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<p>This paper proposes a measure-based approach for modeling and analyzing 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 family of geometric covering objects in a two-dimensional continuous space. In contrast to classical crisp models and traditional fuzzy formulations, where fuzziness is typically 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 max-aggregator is employed, which represents a canonical fuzzy generalization of the set union operation and ensures a direct link to the classical geometric interpretation of coverage. On this basis, a measure-theoretic integral functional of fuzzy coverage is introduced, providing a unified mathematical framework for the maximal coverage problem and, at the same time, establishing 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. It also 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 on 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 of maximal coverage optimization and for the development of adaptive and dynamic models of fuzzy continuous coverage.</p>

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Measure-based Modeling of Continuous Maximal Coverage Problems in a Fuzzy Setting

  • 期刊论文

  • 发表日期：

    2026年7月31日

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  • S. V. Yakovlev,
  • O. B. Matsyi,
  • A. V. Hlushko

摘要

This paper proposes a measure-based approach for modeling and analyzing 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 family of geometric covering objects in a two-dimensional continuous space. In contrast to classical crisp models and traditional fuzzy formulations, where fuzziness is typically 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 max-aggregator is employed, which represents a canonical fuzzy generalization of the set union operation and ensures a direct link to the classical geometric interpretation of coverage. On this basis, a measure-theoretic integral functional of fuzzy coverage is introduced, providing a unified mathematical framework for the maximal coverage problem and, at the same time, establishing 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. It also 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 on 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 of maximal coverage optimization and for the development of adaptive and dynamic models of fuzzy continuous coverage.