{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T14:35:05Z","timestamp":1775745305149,"version":"3.50.1"},"reference-count":29,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,12,2]],"date-time":"2024-12-02T00:00:00Z","timestamp":1733097600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012166","name":"National key Research and development program of China","doi-asserted-by":"publisher","award":["2020YFB1709800"],"award-info":[{"award-number":["2020YFB1709800"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The efficiency and quality of the manufacturing industry are greatly influenced by production scheduling, which makes it a crucial aspect. A well-designed production scheduling scheme can significantly enhance manufacturing efficiency and reduce enterprise costs. This paper presents a tailored optimization model designed to address a more complex production scheduling problem that incorporates parallel machines and preventive maintenance. The proposed solutions aim to achieve a balance between job sequence and machine reliability, considering the minimum maintenance cost rate for determining maintenance cycles of deteriorating machines in real manufacturing scenarios. Furthermore, the objective of minimizing the maximum completion time guides machine assignment and job sequence based on maintenance constraints. The innovation lies in the introduction of a greedy algorithm that utilizes a water injection model to address this NP-hard integrated problem. A pre-distribution model is constructed using the water injection model, and its solution is utilized as input for constructing the production scheduling model, which aids in determining machine assignment and job sequence. This algorithm demonstrates remarkable effectiveness and efficiency, enabling the achievement of an optimal solution. A numerical example is presented to illustrate the computational process, accompanied by an extensive discussion of the results showcasing improved performance. Furthermore, the optimization model developed in this paper can be adapted to tackle the production scheduling problem with modifications tailored for parallel machines.<\/jats:p>","DOI":"10.3390\/axioms13120848","type":"journal-article","created":{"date-parts":[[2024,12,4]],"date-time":"2024-12-04T10:07:10Z","timestamp":1733306830000},"page":"848","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["An Optimization Model for Production Scheduling in Parallel Machine Systems"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-5322-9971","authenticated-orcid":false,"given":"Leting","family":"Zu","sequence":"first","affiliation":[{"name":"Department of Engineering Management, Chongqing University, Chongqing 400044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2589-6643","authenticated-orcid":false,"given":"Wenzhu","family":"Liao","sequence":"additional","affiliation":[{"name":"Department of Engineering Management, Chongqing University, Chongqing 400044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaoxia","family":"Yang","sequence":"additional","affiliation":[{"name":"Department of Engineering Management, Chongqing University, Chongqing 400044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Cavallieri, M.S., Viles, E., and Montoya-Torres, J.R. (Prod. Plan. Control., 2024). Sustainable production scheduling: Insights from real-life applications, Prod. Plan. Control., early access.","DOI":"10.1080\/09537287.2024.2389879"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1764","DOI":"10.1016\/j.cor.2005.05.034","article-title":"Single-machine scheduling with periodic maintenance to minimize makespan","volume":"34","author":"Ji","year":"2007","journal-title":"Comput. Oper. Res."},{"key":"ref_3","first-page":"1099","article-title":"Single-machine integrated scheduling model considering maintenance activities","volume":"20","author":"Gan","year":"2014","journal-title":"Comput. Integr. Manuf. Syst."},{"key":"ref_4","first-page":"524","article-title":"Integrating Production Scheduling and Preventive Maintenance for a Single Machine with Due Window","volume":"49","author":"Ding","year":"2015","journal-title":"J. Shanghai Jiaotong Univ."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"765","DOI":"10.1007\/s00170-016-9093-3","article-title":"Scheduling flexible flow shop with recirculation and machine sequence-dependent processing times: Formulation and solution procedures","volume":"89","author":"Ahonen","year":"2017","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1016\/j.ejor.2016.06.025","article-title":"Distributionally robust single machine scheduling with risk aversion","volume":"256","author":"Chang","year":"2017","journal-title":"Eur. J. Oper. Res."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1093\/comjnl\/bxy030","article-title":"Single-machine scheduling with simultaneous considerations of resource allocation and deteriorating jobs","volume":"62","author":"Liu","year":"2019","journal-title":"Comput. J."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"106072","DOI":"10.1016\/j.cie.2019.106072","article-title":"An energy-efficient single machine scheduling problem with machine reliability constraints","volume":"137","author":"Chen","year":"2019","journal-title":"Comput. Ind. Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2708","DOI":"10.1080\/00207543.2020.1737336","article-title":"A single machine scheduling problem with machine availability constraints and preventive maintenance","volume":"59","author":"Chen","year":"2021","journal-title":"Int. J. Prod. Res."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"446","DOI":"10.1007\/s10878-021-00822-1","article-title":"Single-machine multitasking scheduling with job efficiency promotion","volume":"44","author":"Ji","year":"2022","journal-title":"J. Comb. Optim."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1007\/s40314-024-02717-3","article-title":"Single machine scheduling problems with truncated learning effects and exponential past-sequence-dependent delivery times","volume":"43","author":"Wang","year":"2024","journal-title":"Comput. Appl. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"106642","DOI":"10.1016\/j.cor.2024.106642","article-title":"Target-oriented robust satisficing models for the single machine scheduling problems with release time","volume":"167","author":"Zhang","year":"2024","journal-title":"Comput. Oper. Res."},{"key":"ref_13","first-page":"1","article-title":"Single machine scheduling problem with unexpected failures under flexible maintenance","volume":"59","author":"Yang","year":"2024","journal-title":"J. Oper. Res. Soc."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1196","DOI":"10.1016\/j.cor.2012.12.007","article-title":"Parallel machine scheduling with splitting jobs by a hybrid differential evolution algorithm","volume":"40","author":"Wang","year":"2013","journal-title":"Comput. Oper. Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1080\/0305215X.2013.786062","article-title":"An ant colony optimization heuristic for an integrated production and distribution scheduling problem","volume":"46","author":"Chang","year":"2014","journal-title":"Eng. Optim."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1016\/j.jmsy.2015.07.002","article-title":"Multi-objective optimization of parallel machine scheduling integrated with multi-resources preventive maintenance planning","volume":"37","author":"Wang","year":"2015","journal-title":"J. Manuf. Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jmsy.2016.10.005","article-title":"Minimizing total earliness and tardiness through unrelated parallel machine scheduling using distributed release time control","volume":"42","author":"Cheng","year":"2017","journal-title":"J. Manuf. Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ijpe.2017.01.014","article-title":"Parallel-machine scheduling with machine-dependent maintenance periodic recycles","volume":"186","author":"Li","year":"2017","journal-title":"Int. J. Prod. Econ."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1007\/s00170-016-8477-8","article-title":"A parallel machine batch scheduling problem in a brewing company","volume":"87","author":"Jobish","year":"2016","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1016\/j.asoc.2018.02.018","article-title":"A hybrid ABC-TS algorithm for the unrelated parallel-batching machines scheduling problem with deteriorating jobs and maintenance activity","volume":"66","author":"Lu","year":"2018","journal-title":"Appl. Soft. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"106320","DOI":"10.1016\/j.cie.2020.106320","article-title":"An artificial bee colony with division for distributed unrelated parallel machine scheduling with preventive maintenance","volume":"141","author":"Lei","year":"2020","journal-title":"Comput. Ind. Eng."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"105738","DOI":"10.1016\/j.cor.2022.105738","article-title":"A general variable neighborhood search algorithm for a parallel-machine scheduling problem considering machine health conditions and preventive maintenance","volume":"143","author":"Zhang","year":"2022","journal-title":"Comput. Oper. Res."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"108800","DOI":"10.1016\/j.cie.2022.108800","article-title":"A generic optimization framework for scheduling problems under machine deterioration and maintenance activities","volume":"174","author":"Rudek","year":"2022","journal-title":"Comput. Ind. Eng."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"109631","DOI":"10.1016\/j.cie.2023.109631","article-title":"Knowledge-enhanced reinforcement learning for multi-machine integrated production and maintenance scheduling","volume":"185","author":"Hu","year":"2023","journal-title":"Comput. Ind. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"102406","DOI":"10.1016\/j.rcim.2022.102406","article-title":"Multi-agent deep reinforcement learning based Predictive Maintenance on parallel machines","volume":"78","author":"Kubler","year":"2022","journal-title":"Robot. Comput.-Integr. Manuf."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"913","DOI":"10.3233\/JIFS-161385","article-title":"Joint optimization of preventive maintenance and production scheduling for parallel machines system","volume":"32","author":"Liao","year":"2017","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1134\/S0001434624010048","article-title":"Comparison of Purely Greedy and Orthogonal Greedy Algorithm","volume":"115","author":"Vishnevetskiy","year":"2024","journal-title":"Math. Notes"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1935","DOI":"10.1007\/s13202-023-01658-7","article-title":"Transmission model of transient flow wave signal in intelligent layered water injection system","volume":"13","author":"Ming","year":"2023","journal-title":"J. Pet. Explor. Prod. Technol."},{"key":"ref_29","unstructured":"Huang, F. (2014). Joint Optimization of Production Scheduling and Maintenance for Parallel Machine System. [Master\u2019s Thesis, Shanghai Jiaotong University]."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/12\/848\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:45:19Z","timestamp":1760114719000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/12\/848"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,2]]},"references-count":29,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2024,12]]}},"alternative-id":["axioms13120848"],"URL":"https:\/\/doi.org\/10.3390\/axioms13120848","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,12,2]]}}}