{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,8]],"date-time":"2026-02-08T11:12:43Z","timestamp":1770549163504,"version":"3.49.0"},"reference-count":59,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,8,18]],"date-time":"2021-08-18T00:00:00Z","timestamp":1629244800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the National Key R&amp;D Program of China","award":["2018YFB1702700"],"award-info":[{"award-number":["2018YFB1702700"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51905196"],"award-info":[{"award-number":["51905196"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["71620107002"],"award-info":[{"award-number":["71620107002"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper investigates the nonidentical parallel production line scheduling problem derived from an axle housing machining workshop of an axle manufacturer. The characteristics of axle housing machining lines are analyzed, and a nonidentical parallel line scheduling model with a jumping process operation (NPPLS-JP), which considers mixed model production, machine eligibility constraints, and fuzzy due dates, is established so as to minimize the makespan and earliness\/tardiness penalty cost. While the physical structures of the parallel lines in the NPPLS-JP model are symmetric, the production capacities and process capabilities are asymmetric for different models. Different from the general parallel line scheduling problem, NPPLS-JP allows for a job to transfer to another production line to complete the subsequent operations (i.e., jumping process operations), and the transfer is unidirectional. The significance of the NPPLS-JP model is that it meets the demands of multivariety mixed model production and makes full use of the capacities of parallel production lines. Aiming to solve the NPPLS-JP problem, we propose a hybrid algorithm named the multi-objective grey wolf optimizer based on decomposition (MOGWO\/D). This new algorithm combines the GWO with the multi-objective evolutionary algorithm based on decomposition (MOEA\/D) to balance the exploration and exploitation abilities of the original MOEA\/D. Furthermore, coding and decoding rules are developed according to the features of the NPPLS-JP problem. To evaluate the effectiveness of the proposed MOGWO\/D algorithm, a set of instances with different job scales, job types, and production scenarios is designed, and the results are compared with those of three other famous multi-objective optimization algorithms. The experimental results show that the proposed MOGWO\/D algorithm exhibits superiority in most instances.<\/jats:p>","DOI":"10.3390\/sym13081521","type":"journal-article","created":{"date-parts":[[2021,8,18]],"date-time":"2021-08-18T22:51:00Z","timestamp":1629327060000},"page":"1521","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Modeling and Optimization for Multi-Objective Nonidentical Parallel Machining Line Scheduling with a Jumping Process Operation Constraint"],"prefix":"10.3390","volume":"13","author":[{"given":"Guangyan","family":"Xu","sequence":"first","affiliation":[{"name":"School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China"}]},{"given":"Zailin","family":"Guan","sequence":"additional","affiliation":[{"name":"School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China"}]},{"given":"Lei","family":"Yue","sequence":"additional","affiliation":[{"name":"School of Mechanical and Electrical Engineering, Guangzhou University, Guangzhou 510000, China"}]},{"given":"Jabir","family":"Mumtaz","sequence":"additional","affiliation":[{"name":"College of Mechanical and Electronic Engineering, Wenzhou University, Wenzhou 325000, China"}]},{"given":"Jun","family":"Liang","sequence":"additional","affiliation":[{"name":"School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1080\/0951192X.2010.511654","article-title":"A genetic algorithm for simultaneous lotsizing and sequencing of the permutation flow shops with sequence-dependent setups","volume":"24","author":"Mohammadi","year":"2011","journal-title":"Int. J. Comput. Integr. Manuf."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"5843","DOI":"10.1080\/00207543.2011.632385","article-title":"Sequence-dependent flow shop scheduling problem minimising the number of tardy jobs","volume":"50","author":"Varmazyar","year":"2012","journal-title":"Int. J. Prod. Res."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1016\/j.cie.2018.12.065","article-title":"Multi objective lotsizing and scheduling with material constraints in flexible parallel lines using a Pareto based guided artificial bee colony algorithm","volume":"128","author":"Yue","year":"2019","journal-title":"Comput. Ind. Eng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"358","DOI":"10.1016\/j.cor.2007.10.004","article-title":"A comparison of scheduling algorithms for flexible flow shop problems with unrelated parallel machines, setup times, and dual criteria","volume":"36","author":"Jungwattanakit","year":"2009","journal-title":"Comput. Oper. Res."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2013","DOI":"10.1016\/j.cor.2004.01.003","article-title":"Simulated annealing heuristic for flow shop scheduling problems with unrelated parallel machines","volume":"32","author":"Low","year":"2005","journal-title":"Comput. Oper. Res."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1739","DOI":"10.1007\/s00170-014-6343-0","article-title":"Cyclic hybrid flow shop scheduling problem with limited buffers and machine eligibility constraints","volume":"76","author":"Soltani","year":"2014","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1001","DOI":"10.1007\/s00170-012-4052-0","article-title":"A two-criteria objective function flexible flowshop scheduling problem with machine eligibility constraint","volume":"64","author":"Tadayon","year":"2013","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"781","DOI":"10.1016\/j.ejor.2004.06.038","article-title":"A genetic algorithm for hybrid flowshops with sequence dependent setup times and machine eligibility","volume":"169","author":"Ruiz","year":"2007","journal-title":"Eur. J. Oper. Res."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"5293","DOI":"10.1080\/00207543.2017.1408971","article-title":"A re-entrant hybrid flow shop scheduling problem with machine eligibility constraints","volume":"56","author":"Zhang","year":"2018","journal-title":"Int. J. Prod. Res."},{"key":"ref_10","first-page":"147","article-title":"Leveraging constraint-based approaches formulti-objective flexible flow-shop scheduling with energy costs","volume":"10","author":"Oddi","year":"2016","journal-title":"Intell. Artif."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"701","DOI":"10.1016\/S0098-1354(01)00671-8","article-title":"An MILP continuous-time approach to short-term scheduling of resource-constrained multistage flowshop batch facilities","volume":"25","author":"Henning","year":"2001","journal-title":"Comput. Chem. Eng."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"106814","DOI":"10.1016\/j.cie.2020.106814","article-title":"A multi-constrained supply chain model with optimal production rate in relation to quality of products under stochastic fuzzy demand","volume":"149","author":"Malik","year":"2020","journal-title":"Comput. Ind. Eng."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Malik, A.I., and Sarkar, B. (2019). Coordinating supply-chain management under stochastic fuzzy environment and lead-time reduction. Mathematics, 7.","DOI":"10.3390\/math7050480"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"4860","DOI":"10.1016\/j.eswa.2009.12.029","article-title":"Solving the fuzzy earliness and tardiness in scheduling problems by using genetic algorithms","volume":"37","author":"Wu","year":"2010","journal-title":"Expert Syst. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1016\/S0360-8352(01)00065-1","article-title":"Mixed model assembly line design in a make-to-order environment","volume":"41","author":"Bukchin","year":"2002","journal-title":"Comput. Ind. Eng."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/S0925-5273(99)00097-3","article-title":"Multi-Agent Systems in production planning and control: An application to the scheduling of mixed-model assembly lines","volume":"68","author":"Caridi","year":"2000","journal-title":"Int. J. Prod. Econ."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3095","DOI":"10.1080\/002075497194309","article-title":"A parallel station heuristic for the mixed-model production line balancing problem","volume":"35","author":"Askin","year":"1997","journal-title":"Int. J. Prod. Res."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1016\/j.ijpe.2011.07.022","article-title":"Optimally locating in-house logistics areas to facilitate JIT-supply of mixed-model assembly lines","volume":"135","author":"Emde","year":"2010","journal-title":"Int. J. Prod. Econ."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/j.jmsy.2019.01.001","article-title":"Balancing and cyclical scheduling of asynchronous mixed-model assembly lines with parallel stations","volume":"50","author":"Lopes","year":"2019","journal-title":"J. Manuf. Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1002\/nav.20241","article-title":"Modeling and analysis of a mixed-model assembly line with stochastic operation times","volume":"54","author":"Zhao","year":"2010","journal-title":"Nav. Res. Logist."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Khalid, Q.S., Arshad, M., Maqsood, S., and Kim, S. (2019). Hybrid particle swarm algorithm for products\u2019 scheduling problem in cellular manufacturing system. Symmetry, 11.","DOI":"10.3390\/sym11060729"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1016\/j.ijpe.2003.12.010","article-title":"A beam search heuristic method for mixed-model scheduling with setups","volume":"96","author":"Mcmullen","year":"2005","journal-title":"Int. J. Prod. Econ."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1016\/S0926-5805(01)00083-8","article-title":"GA-based resource-constrained flow-shop scheduling model for mixed precast production","volume":"11","author":"Leu","year":"2002","journal-title":"Autom. Constr."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1007\/s10845-014-0988-2","article-title":"Simultaneous order scheduling and mixed-model sequencing in assemble-to-order production environment: A multi-objective hybrid artificial bee colony algorithm","volume":"28","author":"Wang","year":"2017","journal-title":"J. Intell. Manuf."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"643","DOI":"10.1080\/00207543.2018.1476786","article-title":"A realistic multi-manned five-sided mixed-model assembly line balancing and scheduling problem with moving workers and limited workspace","volume":"57","author":"Bahman","year":"2019","journal-title":"Int. J. Prod. Res."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1007\/s10601-017-9279-9","article-title":"Mixed model line balancing with parallel stations, zoning constraints, and ergonomics","volume":"23","author":"Alghazi","year":"2018","journal-title":"Constraints"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"799","DOI":"10.1007\/s00170-008-1754-4","article-title":"Bicriteria parallel flow line scheduling using hybrid population-based heuristics","volume":"43","author":"Rajeswari","year":"2009","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1047","DOI":"10.1007\/s00170-007-1164-z","article-title":"Parallel line job shop scheduling using genetic algorithm","volume":"35","author":"Haq","year":"2008","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"718","DOI":"10.1016\/j.ejor.2013.03.036","article-title":"A decomposition approach for the general lotsizing and scheduling problem for parallel production lines","volume":"229","author":"Meyr","year":"2013","journal-title":"Eur. J. Oper. Res."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"6252","DOI":"10.1080\/00207543.2019.1675917","article-title":"Hybrid spider monkey optimisation algorithm for multi-level planning and scheduling problems of assembly lines","volume":"58","author":"Mumtaz","year":"2020","journal-title":"Int. J. Prod. Res."},{"key":"ref_31","first-page":"1","article-title":"Multi-objective modeling for preventive maintenance scheduling in a multiple production line","volume":"26","author":"Ebrahimipour","year":"2013","journal-title":"J. Intell. Manuf."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2169","DOI":"10.1109\/ACCESS.2019.2895954","article-title":"Multi-level planning and scheduling for parallel pcb assembly lines using hybrid spider monkey optimization approach","volume":"7","author":"Mumtaz","year":"2019","journal-title":"IEEE Access"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"400","DOI":"10.1016\/j.ins.2018.06.063","article-title":"A many-objective evolutionary algorithm with angle-based selection and shift-based density estimation","volume":"509","author":"Liu","year":"2017","journal-title":"Inf. Sci."},{"key":"ref_34","first-page":"493","article-title":"Messy genetic algorithms: Motivation, analysis, and first results","volume":"3","author":"Goldberg","year":"1989","journal-title":"Complex Syst."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1109\/4235.797969","article-title":"Multiobjective evolutionary algorithms: A comparative case study and the strength Pareto approach","volume":"3","author":"Zitzler","year":"1999","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_36","unstructured":"Giannakoglou, K.C., Tsahalis, D.T., P\u00e9riaux, J., Papailiou, K.D., and Fogarty, T. (2002). SPEA2: Improving the Performance of the Strength Areto Evolutionary Algorithm, International Center for Numerical Methods in Engineering (CIMNE). Evolutionary Methods for Design, Optimization and Control with Applications to Industrial Problems (EUROGEN 2001), Athens, Greece, September."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1109\/4235.996017","article-title":"A fast and elitist multiobjective genetic algorithm: NSGA-II","volume":"6","author":"Deb","year":"2002","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1653","DOI":"10.1016\/j.ejor.2006.08.008","article-title":"Sms-emoa: Multiobjective selection based on dominated hypervolume","volume":"181","author":"Beume","year":"2007","journal-title":"Eur. J. Oper. Res."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1162\/EVCO_a_00009","article-title":"Hype: An algorithm for fast hypervolume-based many-objective optimization","volume":"19","author":"Bader","year":"2011","journal-title":"Evol. Comput."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"712","DOI":"10.1109\/TEVC.2007.892759","article-title":"MOEA\/D: A multiobjective evolutionary algorithm based on decomposition","volume":"11","author":"Zhang","year":"2007","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"561","DOI":"10.1162\/EVCO_a_00038","article-title":"An adaptive evolutionary multi-objective approach based on simulated annealing","volume":"19","author":"Li","year":"2014","journal-title":"Evol. Comput."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/0305215X.2011.632008","article-title":"MOEA\/D-SQA: A multi-objective memetic algorithm based on decomposition","volume":"44","author":"Tan","year":"2012","journal-title":"Eng. Optim."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1016\/j.ins.2015.07.018","article-title":"A new multi-objective particle swarm optimization algorithm based on decomposition","volume":"325","author":"Cai","year":"2015","journal-title":"Inf. Sci."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1845","DOI":"10.1109\/TSMCB.2012.2231860","article-title":"MOEA\/D-ACO: A Multiobjective Evolutionary Algorithm Using Decomposition and Antcolony","volume":"43","author":"Ke","year":"2013","journal-title":"IEEE Trans. Cybern."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"9605","DOI":"10.1007\/s00500-018-3524-z","article-title":"MOEA\/D-GLS: A multiobjective memetic algorithm using decomposition and guided local search","volume":"23","author":"Alhindi","year":"2019","journal-title":"Soft Comput."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"456","DOI":"10.1109\/TEVC.2009.2033671","article-title":"Expensive multiobjective optimization by MOEA\/D with gaussian process model","volume":"14","author":"Zhang","year":"2010","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"474","DOI":"10.1109\/TCYB.2015.2403849","article-title":"Adaptive replacement strategies for MOEA\/D","volume":"46","author":"Wang","year":"2017","journal-title":"IEEE Trans. Cybern."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1023\/A:1021251113462","article-title":"Simple explanation of the no-free-lunch theorem and its implications","volume":"115","author":"Ho","year":"2002","journal-title":"J. Optim. Theory Appl."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1016\/S0360-8352(98)00128-4","article-title":"Multi-objective scheduling with fuzzy due-date","volume":"35","author":"Murata","year":"1998","journal-title":"Comput. Ind. Eng."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"104931","DOI":"10.1016\/j.cor.2020.104931","article-title":"Evolutionary tabu search for flexible due-date satisfaction in fuzzy job shop scheduling","volume":"119","author":"Vela","year":"2020","journal-title":"Comput. Oper. Res."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"172988142092523","DOI":"10.1177\/1729881420925236","article-title":"Modified honey bees mating optimization algorithm for multi-objective uncertain integrated process planning and scheduling problem","volume":"17","author":"Wen","year":"2020","journal-title":"Int. J. Adv. Robot. Syst."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1016\/j.advengsoft.2013.12.007","article-title":"Grey Wolf Optimizer","volume":"69","author":"Mirjalili","year":"2014","journal-title":"Adv. Eng. Softw."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1111\/j.2517-6161.1958.tb00299.x","article-title":"Experiments with Mixtures","volume":"20","year":"1958","journal-title":"J. Roy. Statist. Soc."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"818","DOI":"10.1109\/TSMCA.2008.923086","article-title":"An effective PSO-based hybrid algorithm for multiobjective permutation flow shop scheduling","volume":"38","author":"Li","year":"2008","journal-title":"IEEE Trans. Syst. Man Cybern. Paart A Syst. Hum."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/j.eswa.2015.10.039","article-title":"Multi-objective grey wolf optimizer: A novel algorithm for multi-criterion optimization","volume":"47","author":"Mirjalili","year":"2015","journal-title":"Expert Syst. Appl."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1109\/TEVC.2004.826067","article-title":"Handling multiple objectives with particle swarm optimization","volume":"8","author":"Coello","year":"2004","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1109\/TEVC.2003.810758","article-title":"Performance assessment of multiobjective optimizers: An analysis and review","volume":"7","author":"Zitzler","year":"2003","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"2391","DOI":"10.1109\/TCYB.2014.2307319","article-title":"Consistencies and contradictions of performance metrics in multiobjective optimization","volume":"44","author":"Jiang","year":"2014","journal-title":"IEEE Trans. Cybern."},{"key":"ref_59","first-page":"323","article-title":"A genetic algorithm for hybrid flow-shop scheduling with multiprocessor tasks","volume":"8","author":"Oguz","year":"2005","journal-title":"Complex Syst."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/8\/1521\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:46:41Z","timestamp":1760165201000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/8\/1521"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,18]]},"references-count":59,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["sym13081521"],"URL":"https:\/\/doi.org\/10.3390\/sym13081521","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,8,18]]}}}