{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:18:54Z","timestamp":1760149134603,"version":"build-2065373602"},"reference-count":54,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,7,5]],"date-time":"2023-07-05T00:00:00Z","timestamp":1688515200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>Risk involves identifying several options that the decision-maker can opt for while making a choice either in the direction of risk or reliability. In this approach, risk is defined as the action of the subject which will lead to the loss or guaranteed safety of what has been achieved. As the uncertainty of the external business environment increases for companies, the task of managing risks both individually and as a set of risks becomes more and more relevant. The purpose of this study is to solve the problem of managing multifactorial risks using mathematical methods for determining the optimal risk management trajectories separately for each factor. To determine the optimal risk management trajectories for each factor, a numerical method is used based on the choice of the most effective direction, which is defined as the ratio of risk change to cost change. An information system prototype has been created that can support the management of a set of risks. Approbation of the information system was carried out on an example containing two conceptual risk factors. The proposed prototype builds a three-dimensional risk map by interpolating the risk matrix entered by the risk manager using an additive\u2013multiplicative aggregation procedure, as well as optimal risk management trajectories for all entered risk factors.<\/jats:p>","DOI":"10.3390\/computation11070132","type":"journal-article","created":{"date-parts":[[2023,7,6]],"date-time":"2023-07-06T00:34:30Z","timestamp":1688603670000},"page":"132","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Developing a Numerical Method of Risk Management Taking into Account the Decision-Maker\u2019s Subjective Attitude Towards Multifactorial Risks"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5033-6694","authenticated-orcid":false,"given":"Aleksandr","family":"Alekseev","sequence":"first","affiliation":[{"name":"Administrative Directorate for Organization of Scientific Researches, Perm National Research Polytechnic University, 614990 Perm, Russia"},{"name":"Perm Decision Making Support Center, 614017 Perm, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7674-7846","authenticated-orcid":false,"given":"Zhanna","family":"Mingaleva","sequence":"additional","affiliation":[{"name":"Department of Economics and Industrial Management, Perm National Research Polytechnic University, 614990 Perm, Russia"}]},{"given":"Irina","family":"Alekseeva","sequence":"additional","affiliation":[{"name":"Department of Construction Engineering and Materials Science, Perm National Research Polytechnic University, 614990 Perm, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7232-8297","authenticated-orcid":false,"given":"Elena","family":"Lobova","sequence":"additional","affiliation":[{"name":"Department of Economics and Industrial Management, Perm National Research Polytechnic University, 614990 Perm, Russia"}]},{"given":"Alexander","family":"Oksman","sequence":"additional","affiliation":[{"name":"Department of Economics and Industrial Management, Perm National Research Polytechnic University, 614990 Perm, Russia"}]},{"given":"Alexander","family":"Mitrofanov","sequence":"additional","affiliation":[{"name":"Department of Technical Disciplines, Lysva Branch of the Perm National Research Polytechnic University, 618900 Lysva, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1007\/s10479-015-1853-1","article-title":"Sustainable supply chain management: A modeling perspective","volume":"229","author":"Brandenburg","year":"2015","journal-title":"Ann. Oper. Res."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Lin, T.T., Yen, H.-T., and Hsu, S.-Y. (2022). A Synergy Value Analysis of Sustainable Management Projects: Illustrated by the Example of the Aesthetic Medicine Industry. J. Risk Financ. Manag., 15.","DOI":"10.3390\/jrfm15080348"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Willett, A.H. (1901). The Economic Theory of Risk and Insurance, P.S. King & Son.","DOI":"10.7312\/will93384"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Nosratabadi, S., Mosavi, A., Shamshirband, S., Zavadskas, E.K., Rakotonirainy, A., and Chau, K.W. (2019). Sustainable business models: A review. Sustainability, 11.","DOI":"10.20944\/preprints201810.0378.v3"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Haimes, Y.Y. (2004). Risk Modeling, Assessment, and Management, John Wiley & Sons, Inc.. [2nd ed.]. First published 1998.","DOI":"10.1002\/0471723908"},{"key":"ref_6","unstructured":"Gracheva, M.V., Sekerin, A.B., Afanasyev, A.M., Babaskin, S.Y., Bykova, A.G., Volkov, I.M., Kopytin, K.V., Nikitin, S.A., Pervushin, V.A., and Svetlov, N.M. (2012). Risk-Management of Investment Project, Unity-Dana."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"29","DOI":"10.13106\/jafeb.2018.vol5.no1.29","article-title":"Research on corporate risk reporting: Current trends and future avenues","volume":"5","author":"Mazumder","year":"2018","journal-title":"J. Asian Financ. Econ. Bus."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Yang, S., Ishtiaq, M., and Anwar, M. (2018). Enterprise risk management practices and firm performance, the mediating role of competitive advantage and the moderating role of financial literacy. J. Risk Financ. Manag., 11.","DOI":"10.3390\/jrfm11030035"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1007\/s10479-021-04259-x","article-title":"Business analytics for corporate risk management and performance improvement","volume":"315","author":"Hsu","year":"2021","journal-title":"Ann. Oper. Res."},{"key":"ref_10","first-page":"187","article-title":"Impact of Risk Governance, Associated Practices and Tools on Enterprise Risk Management: Some Evidence from Colombia","volume":"14","author":"Villanueva","year":"2022","journal-title":"Rev. Finanz. Pol\u00edtica Econ\u00f3mica"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Dragomir, V.D. (2020). Corporate Environmental Strategy, Springer.","DOI":"10.1007\/978-3-030-29548-6"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"690","DOI":"10.1108\/IJOPM-08-2015-0508","article-title":"Supply chain integration, risk management and manufacturing flexibility","volume":"38","author":"Chaudhuri","year":"2018","journal-title":"Int. J. Oper. Prod. Manag."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1108\/IJPDLM-01-2017-0043","article-title":"A review of supply chain risk management: Definition, theory, and research agenda","volume":"48","author":"Fan","year":"2018","journal-title":"Int. J. Phys. Distrib. Logist. Manag."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1080\/13675567.2019.1684462","article-title":"Managing environmental uncertainty for improved firm financial performance: The moderating role of supply chain risk management practices on managerial decision making","volume":"23","author":"Singh","year":"2020","journal-title":"Int. J. Logist. Res. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"107972","DOI":"10.1016\/j.ijpe.2020.107972","article-title":"Can supply chain risk management practices mitigate the disruption impacts on supply chains\u2019 resilience and robustness? Evidence from an empirical survey in a COVID-19 outbreak era","volume":"233","author":"Ruel","year":"2021","journal-title":"Int. J. Prod. Econ."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Elmsalmi, M., Hachicha, W., and Aljuaid, A.M. (2021). Prioritization of the Best Sustainable Supply Chain Risk Management Practices Using a Structural Analysis Based-Approach. Sustainability, 13.","DOI":"10.3390\/su13094608"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/978-3-030-22493-6_18","article-title":"Management of Organizational Knowledge as a Basis for the Competitiveness of Enterprises in the Digital Economy","volume":"78","author":"Mingaleva","year":"2020","journal-title":"Lect. Notes Netw. Syst."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Lee, J., and Lee, D.K. (2018). Application of industrial risk management practices to control natural hazards, facilitating risk communication. Int. J. Geo-Inf., 7.","DOI":"10.3390\/ijgi7090377"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1108\/IJAIM-05-2015-0027","article-title":"Risk management and firm value: Recent theory and evidence","volume":"24","author":"Krause","year":"2016","journal-title":"Int. J. Account. Inf. Manag."},{"key":"ref_20","first-page":"953","article-title":"Corporate Social Responsibility and Firm Risk: Controversial Versus Noncontroversial Industries","volume":"8","author":"Eriandani","year":"2021","journal-title":"J. Asian Financ. Econ. Bus."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1002\/smj.3224","article-title":"Risk management and corporate social responsibility","volume":"42","author":"Kim","year":"2021","journal-title":"Strateg. Manag. J."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"101805","DOI":"10.1016\/j.frl.2020.101805","article-title":"Corporate social responsibility, enterprise risk management, and real earnings management: Evidence from managerial confidence","volume":"41","author":"Kuo","year":"2021","journal-title":"Financ. Res. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18820\/24150487\/as25i1.1","article-title":"An exploratory factor analysis of risk management practices: A study among small and medium contractors in Gauteng","volume":"25","author":"Renault","year":"2018","journal-title":"Acta Structilia"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1177\/0972262918803172","article-title":"Corporate risk management, firms\u2019 characteristics and capital structure: Evidence from Bombay Stock Exchange (BSE) Sensex Companies","volume":"22","author":"Joshi","year":"2018","journal-title":"Vision"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1177\/2278533719886995","article-title":"Corporate governance an imperative for stakeholders protection: Evidence from risk management of Indian listed firms","volume":"8","author":"Busru","year":"2020","journal-title":"Bus. Perspect. Res."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"11","DOI":"10.20448\/2002.81.11.20","article-title":"The influence of enterprise risk management practices on organizational performance: Evidence from Kenyan State Corporations","volume":"8","author":"Girangwa","year":"2020","journal-title":"J. Account. Bus. Financ. Res."},{"key":"ref_27","first-page":"6","article-title":"Risk management as a key element of ensuring the implementation of the risk-oriented approach in the activities of business entities","volume":"10","author":"Avdiysky","year":"2017","journal-title":"Econ. Taxes Right"},{"key":"ref_28","first-page":"250","article-title":"Expert Technology for Risk Management in the Implementation of QRM in a High-Tech Industrial Enterprise","volume":"27","author":"Akatov","year":"2019","journal-title":"Manag. Syst. Prod. Eng."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1007\/s10479-017-2452-0","article-title":"Understanding risk management for intentional supply chain disruptions: Risk detection, risk mitigation, and risk recovery","volume":"283","author":"DuHadway","year":"2019","journal-title":"Ann. Oper. Res."},{"key":"ref_30","first-page":"117","article-title":"Board and management-level factors affecting the maturity of IT risk management practices","volume":"33","author":"Vincent","year":"2019","journal-title":"J. Inf. Syst."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1016\/j.econmod.2019.05.007","article-title":"Corporate liquidity and risk management with time-inconsistent preferences","volume":"81","author":"Liu","year":"2019","journal-title":"Econ. Model."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1111\/basr.12166","article-title":"Corporate environmental responsibilities and executive compensation: A risk management perspective","volume":"124","author":"Hao","year":"2019","journal-title":"Bus. Soc. Rev."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"100879","DOI":"10.1016\/j.najef.2018.11.008","article-title":"The role of the board and the audit committee in corporate risk management","volume":"54","author":"Tai","year":"2020","journal-title":"N. Am. J. Econ. Financ."},{"key":"ref_34","first-page":"129","article-title":"Corporate global responsibility and reputation risk management","volume":"1","author":"Cassano","year":"2019","journal-title":"Symphonya Emerg. Issues Manag."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1479","DOI":"10.1007\/s11187-020-00358-4","article-title":"Export promotion programs, export capabilities, and risk management practices of internationalized SMEs","volume":"57","author":"Catanzaro","year":"2021","journal-title":"Small Bus. Econ."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1111\/1467-8500.12257","article-title":"Diffusion of Corporate Risk-Management Characteristics: Perspectives of Chief Audit Executives through a Survey Approach","volume":"77","author":"Christopher","year":"2018","journal-title":"Aust. J. Public Adm."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1016\/0377-2217(94)90290-9","article-title":"Models, modelling and modellers: An application to risk analysis","volume":"75","year":"1994","journal-title":"Eur. J. Oper. Res."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"136","DOI":"10.15593\/2499-9873\/2019.4.09","article-title":"Stability analysis of the rating and control mechanism to agent\u2019s strategic behavior (on example of the risk management policy coordination)","volume":"4","author":"Alekseev","year":"2019","journal-title":"Appl. Math. Control Sci."},{"key":"ref_39","unstructured":"Mingaleva, Z.A., and Lobova, E.S. (2022). \u201cPortfolio analysis of innovative projects, taking into account endogenous and exogenous risks\u201d. Programmy Dlia EVM. Bazy Dannykh. Topologii Integral\u2019nykh Mikroskhem [Computer Programs. Database. Topologies of Integrated Circuits]. Computer program RU 2022664434."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2396","DOI":"10.1002\/ijfe.1913","article-title":"The interaction of risk management tools: Financial hedging, corporate diversification and liquidity","volume":"26","author":"Jiang","year":"2021","journal-title":"Int. J. Financ. Econ."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"114","DOI":"10.15593\/2499-9873\/2020.1.08","article-title":"Control of a complex objects, states of which are describing by the matrix rating mechanisms","volume":"1","author":"Alekseev","year":"2020","journal-title":"Appl. Math. Control Sci."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1007\/978-3-030-85799-8_25","article-title":"A Syndinic approach to enterprise risk management","volume":"315","author":"Mingaleva","year":"2021","journal-title":"Lect. Notes Netw. Syst."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1002\/for.3980010305","article-title":"How probable is probable? A numerical translation of verbal probability expressions","volume":"1","year":"1982","journal-title":"J. Forecast."},{"key":"ref_44","first-page":"121","article-title":"A tool for risk assessment","volume":"3","year":"2013","journal-title":"Saf. Eng."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"56","DOI":"10.2118\/166269-PA","article-title":"The risk of using risk matrices","volume":"6","author":"Thomas","year":"2014","journal-title":"SPE Econ. Manag."},{"key":"ref_46","unstructured":"Alekseev, A. (2017, January 25\u201327). The additive-multiplicative fuzzy aggregation matrix mechanism and equivalent it\u2019s the continuous mechanism. Proceedings of the 12th International Scientific and Practical Conference \u201cModern Complex Control Systems 2017\u201d, Lipetsk, Russia."},{"key":"ref_47","unstructured":"Alekseev, A. (2018, January 23\u201325). On strategy-proofness of matrix integrated rating mechanism. Proceedings of the Socio-Physics and Socio-Engineering, Moscow, Russia."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Alekseev, A., Salamatina, A., and Kataeva, T. (2019, January 15\u201317). Rating and Control Mechanisms Design in the Program \u201cResearch of Dynamic Systems\u201d. Proceedings of the 2019 IEEE 21st Conference on Business Informatics (CBI), Moscow, Russia.","DOI":"10.1109\/CBI.2019.10103"},{"key":"ref_49","first-page":"139","article-title":"A Theory of production","volume":"18","author":"Cobb","year":"1928","journal-title":"Am. Econ. Rev."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Alekseev, A., Noskova, A., and Neifeld, V. (2022, January 9\u201311). The Software Modules \u201cInsider\u201d as a KYC-Solution for Industrial Verification and Bankruptcy Prediction. Proceedings of the 2022 4th International Conference on Control Systems, Mathematical Modeling, Automation and Energy Efficiency (SUMMA), Lipetsk, Russia.","DOI":"10.1109\/SUMMA57301.2022.9974034"},{"key":"ref_51","unstructured":"L\u00f6ffler, G., and Posch, P.N. (2007). Credit Risk Modeling Using Excel and VBA, John Wiley & Sons Ltd."},{"key":"ref_52","unstructured":"Alexander, C. (2008). Market Risk Analysis, Practical Financial Econometrics, John Wiley & Sons Ltd."},{"key":"ref_53","unstructured":"Lewis, N. (2004). Operational Risk with Excel and VBA: Applied Statistical Methods for Risk Management, John Wiley & Sons, Inc."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Tysiak, W., and Sereseanu, A. (2009, January 21\u201323). Monte Carlo simulation in risk management in projects using Excel. Proceedings of the IEEE International Workshop on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications, Cosenza, Italy.","DOI":"10.1109\/IDAACS.2009.5342913"}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/11\/7\/132\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:06:55Z","timestamp":1760126815000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/11\/7\/132"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,5]]},"references-count":54,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2023,7]]}},"alternative-id":["computation11070132"],"URL":"https:\/\/doi.org\/10.3390\/computation11070132","relation":{},"ISSN":["2079-3197"],"issn-type":[{"type":"electronic","value":"2079-3197"}],"subject":[],"published":{"date-parts":[[2023,7,5]]}}}