{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:12:11Z","timestamp":1760058731688,"version":"build-2065373602"},"reference-count":54,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,4,27]],"date-time":"2025-04-27T00:00:00Z","timestamp":1745712000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Cyber-physical systems (CPSs) are fundamental components of Industry 4.0 production environments. Their interconnection is crucial for the successful implementation of distributed and autonomous production plans. A particularly relevant challenge is the optimal scheduling of tasks that require the collaboration of multiple CPSs. To ensure the feasibility of optimal schedules, two primary issues must be addressed: (1) The design of global systems emerging from the interconnection of CPSs; (2) The development of a scheduling formalism tailored to interconnected Industry 4.0 settings. Our approach is based on a Category Theory formalization of interconnections as compositions. This framework aims to guarantee that the emergent behaviors align with the intended outcomes. Building upon this foundation, we introduce a formalism that captures the assignment of operations to cyber-physical systems.<\/jats:p>","DOI":"10.3390\/axioms14050332","type":"journal-article","created":{"date-parts":[[2025,4,28]],"date-time":"2025-04-28T09:39:47Z","timestamp":1745833187000},"page":"332","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Compositional Scheduling in Industry 4.0 Cyber-Physical Systems"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2988-4519","authenticated-orcid":false,"given":"Fernando","family":"Tohm\u00e9","sequence":"first","affiliation":[{"name":"Department of Economics, Instituto de Matem\u00e1tica de Bah\u00eda Blanca (UNS-Conicet), Universidad Nacional del Sur,  Bah\u00eda Blanca B8000, Argentina"}]},{"given":"Daniel","family":"Rossit","sequence":"additional","affiliation":[{"name":"Department of Engineering, Instituto de Matem\u00e1tica de Bah\u00eda Blanca (UNS-Conicet), Universidad Nacional del Sur, Bah\u00eda Blanca B8000, Argentina"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,27]]},"reference":[{"key":"ref_1","first-page":"18","article-title":"A cyber-physical systems architecture for industry 4.0-based manufacturing systems","volume":"3","author":"Lee","year":"2015","journal-title":"Manuf. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"508","DOI":"10.1080\/00207543.2017.1351644","article-title":"Smart manufacturing","volume":"56","author":"Kusiak","year":"2018","journal-title":"Int. J. Prod. Res."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Lee, E.A. (2008, January 5\u20137). Cyber physical systems: Design challenges. Proceedings of the 2008 11th IEEE International Symposium on Object and Component-Oriented Real-Time Distributed Computing (ISORC), Orlando, FL, USA.","DOI":"10.1109\/ISORC.2008.25"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/j.procir.2014.03.115","article-title":"Cyber-physical production systems: Roots, expectations and R&D challenges","volume":"17","author":"Monostori","year":"2014","journal-title":"Procedia CIRP"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"3802","DOI":"10.1080\/00207543.2018.1504248","article-title":"Industry 4.0: Smart scheduling","volume":"57","author":"Rossit","year":"2019","journal-title":"Int. J. Prod. Res."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"109891","DOI":"10.1016\/j.cie.2024.109891","article-title":"Cyber-physical systems security: A systematic review","volume":"188","author":"Harkat","year":"2024","journal-title":"Comput. Ind. Eng."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1016\/j.cirp.2016.06.005","article-title":"Cyber-physical systems in manufacturing","volume":"65","author":"Monostori","year":"2016","journal-title":"CIRP Ann."},{"key":"ref_8","first-page":"111","article-title":"Scheduling research contributions to Smart manufacturing","volume":"15","author":"Rossit","year":"2018","journal-title":"Manuf. Lett."},{"key":"ref_9","first-page":"1","article-title":"Industry 4.0: A survey on technologies, applications and open research issues","volume":"6","author":"Lu","year":"2017","journal-title":"J. Ind. Inf. Integr."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/j.compind.2016.03.001","article-title":"Enterprise information systems state of the art: Past, present and future trends","volume":"79","author":"Romero","year":"2016","journal-title":"Comput. Ind."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Kopetz, H., Bondavalli, A., Brancati, F., Fr\u00f6mel, B., H\u00f6ftberger, O., and Iacob, S. (2016). Emergence in Cyber-Physical Systems-of-Systems (CPSoSs). Cyber-Physical Systems of Systems, Springer.","DOI":"10.1007\/978-3-319-47590-5"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Bakirtzis, G. (2021). Compositional cyber-physical systems theory. arXiv.","DOI":"10.4204\/EPTCS.333.9"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Bakirtzis, G., Vasilakopoulou, C., and Fleming, C.H. (2021). Compositional cyber-physical systems modeling. arXiv.","DOI":"10.4204\/EPTCS.333.9"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1145\/3461669","article-title":"Categorical semantics of cyber-physical systems theory","volume":"5","author":"Bakirtzis","year":"2021","journal-title":"ACM Trans. Cyber Phys. Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.rcim.2011.07.002","article-title":"From cloud computing to cloud manufacturing","volume":"28","author":"Xu","year":"2012","journal-title":"Robot. Comput. Integr. Manuf."},{"key":"ref_16","first-page":"91","article-title":"From cloud manufacturing to cloud remanufacturing: A cloud-based approach for WEEE recovery","volume":"2","author":"Wang","year":"2014","journal-title":"Manuf. Lett."},{"key":"ref_17","first-page":"100472","article-title":"Cloud manufacturing architectures: State-of-art, research challenges and platforms description","volume":"34","author":"Chiappa","year":"2023","journal-title":"J. Ind. Inf. Integr."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"9799","DOI":"10.1109\/ACCESS.2023.3239834","article-title":"A survey of cyber-physical systems from a game-theoretic perspective","volume":"11","author":"Tushar","year":"2023","journal-title":"IEEE Access"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2335","DOI":"10.1007\/s10845-024-02380-9","article-title":"Cyber-physical systems: A bibliometric analysis of literature","volume":"36","author":"Singh","year":"2024","journal-title":"J. Intell. Manuf."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2941","DOI":"10.1080\/00207543.2018.1444806","article-title":"Industry 4.0: State of the art and future trends","volume":"56","author":"Xu","year":"2018","journal-title":"Int. J. Prod. Res."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2233","DOI":"10.1109\/TII.2014.2300753","article-title":"Internet of things in industries: A survey","volume":"10","author":"He","year":"2014","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"209","DOI":"10.31181\/dmame712024891","article-title":"Evaluation and selection of a cybersecurity platform: Case of the power sector in India","volume":"7","author":"Verma","year":"2023","journal-title":"Decis. Mak. Appl. Manag. Eng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"220121","DOI":"10.1109\/ACCESS.2020.3042874","article-title":"Industrial artificial intelligence in industry 4.0-systematic review, challenges and outlook","volume":"8","author":"Peres","year":"2020","journal-title":"IEEE Access"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Sanchez, M.A., Rossit, D., and Tohm\u00e9, F. (2024). Enhancing production system resilience with digital twin-driven management. Int. J. Comput. Integr. Manuf., 1\u201320.","DOI":"10.1080\/0951192X.2024.2428686"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Sood, V.M., Singh, Y., Bhargava, B., and Narang, S.K. (2024). Intelligent Security Solutions for Cyber-Physical Systems, CRC Press.","DOI":"10.1201\/9781003406105"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"176","DOI":"10.1016\/j.jmsy.2020.11.017","article-title":"Cyber-physical systems architectures for industrial internet of things applications in Industry 4.0: A literature review","volume":"58","author":"Pivoto","year":"2021","journal-title":"J. Manuf. Syst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"103650","DOI":"10.1016\/j.cose.2023.103650","article-title":"A comprehensive knowledge map for AI improving security management of cyber-physical system enabled smart manufacturing","volume":"137","author":"Cao","year":"2024","journal-title":"Comput. Secur."},{"key":"ref_28","unstructured":"Guo, D., Zhong, R.Y., Rong, Y., and Huang, G.G. (2021). Synchronization of shop-floor logistics and manufacturing under IIoT and digital twin-enabled graduation intelligent manufacturing system. IEEE Trans. Cybern."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"103923","DOI":"10.1016\/j.compind.2023.103923","article-title":"A review of reference architectures for digital manufacturing: Classification, applicability and open issues","volume":"149","author":"Kaiser","year":"2023","journal-title":"Comput. Ind."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Pinedo, M.L. (2012). Scheduling, Springer.","DOI":"10.1007\/978-1-4614-2361-4"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1023\/A:1022235519958","article-title":"Rescheduling manufacturing systems: A framework of strategies, policies, and methods","volume":"6","author":"Vieira","year":"2003","journal-title":"J. Sched."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1809","DOI":"10.1007\/s10845-017-1350-2","article-title":"Review of job shop scheduling research and its new perspectives under Industry 4.0","volume":"30","author":"Zhang","year":"2019","journal-title":"J. Intell. Manuf."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1797","DOI":"10.1109\/TCYB.2020.3024849","article-title":"Evolving scheduling heuristics via genetic programming with feature selection in dynamic flexible job-shop scheduling","volume":"51","author":"Zhang","year":"2020","journal-title":"IEEE Trans. Cybern."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"103549","DOI":"10.1016\/j.compind.2021.103549","article-title":"Multicriteria decision-making method for scheduling problem based on smart batches and their quality prediction capability","volume":"133","author":"Zimmermann","year":"2021","journal-title":"Comput. Ind."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1016\/j.arcontrol.2021.04.008","article-title":"A decision-making framework for dynamic scheduling of cyber-physical production systems based on digital twins","volume":"51","author":"Villalonga","year":"2021","journal-title":"Annu. Rev. Control"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"103458","DOI":"10.1016\/j.compind.2021.103458","article-title":"Systemic formalisation of Cyber-Physical-Social System (CPSS): A systematic literature review","volume":"129","author":"Yilma","year":"2021","journal-title":"Comput. Ind."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1172","DOI":"10.1080\/0951192X.2021.2022760","article-title":"knowledge representation in Industry 4.0 scheduling problems","volume":"35","author":"Rossit","year":"2022","journal-title":"Int. J. Comput. Integr. Manuf."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"103711","DOI":"10.1016\/j.compind.2022.103711","article-title":"Data-driven and autonomous manufacturing control in cyber-physical production systems","volume":"141","author":"Antons","year":"2022","journal-title":"Comput. Ind."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"103383","DOI":"10.1016\/j.compind.2020.103383","article-title":"Integrated and adaptive AR maintenance and shop-floor rescheduling","volume":"125","author":"Mourtzis","year":"2021","journal-title":"Comput. Ind."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1016\/j.jmsy.2018.10.004","article-title":"Production rescheduling review: Opportunities for industrial integration and practical applications","volume":"49","author":"Uhlmann","year":"2018","journal-title":"J. Manuf. Syst."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"6034","DOI":"10.1080\/00207543.2020.1799105","article-title":"Assessing the performances of a novel decentralised scheduling approach in Industry 4.0 and cloud manufacturing contexts","volume":"59","author":"Grassi","year":"2021","journal-title":"Int. J. Prod. Res."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1016\/j.procir.2019.02.099","article-title":"Coupling of centralized and decentralized scheduling for robust production in agile production systems","volume":"79","author":"Minguillon","year":"2019","journal-title":"Procedia CIRP"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Tan, Y., and Aufenanger, M. (2011, January 26\u201329). A real-time rescheduling heuristic using decentralized knowledge-based decisions for flexible flow shops with unrelated parallel machines. Proceedings of the 2011 9th IEEE International Conference on Industrial Informatics, Lisbon, Portugal.","DOI":"10.1109\/INDIN.2011.6034918"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"15","DOI":"10.11128\/sne.30.tn.10504","article-title":"Development of a Simulation Model to Analyze the Performance of Decentral Rescheduling Algorithms in Production Systems","volume":"30","author":"Sundermeier","year":"2020","journal-title":"Simul. Notes Eur."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Yang, Y., Ren, F., and Zhang, M. (2022, January 2\u20134). A Hybrid Multiagent-Based Rescheduling Mechanism for Open and Stochastic Environments Concerning the Execution Stage. Proceedings of the Australasian Joint Conference on Artificial Intelligence, Sydney, NSW, Australia.","DOI":"10.1007\/978-3-030-97546-3_45"},{"key":"ref_46","unstructured":"Spivak, D.I. (2014). Category Theory for the Sciences, MIT Press."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"4746","DOI":"10.1016\/j.jpaa.2019.02.014","article-title":"Hypergraph categories","volume":"223","author":"Fong","year":"2019","journal-title":"J. Pure Appl. Algebra"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Platzer, A. (2018). Logical Foundations of Cyber-Physical Systems, Springer.","DOI":"10.1007\/978-3-319-63588-0"},{"key":"ref_49","unstructured":"Garey, M.R., and Johnson, D.S. (1979). Computers and Intractability, Freeman."},{"key":"ref_50","first-page":"1666","article-title":"Categories for planning and scheduling","volume":"67","author":"Breiner","year":"2020","journal-title":"Not. Am. Math. Soc."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Tohm\u00e9, F., and Fioriti, A. (2024). Level-Agnostic Representations of Interacting Agents. Mathematics, 12.","DOI":"10.3390\/math12172697"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1007\/s11787-022-00319-3","article-title":"Brain and its universal logical model of multi-agent biological systems","volume":"16","author":"Schumann","year":"2022","journal-title":"Log. Universalis"},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Kr\u00f3l, J., and Schumann, A. (2023). The formal layer of {brain and mind} and emerging consciousness in physical systems. Found. Sci., 1\u201330.","DOI":"10.1007\/s10699-023-09937-6"},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Fong, B., and Spivak, D.I. (2019). An Invitation to Applied Category Theory: Seven Sketches in Compositionality, Cambridge University Press.","DOI":"10.1017\/9781108668804"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/5\/332\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:22:25Z","timestamp":1760030545000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/5\/332"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,27]]},"references-count":54,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2025,5]]}},"alternative-id":["axioms14050332"],"URL":"https:\/\/doi.org\/10.3390\/axioms14050332","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,4,27]]}}}