{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,12]],"date-time":"2025-12-12T15:53:23Z","timestamp":1765554803517,"version":"3.48.0"},"reference-count":38,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2025,12,12]],"date-time":"2025-12-12T00:00:00Z","timestamp":1765497600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["72271051"],"award-info":[{"award-number":["72271051"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["2232018H-07"],"award-info":[{"award-number":["2232018H-07"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>For a no-wait flow shop with continuous-flow characteristics, this study simultaneously considers machine setup times and rated processing speed constraints, aiming to minimize the sum of the maximum completion time and the maximum tardiness. First, lower bounds for the maximum completion time, the maximum tardiness, and the total objective function are developed. Second, a mixed-integer programming (MIP) model is formulated for the problem, and nonlinear elements are subsequently linearized via time discretization. Due to the computational complexity of the problem, two algorithms are proposed: a heuristic algorithm with fixed machine links and greedy rules (HAFG) and a genetic algorithm based on altering machine combinations (GAAM) for solving large-scale instances. The Earliest Due Date (EDD) rule is used as baselines for algorithmic comparison. To better understand the behaviors of the two algorithms, we observe the two components of the objective function separately. The results show that, compared with the EDD rule and GAAM, the HAFG algorithm tends to focus more on optimizing the maximum completion time. The performance of both algorithms is evaluated using their relative deviations from the developed lower bounds and is compared against the EDD rule. Numerical experiments demonstrate that both HAFG and GAAM significantly outperform the EDD rule. In large-scale instances, the HAFG algorithm achieves a gap of about 4%, while GAAM reaches a gap of about 3%, which is very close to the lower bound. In contrast, the EDD rule shows a deviation of about 10%. Combined with a sensitivity analysis on the number of machines, the proposed framework provides meaningful managerial insights for continuous-flow production environments.<\/jats:p>","DOI":"10.3390\/a18120788","type":"journal-article","created":{"date-parts":[[2025,12,12]],"date-time":"2025-12-12T15:27:20Z","timestamp":1765553240000},"page":"788","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Optimization of Continuous Flow-Shop Scheduling Considering Due Dates"],"prefix":"10.3390","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1603-4163","authenticated-orcid":false,"given":"Feifeng","family":"Zheng","sequence":"first","affiliation":[{"name":"Glorious Sun School of Business and Management, Donghua University, Shanghai 200051, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chunyao","family":"Zhang","sequence":"additional","affiliation":[{"name":"Glorious Sun School of Business and Management, Donghua University, Shanghai 200051, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3190-5008","authenticated-orcid":false,"given":"Ming","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Economics & Management, Tongji University, Shanghai 200092, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"4565","DOI":"10.1080\/00207543.2023.2263577","article-title":"A comprehensive literature review of the flow shop group scheduling problems: Systematic and bibliometric reviews","volume":"62","year":"2024","journal-title":"Int. J. Prod. Res."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ejor.2023.02.001","article-title":"A review and classification on distributed permutation flow shop scheduling problems","volume":"312","year":"2024","journal-title":"Eur. J. Oper. Res."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1275","DOI":"10.1080\/00207543.2024.2373426","article-title":"Integrated scheduling of multi-objective lot-streaming hybrid flow shop with AGV based on deep reinforcement learning","volume":"63","author":"Tang","year":"2025","journal-title":"Int. J. Prod. Res."},{"key":"ref_4","first-page":"4899","article-title":"A cooperative scatter search with reinforcement learning mechanism for the distributed permutation flow shop scheduling problem with sequence-dependent setup times","volume":"53","author":"Zhao","year":"2023","journal-title":"IEEE Trans. Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"123278","DOI":"10.1016\/j.eswa.2024.123278","article-title":"Self-adaptive population-based iterated greedy algorithm for distributed permutation flow shop scheduling problem with part of jobs subject to a common deadline constraint","volume":"248","author":"Li","year":"2024","journal-title":"Expert Syst. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1852","DOI":"10.1080\/00207543.2020.1727042","article-title":"The no-wait flow shop with rejection","volume":"59","author":"Koulamas","year":"2021","journal-title":"Int. J. Prod. Res."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/j.cie.2019.02.041","article-title":"Minimizing maximum completion time in mixed no-wait flow shops with sequence-dependent setup times","volume":"130","author":"Cheng","year":"2019","journal-title":"Comput. Ind. Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"107705","DOI":"10.1016\/j.asoc.2021.107705","article-title":"New benchmark algorithms for No-wait flow shop group scheduling problem with sequence-dependent setup times","volume":"111","author":"Cheng","year":"2021","journal-title":"Appl. Soft Comput."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.ins.2020.06.052","article-title":"No-wait two-stage flow shop problem with multi-task flexibility of the first machine","volume":"544","author":"Dong","year":"2021","journal-title":"Inf. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"101617","DOI":"10.1016\/j.swevo.2024.101617","article-title":"Q-learning guided algorithms for bi-criteria minimization of total flow time and maximum completion time in no-wait permutation flow shops","volume":"89","year":"2024","journal-title":"Swarm Evol. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"106009","DOI":"10.1016\/j.cor.2022.106009","article-title":"A branch-and-cut approach for the distributed no-wait flow shop scheduling problem","volume":"148","author":"Avci","year":"2022","journal-title":"Comput. Oper. Res."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"105921","DOI":"10.1016\/j.engappai.2023.105921","article-title":"An effective iterated local search algorithm for the distributed no-wait flow shop scheduling problem","volume":"120","author":"Avci","year":"2023","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"106946","DOI":"10.1016\/j.asoc.2020.106946","article-title":"A discrete artificial bee colony algorithm for the distributed heterogeneous no-wait flow shop scheduling problem","volume":"100","author":"Li","year":"2021","journal-title":"Appl. Soft Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3694","DOI":"10.1109\/TSMC.2024.3370376","article-title":"A novel evolutionary algorithm for scheduling distributed no-wait flow shop problems","volume":"54","author":"Pan","year":"2024","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"6692","DOI":"10.1109\/TII.2022.3192881","article-title":"A population-based iterated greedy algorithm for distributed assembly no-wait flow-shop scheduling problem","volume":"19","author":"Zhao","year":"2022","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"620","DOI":"10.1109\/TSMC.2024.3488205","article-title":"A policy-based meta-heuristic algorithm for energy-aware distributed no-wait flow-shop scheduling in heterogeneous factory systems","volume":"55","author":"Zhao","year":"2024","journal-title":"IEEE Trans. Syst. Man, Cybern. Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2854","DOI":"10.1080\/00207543.2022.2070786","article-title":"A reinforcement learning-driven brain storm optimisation algorithm for multi-objective energy-efficient distributed assembly no-wait flow shop scheduling problem","volume":"61","author":"Zhao","year":"2023","journal-title":"Int. J. Prod. Res."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"101559","DOI":"10.1016\/j.swevo.2024.101559","article-title":"An enhanced estimation of distribution algorithm with problem-specific knowledge for distributed no-wait flow shop group scheduling problems","volume":"87","author":"Zhang","year":"2024","journal-title":"Swarm Evol. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"101179","DOI":"10.1016\/j.swevo.2022.101179","article-title":"A discrete artificial bee colony method based on variable neighborhood structures for the distributed permutation flow shop problem with sequence-dependent setup times","volume":"75","author":"Yu","year":"2022","journal-title":"Swarm Evol. Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"106850","DOI":"10.1016\/j.cor.2024.106850","article-title":"An effective two-stage heuristic for scheduling the distributed assembly flow shops with sequence dependent setup times","volume":"173","author":"Song","year":"2025","journal-title":"Comput. Oper. Res."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"102035","DOI":"10.1016\/j.swevo.2025.102035","article-title":"A knowledge-based two-population optimization algorithm for distributed heterogeneous assembly permutation flow shop scheduling with batch delivery and setup times","volume":"97","author":"Zhao","year":"2025","journal-title":"Swarm Evol. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"106230","DOI":"10.1016\/j.engappai.2023.106230","article-title":"A Q-learning artificial bee colony for distributed assembly flow shop scheduling with factory eligibility, transportation capacity and setup time","volume":"123","author":"Wang","year":"2021","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"115339","DOI":"10.1016\/j.eswa.2021.115339","article-title":"Multi-objective distributed reentrant permutation flow shop scheduling with sequence-dependent setup time","volume":"183","author":"Rifai","year":"2021","journal-title":"Expert Syst. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"107079","DOI":"10.1016\/j.cor.2025.107079","article-title":"A trajectory-based algorithm enhanced by Q-learning and cloud integration for hybrid flexible flow shop scheduling problem with sequence-dependent setup times: A case study","volume":"181","author":"Ozsoydan","year":"2025","journal-title":"Comput. Oper. Res."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/j.ejor.2022.02.019","article-title":"A parameter-Less iterated greedy method for the hybrid flow shop scheduling problem with setup times and due date windows","volume":"303","author":"Missaoui","year":"2022","journal-title":"Eur. J. Oper. Res."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"118068","DOI":"10.1016\/j.eswa.2022.118068","article-title":"Adaptive genetic algorithm for two-stage hybrid flow-shop scheduling with sequence-independent setup time and no-interruption requirement","volume":"208","author":"Qiao","year":"2022","journal-title":"Expert Syst. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1233","DOI":"10.1080\/00207543.2022.2031331","article-title":"A novel shuffled frog-leaping algorithm with reinforcement learning for distributed assembly hybrid flow shop scheduling","volume":"61","author":"Cai","year":"2023","journal-title":"Int. J. Prod. Res."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1016\/j.ejor.2025.02.009","article-title":"Minimising Makespan and total tardiness for the flow shop group scheduling problem with sequence dependent setup times","volume":"324","author":"He","year":"2025","journal-title":"Eur. J. Oper. Res."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"129012","DOI":"10.1016\/j.eswa.2025.129012","article-title":"An iterative greedy algorithm based on neighborhood search for energy-efficient scheduling of distributed permutation flow shop with sequence-dependent setup time","volume":"296","author":"Zhong","year":"2026","journal-title":"Expert Syst. Appl."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"504","DOI":"10.1016\/j.ejor.2021.07.055","article-title":"Two-machine flow shop scheduling with a common due date to maximize total early work","volume":"300","author":"Chen","year":"2022","journal-title":"Eur. J. Oper. Res."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"5488","DOI":"10.1080\/00207543.2021.1962017","article-title":"Scheduling in a no-wait flow shop to minimise total earliness and tardiness with additional idle time allowed","volume":"60","author":"Schaller","year":"2022","journal-title":"Int. J. Prod. Res."},{"key":"ref_32","first-page":"1153","article-title":"Bi-objective hybrid flow shop scheduling with common due date","volume":"21","author":"Li","year":"2021","journal-title":"Oper. Res."},{"key":"ref_33","first-page":"1403","article-title":"Minimizing the weighted sum of maximum earliness and maximum tardiness in a single-agent and two-agent form of a two-machine flow shop scheduling problem","volume":"22","author":"Nasrollahi","year":"2022","journal-title":"Oper. Res."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"107835","DOI":"10.1016\/j.cie.2021.107835","article-title":"Flow shop scheduling with two distinct job due dates","volume":"163","author":"Koulamas","year":"2022","journal-title":"Comput. Ind. Eng."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"109317","DOI":"10.1016\/j.cie.2023.109317","article-title":"Scheduling on proportionate flow shop with job rejection and common due date assignment","volume":"181","author":"Geng","year":"2023","journal-title":"Comput. Ind. Eng."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"7039","DOI":"10.1080\/00207543.2025.2492747","article-title":"Lot-streaming flow shop scheduling under stochastic due dates","volume":"63","author":"Liu","year":"2025","journal-title":"Int. J. Prod. Res."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"105204","DOI":"10.1016\/j.cor.2020.105204","article-title":"Just-in-time scheduling for a distributed concrete precast flow shop system","volume":"129","author":"Xiong","year":"2021","journal-title":"Comput. Oper. Res."},{"key":"ref_38","first-page":"82","article-title":"Scheduling on proportionate flow shop with total late work and job rejection","volume":"25","author":"Geng","year":"2025","journal-title":"Oper. Res."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/18\/12\/788\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,12]],"date-time":"2025-12-12T15:29:27Z","timestamp":1765553367000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/18\/12\/788"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,12]]},"references-count":38,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2025,12]]}},"alternative-id":["a18120788"],"URL":"https:\/\/doi.org\/10.3390\/a18120788","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12,12]]}}}