{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,9]],"date-time":"2026-07-09T18:01:02Z","timestamp":1783620062870,"version":"3.55.0"},"reference-count":43,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,4,29]],"date-time":"2025-04-29T00:00:00Z","timestamp":1745884800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100014434","name":"IMPRESS-U program","doi-asserted-by":"publisher","award":["2023\/05\/Y\/ST6\/00263"],"award-info":[{"award-number":["2023\/05\/Y\/ST6\/00263"]}],"id":[{"id":"10.13039\/501100014434","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper addresses the maximum coverage location problem in a generalized setting, where both facilities (service areas) and regional demand are modeled as continuous entities. Unlike traditional formulations, our approach allows for arbitrary shapes for both service areas and demand regions, with additional constraints on facility placement. The key novelty of this work is its ability to handle complex, irregularly shaped service areas, including approximating them as unions of centrally symmetric shapes. This enables the use of an analytical approach based on spatial symmetry, which allows for efficient estimation of the covered area. The problem is formulated as a nonlinear optimization task. We analyze the properties of the objective function and leverage the Shapely library in Python 3.13.3 for efficient geometric computations. To improve computational efficiency, we develop an extended elastic model that significantly reduces processing time. This model generalizes the well-known quasi-physical, quasi-human algorithm for circle packing, extending its applicability to more complex spatial configurations. The effectiveness of the proposed approach is validated through test cases in which service areas take the form of circles, ellipses, and irregular polygons. Our method provides a robust and adaptable solution for various settings of practically interesting continuous maximum coverage location problems involving irregular regional demand and service areas.<\/jats:p>","DOI":"10.3390\/sym17050676","type":"journal-article","created":{"date-parts":[[2025,5,2]],"date-time":"2025-05-02T11:35:13Z","timestamp":1746185713000},"page":"676","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Continuous Maximum Coverage Location Problem with Arbitrary Shape of Service Areas and Regional Demand"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1707-843X","authenticated-orcid":false,"given":"Sergiy","family":"Yakovlev","sequence":"first","affiliation":[{"name":"Institute of Mathematics, Lodz University of Technology, 90-924 Lodz, Poland"},{"name":"Institute of Computer Science and Artificial Intelligence, V.N. Karazin Kharkiv National University, 61022 Kharkiv, Ukraine"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2381-7999","authenticated-orcid":false,"given":"Sergiy","family":"Shekhovtsov","sequence":"additional","affiliation":[{"name":"System Engineering Department, Kharkiv National University of Radioelectronics, 61166 Kharkiv, Ukraine"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2780-7993","authenticated-orcid":false,"given":"Lyudmyla","family":"Kirichenko","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, Lodz University of Technology, 90-924 Lodz, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Olha","family":"Matsyi","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, Lodz University of Technology, 90-924 Lodz, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-6629-0653","authenticated-orcid":false,"given":"Dmytro","family":"Podzeha","sequence":"additional","affiliation":[{"name":"Mathematical Modelling and Artificial Intelligence Department, National Aerospace University \u201cKharkiv Aviation Institute\u201d, 61072 Kharkiv, Ukraine"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2623-3294","authenticated-orcid":false,"given":"Dmytro","family":"Chumachenko","sequence":"additional","affiliation":[{"name":"Mathematical Modelling and Artificial Intelligence Department, National Aerospace University \u201cKharkiv Aviation Institute\u201d, 61072 Kharkiv, Ukraine"},{"name":"Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139, USA"},{"name":"Ubiquitous Health Technology Lab, University of Waterloo, Waterloo, ON N2L 2G5, Canada"},{"name":"Balsillie School of International Affairs, Waterloo, ON N2L 6C2, Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,29]]},"reference":[{"key":"ref_1","unstructured":"Hamacher, H.W., and Drezner, Z. (2002). Facility Location: Applications and Theory, Springer."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Eiselt, H.A., and Marianov, V. (2011). Foundations of Location Analysis, Springer.","DOI":"10.1007\/978-1-4419-7572-0"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1363","DOI":"10.1287\/opre.19.6.1363","article-title":"The location of emergency service facilities","volume":"19","author":"Toregas","year":"1971","journal-title":"Oper. Res."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1111\/j.1435-5597.1974.tb00902.x","article-title":"The maximal covering location problem","volume":"32","author":"Church","year":"1974","journal-title":"Pap. Reg. Sci. Assoc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1057\/jors.1986.134","article-title":"Recent applications of the maximal covering location planning (MCLP) model","volume":"37","author":"Chung","year":"1986","journal-title":"J. Oper. Res. Soc."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Laporte, G., Nickel, S., and Saldanha-da-Gama, F. (2015). Covering location problems. Location Science, Springer International Publishing. [2nd ed.].","DOI":"10.1007\/978-3-319-13111-5"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Church, R., and Murray, A. (2018). Location Covering Models, Springer.","DOI":"10.1007\/978-3-319-99846-6"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"882","DOI":"10.1016\/j.ejor.2018.12.021","article-title":"Benders decomposition for very large scale partial set covering and maximal covering location problems","volume":"275","author":"Cordeau","year":"2019","journal-title":"Eur. J. Oper. Res."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"692","DOI":"10.1016\/j.ejor.2019.11.036","article-title":"On the fuzzy maximal covering location problem","volume":"283","author":"Blanco","year":"2020","journal-title":"Eur. J. Oper. Res."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"214","DOI":"10.1016\/j.cor.2015.04.005","article-title":"On covering location problems on networks with edge demand","volume":"74","author":"Berman","year":"2016","journal-title":"Comput. Oper. Res."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/j.omega.2018.08.013","article-title":"The planar multiple obnoxious facilities location problem: A Voronoi based heuristic","volume":"87","author":"Drezner","year":"2019","journal-title":"Omega"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"105181","DOI":"10.1016\/j.cor.2020.105181","article-title":"Minmax regret maximal covering location problems with edge demands","volume":"130","author":"Kalcsics","year":"2021","journal-title":"Comput. Oper. Res."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1080\/13658810601169857","article-title":"Coverage optimization in continuous space facility siting","volume":"21","author":"Murray","year":"2007","journal-title":"Int. J. Geogr. Inf. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.ejor.2015.09.006","article-title":"A mean-shift algorithm for large-scale planar maximal covering location problems","volume":"250","author":"He","year":"2016","journal-title":"Eur. J. Oper. Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"105310","DOI":"10.1016\/j.cor.2021.105310","article-title":"Continuous maximal covering location problems with interconnected facilities","volume":"132","author":"Blanco","year":"2021","journal-title":"Comput. Oper. Res."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1016\/j.ejor.2020.09.029","article-title":"New exact algorithms for planar maximum covering location by ellipses problems","volume":"291","author":"Tedeschi","year":"2021","journal-title":"Eur. J. Oper. Res."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1040","DOI":"10.1016\/j.ejor.2022.10.037","article-title":"Multi-type maximal covering location problems: Hybridizing discrete and continuous problems","volume":"307","author":"Blanco","year":"2023","journal-title":"Eur. J. Oper. Res."},{"key":"ref_18","unstructured":"Chen, H., Wu, D., and Ortega, A. (2023). Planar Maximum Coverage Location Problem with Partial Coverage, Continuous Spatial Demand, and Adjustable Quality of Service. Mathematics, 11."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"675","DOI":"10.1007\/s10589-009-9258-1","article-title":"Covering a polygonal region by rectangles","volume":"48","author":"Stoyan","year":"2011","journal-title":"Comput. Optim. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1287\/ijoc.2016.0722","article-title":"Planar maximum coverage location problem with partial coverage and rectangular demand and service zones","volume":"29","author":"Bansal","year":"2017","journal-title":"Inf. J. Comput."},{"key":"ref_21","unstructured":"Gillies, S. (2025, April 06). The Shapely User Manual. Available online: https:\/\/shapely.readthedocs.io\/en\/stable\/manual.html."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"35","DOI":"10.32620\/reks.2022.2.03","article-title":"Formalization and solution of the maximum area coverage problem using library Shapely for territory monitoring","volume":"2","author":"Yakovlev","year":"2022","journal-title":"Radioelectron. Comput. Syst."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Yakovlev, S., Kartashov, O., and Podzeha, D. (2022). Mathematical models and nonlinear optimization in continuous maximum coverage location problem. Computation, 10.","DOI":"10.3390\/computation10070119"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1007\/s10559-023-00547-5","article-title":"The Concept of Modeling Packing and Covering Problems Using Modern Computational Geometry Software","volume":"59","author":"Yakovlev","year":"2023","journal-title":"Cybern. Syst. Anal."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"595","DOI":"10.1007\/BF02916995","article-title":"Two personification strategies for solving circles packing problem","volume":"42","author":"Huang","year":"1999","journal-title":"Sci. China Ser. E Technol. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"416","DOI":"10.1016\/j.cie.2016.09.029","article-title":"A heuristic quasi-physical algorithm with coarse and fine adjustment for multi-objective weighted circles packing problem","volume":"101","author":"Liu","year":"2016","journal-title":"Comput. Ind. Eng."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1016\/j.cor.2017.12.002","article-title":"An efficient quasi-physical quasi-human algorithm for packing equal circles in a circular container","volume":"92","author":"He","year":"2018","journal-title":"Comput. Oper. Res."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1016\/j.ejor.2012.10.022","article-title":"Iterated tabu search for the circular open dimension problem","volume":"225","author":"Fu","year":"2013","journal-title":"Eur. J. Oper. Res."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/j.ejor.2021.08.044","article-title":"Iterated dynamic thresholding search for packing equal circles into a circular container","volume":"299","author":"Lai","year":"2022","journal-title":"Eur. J. Oper. Res."},{"key":"ref_30","first-page":"69","article-title":"Phi-Function for complex 2D objects","volume":"2","author":"Stoyan","year":"2004","journal-title":"4OR Q. J. Belg. Fr. Ital. Oper. Res. Soc."},{"key":"ref_31","first-page":"363","article-title":"Mathematical models of placement optimization: Two- and three-dimensional problems and applications","volume":"Volume 73","author":"Fasano","year":"2013","journal-title":"Modeling and Optimization in Space Engineering"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1007\/s10479-008-0456-5","article-title":"Tools of mathematical modelling of arbitrary object packing problems","volume":"179","author":"Bennell","year":"2010","journal-title":"Ann. Oper. Res."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Butenko, S., Pardalos, P.M., and Shylo, V. (2017). Placement problems for irregular objects: Mathematical modeling, optimization and applications. Optimization Methods and Applications, Springer Nature.","DOI":"10.1007\/978-3-319-68640-0"},{"key":"ref_34","unstructured":"Fletcher, R. (1987). Practical Methods for Optimization, John Wiley Sons."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Yakovlev, S., Kiseleva, O., Chumachenko, D., and Podzeha, D. (2023). Maximum Service Coverage in Business Site Selection Using Computer Geometry Software. Electronics, 12.","DOI":"10.3390\/electronics12102329"},{"key":"ref_36","unstructured":"Nocedal, J., and Wright, S.J. (2006). Numerical Optimization, Springer. [2nd ed.]."},{"key":"ref_37","unstructured":"Alidaee, B., and Wang, H. (2025, April 06). Multilevel Facility Location Optimization: A Novel Integer Programming Formulation and Approaches to Heuristic Solutions. Available online: https:\/\/arxiv.org\/abs\/2406.07382."},{"key":"ref_38","unstructured":"Lee, H., Park, S., and Tran, Q.N. (2022). A Fast Adaptive Metaheuristic Algorithm for Large-Scale Facility Location Optimization. arXiv."},{"key":"ref_39","first-page":"113704","article-title":"Multi-objective coverage optimization in wireless sensor networks using a hybrid metaheuristic algorithm","volume":"160","author":"Osman","year":"2020","journal-title":"Expert Syst. Appl."},{"key":"ref_40","first-page":"107670","article-title":"Metaheuristics for solving maximal covering location problems: A comprehensive review and comparative analysis","volume":"111","author":"Abbas","year":"2021","journal-title":"Appl. Soft Comput."},{"key":"ref_41","first-page":"104918","article-title":"A novel metaheuristic method for optimal facility location in dynamic environments","volume":"112","author":"Kaveh","year":"2022","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_42","first-page":"9021","article-title":"Metaheuristics for the continuous coverage problem with irregular demand distributions","volume":"27","author":"Silva","year":"2023","journal-title":"Soft Comput."},{"key":"ref_43","first-page":"1039","article-title":"A Comparative Study of Metaheuristic Optimization Algorithms for Solving Real-World Engineering Design Problems","volume":"139","author":"Altay","year":"2023","journal-title":"Comput. Model. Eng. Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/5\/676\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:24:09Z","timestamp":1760030649000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/5\/676"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,29]]},"references-count":43,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2025,5]]}},"alternative-id":["sym17050676"],"URL":"https:\/\/doi.org\/10.3390\/sym17050676","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,29]]}}}