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International Theoretical Science Journal
Volume 62 >>> № 4 JULY — AUGUST 2026
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DOI 10.34229/KCA2522-9664.26.4.5
UDC 519.85:510.644

S.V. Yakovlev
V.N. Karazin Kharkiv National University, Kharkiv, Ukraine;
Lodz University of Technology, Lodz, Poland,
 s.yakovlev@karazin.ua

O.B. Matsyi
V.N. Karazin Kharkiv National University, Kharkiv, Ukraine,
 matsiy@karazin.ua

A.V. Hlushko
JSC OTP Bank, Kharkiv, Ukraine,  Andrii_v.hlushko@otpbank.com.ua


MEASURE-BASED MODELING OF CONTINUOUS MAXIMAL COVERAGE PROBLEMS
IN A FUZZY SETTING

Abstract. This paper proposes a measure-based approach to the modeling and analysis of continuous maximal geometric coverage problems in a fuzzy setting. Coverage is interpreted not as a binary geometric condition, but as a global integral characteristic of the domain that quantitatively reflects the degree and quality of its coverage by a set of geometric covering objects in a two-dimensional continuous space. In contrast to classical crisp models and traditional fuzzy formulations, where fuzziness is usually associated with demand parameters or external constraints, the proposed approach treats fuzziness as an intrinsic property of the covering objects themselves. To aggregate local coverage degrees, a maximum aggregator is employed, which represents a canonical fuzzy generalization of the set union operation and ensures a direct link with the classical geometric interpretation of coverage. On this basis, a measure-based integral functional of fuzzy coverage is introduced, which provides a unified mathematical framework for the maximal coverage problem and simultaneously establishes a foundation for further generalizations to full coverage problems. A key methodological component of the proposed approach is the level-set representation of the integral coverage functional, which reduces the evaluation of fuzzy coverage to a family of classical geometric problems involving the computation of union areas of geometric sets at fixed intensity levels. This representation ensures a transparent geometric interpretation of the model and enables the application of a wide range of exact, approximate, and stochastic computational strategies without altering the overall problem formulation. It is shown that the proposed approach is consistent with the classical crisp model of continuous maximal coverage, imposes no restrictions on the shape of covering objects or the analytical form of membership functions, and naturally extends to weighted and spatially heterogeneous domains. The developed methodology provides a coherent conceptual basis for further studies on maximal coverage optimization as well as for the development of adaptive and dynamic models of fuzzy continuous coverage.

Keywords: continuous coverage, fuzzy coverage, measure-based approach, integral functional, level sets, maximal coverage, computational geometry.


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REFERENCES

    1. Daskin M.S. Network and discrete location: Models, algorithms, and applications. New York: Wiley, 1995. 320 p.
    2. Church R., ReVelle C. The maximal covering location problem. Papers of the Regional Science Association. 1974. Vol. 32, N 1. P. 101–118. https://doi.org/10.1007/BF01942293.
    3. Berman O., Krass D. The generalized maximal covering location problem. Computers & Operations Research. 2002. Vol. 29, Iss. 6. P. 563–581. https://doi.org/10.1016/S0305-0548(01)00079-X.
    4. Nematian J., Stanimirovi P.S., Ghorbani S., et al. A maximal covering location problem under uncertainty. Mathematics. 2025. Vol. 13, Iss. 22. Article number 3653. https://doi.org/10.3390/math13223653.
    5. Atta S., Sinha Mahapatra P.R., Mukhopadhyay A. Solving a new variant of the capacitated maximal covering location problem with fuzzy coverage area using metaheuristic approaches. Computers & Industrial Engineering. 2022. Vol. 170. Article number 108315. https://doi.org/10.1016/j.cie.2022.108315.
    6. ReVelle C., Eiselt H.A., Daskin M.S. A bibliography for some fundamental problem categories in discrete location science. European Journal of Operational Research. 2008. Vol. 184, Iss. 3. P. 817–848. https://doi.org/10.1016/j.ejor.2006.12.044.
    7. Shaikh M.S., Wang C., Xie S., et al. Coverage and connectivity maximization for wireless sensor networks using improved chaotic grey wolf optimization. Scientific Reports. 2025. Vol. 15. Article number 15706. https://doi.org/10.1038/s41598-025-00184-2.
    8. Muhammad H., Nam H. Sensor node deployment optimization for continuous coverage. Sensors. 2025. Vol. 25, Iss. 12. Article number 3620. https://doi.org/10.3390/s25123620.
    9. Rahmani A.M., Ali S., Yousefpoor M.S., et al. An area coverage scheme based on fuzzy logic and shuffled frog-leaping algorithm in heterogeneous wireless sensor networks. Mathematics. 2021. Vol. 9, Iss. 18. Article number 2251. https://doi.org/10.3390/math9182251.
    10. Qin N.N., Chen J.L. An area coverage algorithm for wireless sensor networks based on differential evolution. International Journal of Distributed Sensor Networks. 2018. Vol. 14, N 8. https://doi.org/10.1177/1550147718796734.
    11. Boualem A., De Runz C., Ayaida M., Akdag H. A fuzzy/possibility approach for area coverage in wireless sensor networks. Soft Computing. 2023. Vol. 27, Iss. 14. P. 9367–9382. https://doi.org/10.1007/s00500-023-08406-3.
    12. Snyder L.V. Facility location under uncertainty: A review. IIE Transactions. 2006. Vol. 38, Iss. 7. P. 547–564. https://doi.org/10.1080/07408170500216480.
    13. Yin R., Chen M., Wu J., Liu Y. Coverage-based variable precision fuzzy rough sets with applications to emergency decision-making. International Journal of Computational Intelligence Systems. 2025. Vol. 18. Article number 43. https://doi.org/10.1007/s44196-024-00728-w.
    14. Arana-Jimnez M., Blanco V., Fernїndez E. On the fuzzy maximal covering location problem. European Journal of Operational Research. 2020. Vol. 283, Iss. 2. P. 692–705. https://doi.org/10.1016/j.ejor.2019.11.036.
    15. Yakovlev S., Shekhovtsov S., Kirichenko L., et al. Continuous maximum coverage location problem with arbitrary shape of service areas and regional demand. Symmetry. 2025. Vol. 17, Iss. 5. Article number 676. https://doi.org/10.3390/sym17050676.
    16. Yakovlev S.V. The concept of modeling packing and covering problems using modern computational geometry software. Cybernetics and Systems Analysis. 2023. Vol. 59, N 1. P. 108–119. https://doi.org/10.1007/s10559-023-00547-5.
    17. Yakovlev S., Kartashov O., Podzeha D. Mathematical models and nonlinear optimization in continuous maximum coverage location problem. Computation. 2022. Vol. 10, Iss. 7. Article number 119. https://doi.org/10.3390/computation10070119.
    18. Yakovlev S., Kiseleva O., Chumachenko D., Podzeha D. Maximum service coverage using computational geometry software. Electronics. 2023. Vol. 12, Iss. 10. Article number 2329. https://doi.org/10.3390/electronics12102329.
    19. Yakovlev S., Kirichenko L., Matsyi O., Kirpich A., Chumachenko D. Optimization of mobile medical service locations based on predictive analytics in crisis scenarios. Proc. IADIS International Conferences Information Systems 2025 and e-Society 2025 (12–14 March 2025, Porto, Portugal). Porto, 2025. P. 538–541.
    20. Bulat A., Kiseleva E., Yakovlev S., et al. Solving the problem of fuzzy partition-distribution with determination of the location of subset centers. Computation. 2024. Vol. 12, Iss. 10. Article number 199. https://doi.org/10.3390/computation12100199.
    21. Zadeh L.A. Fuzzy sets. Information and Control. 1965. Vol. 8, Iss. 3. P. 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X.
    22. Mendel J.M. Fuzzy logic systems for engineering: a tutorial. Proceedings of the IEEE. 1995. Vol. 83, Iss. 3. P. 345–377. https://doi.org/10.1109/5.364485.
    23. Zimmermann K. Fuzzy set covering problem. International Journal of General Systems. 1991. Vol. 20, Iss. 1. P. 127–131. https:///doi.org/10.1080/03081079108945020.
    24. Elhabyan R., Shi W., St-Hilaire M. Coverage protocols for wireless sensor networks: Review and future directions. Journal of Communications and Networks. 2019. Vol. 21, Iss. 1. P. 45–60. https://doi.org/10.1109/JCN.2019.000005.
    25. Drakuli D., Takai A., Mari M. Fuzzy covering location problems with different aggregation operators. Filomat. 2017. Vol. 31, Iss. 2. P. 513–522. https://doi.org/10.2298/FIL1702513D.
    26. Bhattacharya A., Pal M. Fuzzy covering problem of fuzzy graphs and its application to investigate the Indian economy in new normal. Journal of Applied Mathematics and Computing. 2022. Vol. 68. P. 479–510. https://doi.org/10.1007/s12190-021-01539-4.
    27. Kiseleva E.M., Prytomanova O.M. Fuzzy problem of optimal partitioning of sets with constraints on the location of subset centers. System Research and Information Technologies. 2020. N 1. P. 78–89. https://doi.org/10.20535/SRIT.2308-8893.2020.1.07.
    28. Bulat A., Kiseleva E., Hart L., Prytomanova O. Generalized models of logistics problems and approaches to their solution based on the synthesis of the theory of optimal partitioning and neuro-fuzzy technologies. In: Zgurovsky M., Pankratova N. (Eds.). System Analysis and Artificial Intelligence. Studies in Computational Intelligence, vol 1107. https://doi.org/10.1007/978-3-031-37450-0_21.
    29. Leichenko K., Skorobohatko S., Fesenko H., et al. Assessment of the reliability of wireless sensor networks for forest fire monitoring systems considering fatal combinations of multiple sensor failures. Cybernetics and Systems Analysis. 2025. Vol. 61, N 1. P. 137–147. https://doi.org/10.1007/s10559-025-00753-3.
    30. Skorobohatkо S., Fesenko H., Kharchenko V., et al. Architecture and reliability models of hybrid sensor networks for environmental and emergency monitoring systems. Cybernetics and Systems Analysis. 2024. Vol. 60, N 1. P. 293–304. https://doi.org/10.1007/s10559-024-00670-x.
    31. Pirkul H., Schilling D.A. The maximal covering location problem with capacities on total workload. Management Science. 1991. Vol. 37, Iss. 2. P. 125–250. https://doi.org/10.1287/10.1287/mnsc.37.2.233.



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