities of experts, analysts and organizations performing security auditing and risk analysis of malfunction of the systems of a particular class.
Conceptual approaches (frameworks) provide a qualitative assessment of the resilience of complex systems and are typically used to conceptualize the resilience framework. For this purpose, conceptual, ontological or other types of graph models are designed that reflect the semantic matter of the resilience category and components, as well as the relationship between the system resilience and the factors influencing it. Since these models are qualitative, they are sufficiently rarely used in resilience assessment of industrial and technical systems, but they show effectiveness in analyzing the resilience of organizational and socio-economic systems.
Index-based approaches are based on a comprehensive analysis of the target quantitative and qualitative indicators of the operating and service conditions of the system, which makes it possible to examine and evaluate the dynamic characteristics of the system resilience including influencing and contributing factors. In this case, the integral resilience index is calculated by aggregating selected indicators and identified factors using the weighted average estimation method. Index-based approaches belong to the class of semi-quantitative methods, which ensures their applicability for assessing the robustness and resilience of both engineering and technical systems, as well as socio-economic facilities and infrastructures. The generic structure of index-based methods takes into account the weighting coefficients for the resilience indicators and system components and relies on quantitative and qualitative historical and operational data on the system behavior before and after the occurrence of critical events. To minimize the uncertainty of the raw data and reduce the dependence of the output results of index-based methods for measuring resilience on the quality of input information, fuzzy modeling and expert judgements are used. Fuzzy estimation methods also make it possible to qualitatively take into account observable and unobservable factors that have a direct or indirect impact on the target indicators of system resilience (reliability, robustness, recoverability, adaptability, etc.). Meanwhile, it is worth noting that the state-of-the-art index-based methods scientifically are not sufficiently developed in terms of identifying and quantifying these influencing factors. Thus, to date, the following index-based approaches are widely known [1, 2]:
- Resilience Framework BRIC (Baseline Resilience Indicators for Communities) [3], USA;
- Resilience Framework P.E.O.P.L.E.S (Population and demographics, Environmental and Ecosystem, Organized governmental services, Physical infrastructure, Lifestyle and community competence, Economic development, Social-cultural capital) [4], USA;
- The Benchmark Resilience Tool (BRT) [5], New Zeeland;
- Guidelines for Critical Infrastructures Resilience Evaluation ("Guidelines"/CIRE) [6], Italy;
- The Critical Infrastructure Resilience Index (CIRI) [7], Norway;
- Resilience Management/Measurement Index (RMI) [8], USA;
- Organizational Resilience Health Check (ORHC) [9], Australia;
- Resilience Analysis Grid (RAG) [10], Denmark;
- "Swiss Approach" [11], Switzerland;
- Resilience Matrix-based Integral Estimation Approach (RMIEA) [12], Russia;
- Resilience Multi-dimensional Data Cube (ROLAP-cube) [13], Germany.
Applications of these methods have been successfully reflected in the practice of managing the safety and resilience of critical infrastructures and large-scale systems at the regional level. However, in practice, difficulties arise in obtaining an integral estimation of the system resilience, since there are many dimensions of resilience domains (organizational, technological, ecological, socio-economic, etc.), and their matter and characteristics differ significantly. At the same time, for the completeness of the assessment it is necessary to join everything into a comprehensive whole on the basis of a uniform, clear methodology suitable for all classes of systems, types of critical infrastructures and domains of resilience. To solve this problem a fuzzy index-based method was designed, which is proposed in this study and combines the advantages of well-known risk analysis practices of complex systems and resilience assessment methodologies.
The choice of the mathematical apparatus of fuzzy set theory for developing a method for integral estimation of the regional critical infrastructures resilience is conditioned by the fact that the fuzzy set approach allows taking into account not only the uncertainty and fuzziness of the input data, which are characteristic of the class of systems under examination having a complex heterogeneous structure and component composition, as well as depending on factors of various nature, but also the preferences of experts when making decisions on the targeted management of critical infrastructure. The integral critical infrastructure resilience index based on fuzzy estimates is calculated by combining fuzzy variables and fuzzy logic rules into a unified model that allows assessing the risk level of critical infrastructure and determining the best strategy to improve and enhance the resilience of this infrastructure (system). This index is presented in the form of a certain numerical value that reflects the magnitude of critical infrastructure resilience under the influence of external factors and given operating characteristics of its system elements.
Problem statement
The resilience of dynamic systems such as critical infrastructures is characterized by a multi-purpose directedness, that is, the need to simultaneously achieve several goals subject to situational factors. This typically requires evaluating and optimizing multiple metrics (indices) that meet those goals and specified constraints. Moreover, these indicators may not be consistent and be in mutual contradiction with each other, when improvement in one of the indicators leads to deterioration in another, and vice versa, and meeting the requirements of all criteria is not possible in some conditions. However, criteria and restrictions are not always defined precisely. For such cases, the search for optimal solutions is impossible without taking into account qualitative information on the preferences of various criteria, the expected manner of the system behavior (increasing or decreasing the parameters of the quality objective functions), the dynamics and acceptable limits of system efficiency indices, etc. Thereby, the given problem of analysis and assessment of the critical infrastructures resilience is a fuzzy multi-criteria problem.
The main problems in analyzing and assessing the resilience of complex systems at the stages of the crisis management life cycle (risk assessment, prevention, preparedness, warning, response, recovery, mitigation and learning) [14, 15] are the complete or partial lack of information on the quantitative characteristics of the system status, as well as the subjectivity of experts in determining and assessing these indicators. Thus, when predicting the occurrence of the adverse events in the process of situational control of the system resilience, it is often necessary to evaluate only the expected values of performance quality indicators of the system, that is, the most likely loss of functionality in the event of malfunctions and disruptions in its operation. In real practice, this is usually implemented at a qualitative level, since a considerable part of the system resilience characteristics due to their subjective nature is less amenable to quantitative measurement than generally adopted reliability, safety and fault tolerance indicators of technical systems.
To calculate the integral resilience index of a system (critical infrastructure), which depends on many parameters, it is necessary, first of all, to evaluate a set of basic indicators that characterize the system ability to self-preserve and maintain this protective property, as well as to estimate them taking into account the context of the situation, influencing factors, type (soft/hard) and domain (technological, organizational, personal, cooperative) of resilience. A set of basic indicators include: anticipation and prevention ability, absorbability and responsiveness, robustness and recoverability, adaptability and self-organization ability. Such an integral assessment is obtained on the basis of a mixed (additive-multiplicative) convolution of partial criteria and indices that form these indicators at the lower level of decomposition. In addition, the calculation of the integral index of system resilience should take into account the assessment of the level and ratio of the following expected effects from the implementation of control actions aimed at enhancing the system resilience: system performance (useful effect), resource and cost savings, risk of unexpected failures and losses. The backbone components of a multi-level metrics system for integral estimation of the regional critical infrastructures resilience are represented in Table. 1.
Next, the formal problem statement of system resilience assessment will be considered.
Let it be given the following initial problem specification:
RCP = {rcp1,rc2,...,rcpn} is a set of strategies (control programs) to maintain the resilience of regional critical infrastructure, optimizing the performance characteristics of the system elements of this infrastructure;
RC = {RCAP,RCar,RCrec,RCAD} is a set of backbone (target) capacities of system operation performance, characterizing the ability of critical infrastructure to self-preservation and resilience maintenance under the influence of multiple threats (influencing factors), where: RCAP a is system anticipation and prevention ability, RCAR a is system absorbability and responsiveness, RCREC is a system robustness and recoverability, RCAD is a system adaptability and self-organization ability;
P , S, R is a set of expected effects from the implementation of strategies (control programs) RCP aimed at enhancing the system resilience, where: P is a useful effect (system performance), S is a resource effect (amount of resources saved, operationability), R is a cumulative effect of the probabilities of unforeseen events emergence (risk level of system functionality losses, the amount of possible damage);
IND = {lNDRCp ,INDRCar , INDRCrec , INDRCad } is a set of quantitative and qualitative indicators that are
used for multifactor analysis and index-based estimation of target quality functions of the system, as well as multicriteria analysis of applied strategies for managing the resilience of critical infrastructure, taking into account the required (predicted) values of performance characteristics and efficiency indices P ,S, R , where:
INDRCap ={IX1 ,IX2,...,IXk. } is a set of partial indicators of the lower level of assessment decomposition, on the basis of which an overall index of the system anticipation and prevention capability
RCAP = { INDRCp | (( : S1AP : RAP) — opt} is calculated in terms of expected effects and strategies RCP are evaluated, accordingly;
INDrCr ={IX , IX2,..., LKf} is a set of partial indicators of the lower level of assessment decomposition, on the basis of which an overall index of the system responsiveness and absorption capacity RCar = { INDRCar | (PAR : S1AR : Rar ) — opt} is calculated in terms of expected effects and strategies RCP are evaluated, accordingly;
INDrCrec = {IX1, IX2,..., IXm,} is a set of partial indicators of the lower level of assessment decomposition, on the basis of which an overall index of the system robustness and recoverability RCREC = { INDRCsec | (PREC : S1REC : RREC) — opt} is calculated in terms of expected effects and strategies RCP are evaluated, accordingly;
INDrCd = {IX1,IX2,...,IX*} is a set of partial indicators of the lower level of assessment decomposition, on the basis of which an overall index of the system adaptability (adjustment shift) RCAD = { INDrCd | (PAD : S1AD : RAD) — opt} is calculated in terms of expected effects and strategies RCP are evaluated, accordingly.
Table 1
Backbone components of integral estimation of the regional critical infrastructures resilience
Resilience Type & Domain Soft (socio-economic systems) / Hard (engineering systems) Technological, Organizational, Personal, Cooperative, Societal, Ecological, Cyber, etc.
Expected Contribution Resilience Capacity
Anticipation & Prevention Ability ( RCap ) Absorbability & Responsiveness ( RCar ) Recoverability ( RCrec ) Adaptability ( RCad )
Performance & Useful Effect (P ) - Preparedness Degree - Reliability - Detection ability - Prognostic & Health Management (PHM) - Robustness - Fragility - Stress rate testing - Damage Level & Limitation Exercise (Limits of disruption, deviation & negative disturbance) - Maintainability (technological repairability) - Supportability - Modularity - Segregability - Decomposability - Restoration Index - Resistance/Resistivity - Downtime reduction - Technological Upgradability - Technological Transformability - Integrability - Interoperability - Composability - Reconfiguration Ability
Resource & Cost Saving (s) - Planned Maintenance - Joint Activity Cooperation Plan (Agreements) - Internal Redundancy - Resource Deployment - Coordination (Degree of Concordance) - Communication Plan & Facilitation Ability - Facilities & Assets Reduction - Safety Margin - External Redundancy - Unplanned Maintenance - Reduced Service Level & Costs - Personnel Availability - Recovery Time - Personnel Availability - Spare Parts Availability - Resource Storage Capacity - Long-term/short-term Reconstruction
Loss & Failure Risk (R) - Protectability - Operability - Error & Disturb Sensitivity - Vulnerability - Independency - Situational Awareness - Functionality - Feasibility - Autonomy - Insurance - Restart ability - Self-organization ability - Creativity & Improvisation
Resilience Life-Cycle Analysis Phase Risk Monitoring & Identification (Early Warning, Alarm, Pre-Event Risk Auditing) Risk Initiation & Implementation (Acting Threat & Impact Absorption, System Response) Risk Treatment & Evaluation (Elimination & Reduction of Post-Event Consequences) Risk Mitigation & Learning (System Features & Capabilities Strengthening)
To determine the values of indicators an assessment procedure based on expert judgements and para-metrization is used. Let E be the number of experts involved in the system examination and expertise.
Then the problem of integral estimation of the critical infrastructure resilience is formulated in the form of two interconnected subproblems, by solving which it is possible to determine a set of effective strategies for managing the system resilience under given conditions and restrictions.
The first problem expects ranking the elements of multiple strategies set RCP according to aggregated indicators INDRCp,INDrCr ,INDRC ,INDrCd c IND taking into account the preferences of E experts and
the competence of expert judgments (assessments).
The second problem using the output results of the first problem solved expects evaluating the optimal strategy RCP* c RCP for ensuring and maintaining the resilience of the system (critical infrastructure) on the basis of calculating target efficiency indices of system operation performance RCAP, RCar , RCrec , RCAD
for each strategy rcpi e RCP, i = 1,n in terms of the ratio of design values of the expected effects P , S, R from the implementation of the strategy and the obtained aggregated estimates of the partial indicators of system resilience INDrCp ,INDrCr ,INDRC ,INDrCd c IND , that is, for each strategy rcpi e RCP, i = 1,n it
is required to calculate the overall resilience index ORI of the system by a set of parameter estimates of the quality objective functions (resilience capacities) RCAP,RCAR,RCREC,RCAD decomposed subject to the types and domains of resilience involved in the analysis, as well as subject to the selected context and situational factors. Based on the output results of calculations and comparison of the obtained integral estimates of indicators for each strategy, it is needed to evaluate the optimal program for managing the system resilience acceptable and balanced for the given conditions and restrictions.
Problem-solving procedure and method structure
The general structure of the method for integral multi-level assessment of the resilience of regional critical infrastructures is shown in Fig. 1.
Fig. 1. The hierarchy framework (structure) of proposed methodology for estimation of regional critical infrastructure resilience
To solve the first subproblem the standard method proposed in study [16], well-tested on the problems of assessing the effectiveness of innovation and investment projects, is used. Guided by this method, sets of
quantitative and qualitative indicators IND = {lNDRCp, INDRCr, INDRCrec , INDRCd } characterizing the basic
performance criteria of the system operation are represented as fuzzy sets and are specified using membership functions in the form of triangular or trapezoidal fuzzy numbers on universal sets of alternative strategy options RCP. The membership functions of fuzzy sets are determined on the basis of expert data on paired comparisons of alternatives using the T.L. Saaty analytic hierarchy process [17]. The ranking of alternatives based on the intersection of fuzzy sets is implemented according to the R.E. Bellman and L.A. Zadeh principal scheme known in decision theory [18].
When assessing indicators, experts determine the minimum possible, the most optimistic (or the range of the most optimistic values) and the maximum possible values of estimates of these indicators (influencing factors) at the appropriate level of metrics decomposition of the system resilience indicators.
To transform fuzzy explicit expert judgments on the state of system resilience indicators into crisp characteristics for the purpose of follow-up processing its scores in a defuzzified form, a defuzzification procedure for the obtained fuzzy expert estimates is applied to fuzzy sets, based on the principle of selecting a control action corresponding to the maximum value of the membership function and the centroid method of defuzzification (mean-center method) [19].
All obtained triangular fuzzy estimates for each indicator at the level k must be aggregated. Therefore, when the number of experts is E and when the triangular fuzzy estimate of the e -th expert judgment on the
current state of the j -th indicator at the level k is iXcpe = (ixj, ix2cije, ix]ije), e = 1,E, the integral triangular
fuzzy estimate of all expert judgments (IXcij) on the state of the j -th indicator at the level k can be obtained based on following formulation:
IX c
= (, К, К ) =
min
E
У ix2 L-! ci
, max
E
R}
e = 1, E,
where «~» is a sign denoting a fuzzy set; ixj, ix2a]e, ixj are the minimum possible, the most optimistic and the maximum possible values of the e -th expert judgment of the current state of the j -th indicator at the level k , respectively.
Aggregated triangular fuzzy numbers are fuzzy expert judgments (estimates) of the current state of indicators at the level k. As a result of defuzzification, these fuzzy estimates are converted into crisp estimates. The crisp value of a triangular fuzzy number is evaluated by its crisp probabilistic average value. Then, a crisp estimate of the current state of the j -th indicator (IX ) at the level k can be obtained using the
following formulation:
К = M (.=
(ix1.. + 4ix2. + ix3.
\ ci CIJ CIJ
)
The score of each indicator at the level (k -1) is calculated as a weighted sum of the scores of the partial indices generating it at the lower level k using the formulation:
^ n,m A m
(X )k-1 = I WjIXcj , X Wj = L
v',j=1 A j-1
where m is the number of partial indices associated with i -th indicator at the level k -1; wj and IX are the weight and estimate of the j -th index at the level k, respectively, and take values in the range [0, 1].
In a similar way, index scores are obtained for higher levels of indicators (k-2), (k-3),... by bottom-up aggregating till top of the assessment hierarchy:
f l ,n
(X )k-2 = У vX
V c,i=1
, У •
У к-1
1
RELIABILITY AND QUALITY OF COMPLEX SYSTEMS. 2024;(1)
(XL = [iu^c] , & =
c=l y k-2 c=1
At the top level of hierarchical system of the assessment criteria and index decomposition, the generalized estimate of the overall index of critical infrastructure resilience is calculated as the weighted average of the estimates of all characteristics of the considered dimensions (domains) of system resilience using the following formulations:
l RD
ORI =-i p IX
RD p
p=i
1 RD 1 l 1 n 1 m ORI =— i p 1 i u - i v. - i wIX
RD p=1 1 c=1 n i=1 m j=1
" cij
m j=1
i uc = i vi = i wj = 1,
c=1 i=1 j=1
where IX and p are the final estimates and weights of the p -th resilience domain of critical infrastructure at the upper level of system decomposition, respectively; IXcij. are estimates of partial indices of the critical infrastructure resilience at levels (k - 2), (k-1), (k), respectively; 1, n, m,... are the numbers of estimated resilience indices of critical infrastructure at levels (k-2), (k-1), (k), respectively; uc,vi,wj,... are the weights of the partial resilience indices of critical infrastructure agreed upon by experts and determined at levels (k - 2), (k-1), (k), respectively; RD is a number of considered resilience domains for the critical
infrastructure under examination.
The measure of agreement of defuzzified expert judgments (estimates) is determined on the basis of calculating the coefficients of concordance and variability (spread in values), taking into account not only paired comparisons, but also the general structure of the assessments. If the values of these coefficients are more than 0.7 and less than 0.2, respectively, then the expert judgments are considered agreed upon. Otherwise, to achieve and provide agreement among expert judgments, the Delphi technique or its modifications are used.
To determine the ranks of system performance criteria (resilience indicators) at all levels of index-based assessment, experts use a matrix of paired comparisons of these criteria, which has the property of transitivity and is designed on a nine-point T.L. Saaty scale.
The reliability of estimates of the criteria ranks depends on the completeness and quality of expert data used in generating the matrix of paired comparisons, and is determined on the basis of relative agreement index calculation, which characterizes the degree of violation of the numerical (cardinal) and transitive (ordinal) agreement of paired comparisons. According to the study [17], in practice, acceptable agreement of paired comparisons should be no more than 10 % (for a range of problems up to 20 % is allowed). Otherwise, experts need to reconsider their judgements (estimates).
After determining the membership functions, ranks of criteria and ranks of experts, running the aggregation of estimated quality indices into a quantitatively measurable integral performance criterion is carried out. According to research [16], for this purpose it is advisable to use R.R. Yager convolution [20], designed for the case of unequal criteria and most fully reflecting the qualitative nature of setting the preferences of experts when engineering an integral performance index model, in comparison with additive and multiplicative convolutions. In addition, it is noted that when using this type of convolution, the compensation effect is leveled out, when inadequate ratings for ones criteria can be compensated for by high ratings for others.
As a result of performing all the above operations, a rating of system resilience management programs is constructed by each expert individually or by a special expert team. In the case of a group examination, the judgements of each expert must be raised to the power of his competence, i.e., adjusted. For this purpose, a paired comparisons matrix of the expert judgements competence is also constructed and the eigenvector of this matrix is evaluated, a multiplicative convolution of the results obtained at the previous steps is performed.
In this way, an overall quality rating of system resilience management programs is generated subject to the preferences and judgements (estimates) of all experts. The optimal strategy according to the maximin approach [21] is such a control program that provides a guaranteed result subject to the significance measure of criteria, ranks and competence of expert judgements, i.e., obtaining a Pareto optimal problem solution while meeting the requirements of specified performance criteria and restrictions.
The algorithm for calculating the overall index of system resilience based on the stages of fuzzy calculations discussed above is schematically outlined in Fig. 2.
Fig. 2. The established procedure (algorithm) for overall resilience index (ORI) fuzzy calculation
Formal manipulations and procedures for calculating membership functions, the measure of concordance of the expert judgements, criteria ranks and expert ranks, as well as determining the competence of expert judgements are quite general for solving a wide range of problems and are expounded in detail in the work [16]. This mathematical apparatus found an application to the problem assigned in our study and was adopted to solve it in terms of ranking alternative strategies for managing the resilience of critical infrastructures in dynamically changing conditions according to integral estimates of the performance indicators of the system operation, obtained on the basis of fuzzy calculations.
Let us move on to solving the second subproblem. Based on the obtained aggregated values of system resilience indicators INDRC ,INDRC ,INDRC ,INDRC c IND for all levels of index-based assessment, ex-
rCap ' rcREc ' rcAD '
pressed in defuzzified form, for each system resilience domain involved in the issue consideration, target indicators (quality functions / capacities) of the system operation performance RCAP,RCar ,RCrec ,RCAD are calculated based on a set of parameter estimates of the expected effects P , S, R from the implementation of i -th resilience management strategy rcpi e RCP, i = 1,n . Next, a additive-multiplicative convolution is applied to the obtained calculated values of target indicators RCAP,RCar ,RCrec ,RCAD for all domains of system resilience subject to the ranks of these indicators (i.e., the contribution of each target index to the generalized estimate).
A computational model of the integral index of critical infrastructure resilience OR based on the coupling and aggregation of the target performance indicators (basic capacities) of system operation RCap ,RCar ,RCrec,RCad subject to the resilience domains and influencing factors for each alternative management strategy can be formally expressed in the following general view:
ORI = SS + ^(1 - SS),
n _
^ = n SR , i = 1, n , n ,
i=1
SR = p1 • rcap + p2 • rcar + p3 • RCREC + p4 • rcad ,
where ORI is the integral index of the critical infrastructure resilience; SS is the integral index of the critical infrastructure security calculated on the basis of the approach and mathematical models proposed in safety studies [22, 23]; X is the stability of critical infrastructure to the affects of observable and unobservable factors, i.e., multiple threats (efficiency of system self-preservation and self-recovery); SRt is the generalized indicator of the critical infrastructure resilience for the i -th resilience domain; n is a number of considered resilience domains for the critical infrastructure under examination; RC'AP, RC'ar, RC'rec, RCAD are the aggregated target performance indicators of the system operation calculated for the i -th resilience domain through the expected effects P, S, R from the implementation of the chosen management strategy
rcp e RCP ; pj, j = 1, m are the ranks of aggregated target performance indicators of the system operation
(the measure of significance of the indicator in terms of targeting and task-setting by a critical infrastructure operator or an security expert/analyst when implementing situational control of the system in specific conditions); m is a number of performance criteria of system operation for given conditions, ranked by the measure of significance (in our case m = 4).
The ranks of aggregated target indicators of the system operation performance p j are calculated using the P.C. Fishburn formulation [24]:
2 • (m - j +1)
p j =-
m • (m +1)
The computational model of the critical infrastructure (system) prevention capability index is specified by the following formal expressions:
RCap =(ap + Sap )(1 - Rap ).
System performance at the stage of monitoring and early identification of threats within the resilience management life cycle of critical infrastructure is calculated as follows:
1 k1
PAP = k i pb
'h a=1
saving rate at this stage:
1 k2
S AP = k Zi
2 g =1
risk of failures and losses at this stage:
Rap = ri rh = ^i (( • Dh), rh = V„ • Dh,
k3 h=1 k3 h=1
where pb, b = 1,k1, sg, g = 1,k2, rh, h = 1,k3 are defuzzified estimates ofthe indicators {IX1,IX2,...,IXk1}, {IX1,IX2,...,IXk2}, {IX1,IX2,...,IXk3}e INDRC , respectively, obtained at the step when solving the first subproblem; Vh is a probability of risk initiation; Dh is a degree of risk impact (influence) on the effectiveness of the implementation of the system resilience management strategy (Vh and Dh are also assessed by experts when solving the first subproblem); k = k1 + k2 + k3 is a total number of quantitative and qualitative indicators used to assess the prevention capacity of the critical infrastructure.
Computational estimation models of absorption capacity, responsiveness, recoverability and adaptability indicators have a similar form:
- System absorbability and responsiveness index:
RCar =(Pr + Sar )x(1 - Rar ). System operationability at the stage of initialization, implementation and absorption of threats is cal-
1 1
culated as follows: P'AR = — Xpb .
l1 b=1
1 l2
Resource effect: S AR = — X sg .
l2 g=1
1 l3 1 l3
Amount of possible damage: rar = -X^ = 7X( ■ Dh) , >h = ^^ • Dh.
'3 A=1 '3 A=1
pb, b = 1, l1, sg, g = 1, l2, rA, h = 1, l3 are defuzzified estimates of the indicators
{IXx, IX2,..., IXlx}, {IXj , IX2,..., IXj2}, {IX\, IX2,..., IXj3} e INDrCr , respectively, obtained at the step when
solving the first subproblem. l = lx +12 +13 is a total number of quantitative and qualitative indicators used to assess the absorption and reactive capacity of the system. - System recoverability index:
RCREC = (PRREC + SREC )(1 — RREC ) •
Functional efficiency of the system operation at the stage of assessment, neutralization and elimination
1 m
the consequences of threats in the process of functionality restoration: PREC = — X pb .
m1 b=1
1 m2
Cost effectiveness: S REC=— isg .
m2 g =1
1 "j 1 m3
Level of loss of system functionality: Rrec = — i rh = — i (Vh ■ Dh), rh = Vh ■ Dh.
m3 h=1 m3 h=1
pb, b = 1,m , sg, g = 1,m2, rh, h = 1,m3 are defuzzified estimates of the indicators
{IX1,IX2,...,IXm1}, {IX1,IX2,...,IXm2}, {IX1,IX2,...,IXm3}e INDrcrec , respectively, obtained at the step when
solving the first subproblem. m* = mj + m2 + m3 is a total number of quantitative and qualitative indicators used to assess the recovery capacity of the critical infrastructure. - System adaptability index:
RCad =((d + Sad )(1 - Rad ). Predicted controllability at the stage of mitigation of the damage caused and enhancement of the risk
1 91
resistance of the system after the occurrence of adverse events: PaD =— X pb .
91 b=1
Resource intensity: S AD =— Xsg .
q2 g=1
Cumulative effect of system destruction (degradation) probabilities:
1 ?3 1 ?3
Rad = -I rh = -X (V • D ), r = V • Dh.
q3 h=1 q3 h=1
pb, b = 1,q1 , sg, g = 1,q2, rh, h = 1,q3 are defuzzified estimates of the indicators {IX 1,IX2,...,IXq1}, [iXi, IX2,..., IX q 2}, {IX1
,IX2,...,IX q3}g INDRCd , respectively, obtained at the step when solving the first sub*
problem. q = qx + q2 + q3 is a total number of quantitative and qualitative indicators used to assess the adaptive capacity of the system.
When assessing the riskiness of failures or losses origination, it is necessary to take into account risk neutralization measures, e.g., redundancy (duplication), insurance, improvisation, etc.
The integral safety index of critical infrastructure is calculated using the formalism proposed earlier in the study [2]:
RD p=1
1 RD 1 l 1 n 1 m
SS — n 1 n u1 n 1 n WjlX,
n ¡=
i=1
C¡J
m j=1
É uc=Ë =Ë wj=1,
C=1 i=1 j=1
where RD is a number of considered resilience domains for the critical infrastructure under examination; IXcij are the estimates of the partial safety indicators of critical infrastructure at levels (k-2), (k-1), (k) , respectively; l, n, m,... are the numbers of estimated safety indicators of critical infrastructure at levels (k-2), (k-1), (k), respectively; uc,vi,wp,... are the weights of the partial safety indicators of critical
infrastructure agreed upon by experts and determined at levels (k- 2), (k-1), (k), respectively.
The greater the calculated value of the overall resilience index ORI, the more effective the management strategy is considered to ensure its achievement. Based on the results of calculations and comparison of alternatives, such a strategies can be classified as a set of optimal control programs RCP c RCP. At the same time, it should be noted that a strategy is considered effective if it simultaneously provides minimal costs for maintaining system operability and restoring system functions, and also does not cause significant damage to environmental objects in the process of prevention, absorption, development, neutralization of multiple threats and adaptation to new ones operating conditions.
General characteristics, specificity and formal representation of quantitative and qualitative estimation models of the partial resilience indicators of organizational and technical systems, on the basis of which generalized indicators (backbone capacities) of the critical infrastructures operation performance are calculated through the balance correlation of expected effects (performance, saving rate, riskiness) and which are parameters of a unified optimization model of the overall resilience index, will be discussed in detail in the next part of our study: Part 2 (Resilience capacity models and backbone capabilities). The study will be continued in the next submitted article. The manifold of these indicators is given in Table 1 and includes such characteristics as preparedness, reliability, fragility, tensity, cumulated fatigue, vulnerability, maintainability, supportability, feasibility, limits of destruction, safety margin, resistance, segregability, redundancy, coordi-nability, reconfiguration ability, integrability, autonomy, etc., including permissible errors in measuring system performance characteristics.
Conclusion
Analysis and assessment of system resilience is one of the most responsible and important stages in the control cycle of the soft critical infrastructures security and stability. Defining the applied context of a critical situation, operational data gathering and applying template action plans to neutralize appreciable threats are not enough to ensure effective resilience management and maintain the required (acceptable for specific conditions) level of fault-tolerance of critical infrastructures. Consequently, developed means of adequate information and analytical support for this stage providing optimization of strategic decision-making processes for managing the resilience of critical infrastructures are needed.
For these purposes, an index-based method for the integral estimation of the regional critical infrastructures resilience in the course of the research has been developed. The method is based on the general
conception of resilience and fuzzy set theory, as well as on a combination of soft computing and expertise of the level and ratio of aggregated reliability, safety and stability indices of this class of systems. Based on incomplete initial data the method allows to quantify the systemic risks of the impact of various types of threats on the resilience of critical infrastructure, as well as the useful effects and possible losses, when choosing and implementing anti-crisis measures at the all stages of the life cycle of responding to critical situations taking into account influencing factors.
A distinctive feature of the proposed method is its universality, that is, its applicability to all types of critical infrastructures, both soft and hard. The method allows taking into consideration the imperatives and multi-criteria nature of resilience, uncertainty and high variability of its dynamic characteristics in the process of analysis and assessment at the all stages of the life cycle of system resilience management. This enables operational managers of the situational centers of the region to timely identify critical areas in the operation of infrastructure systems that are most sensitive to local failures and require special attention in terms of monitoring their performance and maintaining protectability in accordance with best security practices and standards.
Other advantages of the method are: the ability to combine both quantitative and qualitative measurements of the system resilience characteristics in one computational estimation model of the integral resilience index, based on the independent expert judgments; the ability to take into account the various significance and antagonistic nature of system resilience indicators and the contribution of each indicator to the aggregated assessment; the ability to rank system resilience indicators depending on the research context, goal-setting, type of resilience, situational factors and class of the system; the inability of manifestation of the "compensation effect", when unacceptable ratings for some indicators can be compensated for by high ratings for other indicators.
The research materials showed that the developed method for integral resilience estimation of critical infrastructures does not contradict the well-known approaches adopted in world practice, but only complements and improves them in terms of flexibility in setting parameters, adaptability to changing conditions and transparency of the assessment procedure, as well as ease of end results interpretation, which so important for practical use.
The novelty of the method lies in the fact that it allows on the basis of fuzzy logic not only to formalize and numerically solve the problem of an integral estimation of the resilience of critical infrastructures based on a set of aggregated quantitative and qualitative indicators of system safety, reliability and stability generally accepted in domestic and foreign standard practices subject to the factors influencing them, but also take into account the multi-aspect matter (multidimensionality), dynamics and uncertainty of these indicators in changing environment.
In practice, the method can be used by system analysts and security experts to prepare design decisions and measures to counter actual threats to the operating of critical infrastructures in conditions of incomplete situational awareness (operational context), inaccuracy of input data and unexpected effects of hidden factors impact on the performance of elements and system in general.
Like for most existing techniques, the bottleneck of the fuzzy-set approach to measuring the integral level of system resilience is that it uses expert judgements. Accordingly, this approach has all the disadvantages characteristic to expert analysis methods, such as: the high cost of organizing a qualified expertise, the error probability due to the narrow-mindedness of experts concerning all aspects of the problem, subjectivity and bias in expert judgments, the difficulty of concordance of expert opinions, the need for a big data and determining the type of membership functions for designing an adequate fuzzy estimation model in some areas, selecting and determining the number of experts. In addition, the accuracy of the results of resilience analysis and measurement, as well as the validity of managerial decisions made on the basis of these results depend on the completeness of the set and decomposability of quantitative and qualitative assessment criteria for estimating the system resilience
The practical value of the research efforts lies in the development of tools for automating resilience analysis and assessment procedures of the critical infrastructures at the initial stages of the life cycle of situ-ational control and response to adverse events. Based on the obtained estimates, guidelines to managers and decision-makers can be generated to improve the specific indicators of system resilience that deviate from standard or specified values. This will also identify strengths and weaknesses in the system resilience both from a technical and organizational point of view.
References
1. Reitan N.K. et al. Evaluation of resilience concepts applied to critical infrastructure using existing methodologies.
IMPROVER Project Report: Deliverable 2.3. 2016:97.
2. Masloboev A.V. Methodological approach to ensuring the viability of ecological and economic systems of the Russian Arctic (Part 2. Method and evaluation criteria). Nadezhnost' i kachestvo slozhnykh system = Reliability and quality of complex systems. 2023;(2):115-126. (In Russ.)
3. Cutter S.L., Burton C.G., Emrich C.T. Disaster Resilience Indicators for Benchmarking Baseline Conditions. Journal of Homeland Security and Emergency Management. 2010;7(1):art. 51.
4. Renschler C.S., Frazier A.E., Arendt L.A. et al. A Framework for Defining and Measuring Resilience at the Community Scale: The P.E.O.P.L.E.S. Resilience Framework. Technical Report No. MCEER-10-0006. Gaithersburg, MD, U.S. 2010. 106 p.
5. Lee A.V., Vargo J., Seville E. Developing a Tool to Measure and Compare Organisations' Resilience. Natural Hazards Review. 2013;14(1):29-41.
6. Bertocchi G., Bologna S., Carducci G. et al. Guidelines for Critical Infrastructures Resilience Evaluation. AIIC Technical Report. Italian Association of Critical Infrastructures Experts. 2016:102.
7. Pursiainen C., Rod B., Baker G. et al. Critical infrastructure resilience index. Proceedings of the 26th European Safety and Reliability Conference. Glasgow, Scotland: CRC Press, 2017:2183-2190.
8. Petit F.D. et al. Resilience Measurement Index: An Indicator of Critical Infrastructure Resilience. Report ANL/DIS-13-01. Decision and Information Sciences Division, Argonne National Laboratory, U.S. Department of Energy. 2013:70.
9. Australian Government Organisational Resilience HealthCheck - Australian Government Organisational Resilience Good Business Guide. 2016:30.
10. Hollnagel E. How Resilient Is Your Organisation? An Introduction to the Resilience Analysis Grid (RAG). Sustainable Transformation: Building a Resilient Organization. Toronto, Canada. 2010:7.
11. Prior T., Hagmann J. Measuring Resilience: Benefits and Limitations of Resilience Indices. SKI Focus Report 8. Zurich: Risk and Resilience Research Group Center for Security Studies (CSS), 2012:26.
12. Masloboev A.V., Bystrov V.V. Conceptual model of critical infrastructure viability in the context of modern security theory of complex systems. Ekonomika. Informatika = Economy. Computer science. 2020;47(3):555-572. (In Russ.)
13. Jovanovic A.S., Schmid N., Klimek P., Choudhary A. Use of Indicators for Assessing Resilience of Smart Critical Infrastructures. IRGC Resource Guide on Resilience. Lausanne: EPFL International Risk Governance Center, 2016:136-140.
14. Severtsev N.A., Yurkov N.K. Bezopasnost' dinamicheskikh sistem na etapakh zhiznennogo tsikla = Safety of dynamic systems at life cycle stages. Penza: Izd-vo PGU, 2023:568. (In Russ.)
15. Pursiainen C. The Crisis Management Cycle. UK, London: Routledge, 2017:194.
16. Gareev T.F. Evaluation of the effectiveness of innovations using fuzzy numbers. Vestnik Kazanskogo gosudar-stvennogo agrarnogo universiteta = Bulletin of the Kazan State Agrarian University. 2008;(4):14-17. (In Russ.)
17. Saaty T.L. Fundamentals of Decision Making and Priority Theory with the Analytic Hierarchy Process. Pittsburgh: RWS Publications, 1994:527.
18. Bellman R.E., Zadeh L.A. Decision-making in fuzzy environment. Management Science. 1970;17(4):141-164.
19. Bogatikov V.N. et al. Razrabotka intellektual'nogo upravleniya v mnogourovnevykh promyshlennykh sistemakh v usloviyakh nepolnoy informatsii na osnove nechetkoy formalizatsii predstavleniy o parametrakh tekhnolog-icheskikh protsessov = Development of intelligent control in multilevel industrial systems in conditions of incomplete information based on fuzzy formalization of ideas about the parameters of technological processes. Novomoskovsk: RKhTU im. D.I. Mendeleeva, 2022:374. (In Russ.)
20. Yager R.R. Multiple objective decision-making using fuzzy sets. International Journal of Man-Machine Studies. 1977;9(4):375-382.
21. Von Neumann J., Morgenstern O. Theory of Games and Economic Behavior. Princeton, NJ, USA: Princeton University Press, 1944:625.
22. Masloboev A.V. Technology of modeling and integrated assessment of regional security indicators. Infor-matsionno-tekhnologicheskiy vestnik = Information Technology Bulletin. 2020;(1):127-139. (In Russ.)
23. Yurkov N.K., Mikhaylov V.S. Integral'nye otsenki v teorii nadezhnosti. vvedenie i osnovnye rezul'taty = Integral estimates in reliability theory. introduction and main results. Moscow: Tekhnosfera, 2020:152. (In Russ.)
24. Fishburn P.C. Utility Theory for Decision Making. USA, New York: John Wiley&Sons, 1970:234.
Список литературы
1. Reitan N. K. [et al.]. Evaluation of resilience concepts applied to critical infrastructure using existing methodologies. IMPROVER Project Report: Deliverable 2.3. 2016. 97 p.
2. Маслобоев А. В. Методический подход к обеспечению жизнеспособности эколого-экономических систем российской Арктики (Часть 2. Метод и критерии оценки) // Надежность и качество сложных систем. 2023. № 2. С. 115-126.
3. Cutter S. L., Burton C. G., Emrich C. T. Disaster Resilience Indicators for Benchmarking Baseline Conditions // Journal of Homeland Security and Emergency Management. 2010. Vol. 7, № 1. Article 51.
4. Renschler C. S., Frazier A. E., Arendt L. A. [et al.]. A Framework for Defining and Measuring Resilience at the Community Scale: The P.E.O.P.L.E.S. Resilience Framework. Technical Report No. MCEER-10-0006. Gaithersburg, MD, U.S. 2010. 106 p.
5. Lee A. V., Vargo J., Seville E. Developing a Tool to Measure and Compare Organisations' Resilience // Natural Hazards Review. 2013. Vol. 14, № 1. P. 29-41.
6. Bertocchi G., Bologna S., Carducci G. [et al.]. Guidelines for Critical Infrastructures Resilience Evaluation // AIIC Technical Report. Italian Association of Critical Infrastructures Experts. 2016. 102 p.
7. Pursiainen C., Rod B., Baker G. [et al.]. Critical infrastructure resilience index // Proceedings of the 26th European Safety and Reliability Conference. Glasgow, Scotland : CRC Press, 2017. P. 2183-2190.
8. Petit F. D. [et al.]. Resilience Measurement Index: An Indicator of Critical Infrastructure Resilience. Report ANL/DIS-13-01. Decision and Information Sciences Division, Argonne National Laboratory, U.S. Department of Energy. 2013. 70 p.
9. Australian Government Organisational Resilience HealthCheck - Australian Government Organisational Resilience Good Business Guide. 2016. 30 p.
10. Hollnagel E. How Resilient Is Your Organisation? An Introduction to the Resilience Analysis Grid (RAG) // Sustainable Transformation: Building a Resilient Organization. Toronto, Canada. 2010. 7 p.
11. Prior T., Hagmann J. Measuring Resilience: Benefits and Limitations of Resilience Indices. SKI Focus Report 8. Zurich: Risk and Resilience Research Group Center for Security Studies (CSS), 2012. 26 p.
12. Маслобоев А. В., Быстров В. В. Концептуальная модель жизнеспособности критических инфраструктур в контексте современной теории безопасности сложных систем // Экономика. Информатика. 2020. Т. 47, № 3. С. 555-572.
13. Jovanovic A. S., Schmid N., Klimek P., Choudhary A. Use of Indicators for Assessing Resilience of Smart Critical Infrastructures // IRGC Resource Guide on Resilience / ed by I. Linkov, M. V. Florin. Lausanne : EPFL International Risk Governance Center, 2016. P. 136-140.
14. Северцев Н. А., Юрков Н. К. Безопасность динамических систем на этапах жизненного цикла. Пенза : Изд-во ПГУ, 2023. 568 с.
15. Pursiainen C. The Crisis Management Cycle. UK, London : Routledge, 2017. 194 p.
16. Гареев Т. Ф. Оценка эффективности инноваций с использованием нечетких чисел // Вестник Казанского государственного аграрного университета. 2008. № 4 (10). С. 14-17.
17. Saaty T. L. Fundamentals of Decision Making and Priority Theory with the Analytic Hierarchy Process. Pittsburgh : RWS Publications, 1994. 527 p.
18. Bellman R. Е., Zadeh L. A., Decision-making in fuzzy environment // Management Science. 1970. Vol. 17, № 4. P. 141-164.
19. В. Н. Богатиков [и др.]. Разработка интеллектуального управления в многоуровневых промышленных системах в условиях неполной информации на основе нечеткой формализации представлений о параметрах технологических процессов. Новомосковск : РХТУ им. Д. И. Менделеева, 2022. 374 с.
20. Yager R. R. Multiple objective decision-making using fuzzy sets // International Journal of Man-Machine Studies. 1977. Vol. 9, № 4. P. 375-382.
21. Von Neumann J., Morgenstern O. Theory of Games and Economic Behavior. Princeton, NJ, USA : Princeton University Press, 1944. 625 p.
22. Маслобоев А. В. Технология моделирования и интегральной оценки показателей региональной безопасности // Информационно-технологический вестник. 2020. № 1 (23). С. 127-139.
23. Юрков Н. К., Михайлов В. С. Интегральные оценки в теории надежности. введение и основные результаты. М. : Техносфера, 2020. 152 с.
24. Fishburn P. C. Utility Theory for Decision Making. USA, NY : John Wiley&Sons, 1970. 234 p.
Информация об авторах / Information about the authors
Андрей Владимирович Маслобоев
доктор технических наук, доцент, ведущий научный сотрудник лаборатории информационных технологий управления промышленно-природными системами, Институт информатики и математического моделирования имени В. А. Путилова Федерального исследовательского центра «Кольский научный центр Российской академии наук»; главный научный сотрудник лаборатории природоподобных технологий и техносферной безопасности Арктики, Центр наноматериаловедения Федерального исследовательского центра «Кольский научный центр Российской академии наук» (Россия, г. Апатиты, ул. Ферсмана, 14) E-mail: masloboev@iimm.ru
Andrey V. Masloboev
Doctor of technical sciences, associate professor, leading researcher of the laboratory of information technologies for industrial-natural system management, Putilov Institute for Informatics and Mathematical Modeling
of the Federal Research Centre "Kola Science Centre of the Russian Academy of Sciences"; chief researcher of the laboratory of nature-inspired technologies and environmental safety of the Arctic, Nanomaterials Research Centre of the Federal Research Centre "Kola Science Centre of the Russian Academy of Sciences" (14 Fersmana street, Apatity, Russia)
Автор заявляет об отсутствии конфликта интересов / The author declares no conflicts of interests.
Поступила в редакцию/Received 17.11.2023 Поступила после рецензирования/Revised 10.12.2023 Принята к публикации/Accepted 20.12.2023
