{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:44:28Z","timestamp":1760147068443,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,1,2]],"date-time":"2023-01-02T00:00:00Z","timestamp":1672617600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The purpose of this research is to investigate a novel three-stage flow shop scheduling problem with an ambiguous processing time. The uncertain information is characterized by Pentagonal fuzzy numbers. To solve the problem, in this paper, two different strategies are proposed; one relies on the idea of a ranking function, and the other on the close interval approximation of the pentagonal fuzzy number. For persons that need to be more specific in their requirements, the close interval approximation of the Pentagonal fuzzy number is judged to be the best appropriate approximation interval. Regarding the rental cost specification, these methods are used to reduce the rental cost for the concerned devices. In addition, a comparison of our suggested approach\u2019s computed total processing time, total machine rental cost, and machine idle time to the existing approach is introduced. A numerical example is shown to clarify the benefits of the two strategies and to help the readers understand it better.<\/jats:p>","DOI":"10.3390\/sym15010130","type":"journal-article","created":{"date-parts":[[2023,1,2]],"date-time":"2023-01-02T04:07:09Z","timestamp":1672632429000},"page":"130","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["A Novel Approach for Minimizing Processing Times of Three-Stage Flow Shop Scheduling Problems under Fuzziness"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4420-7287","authenticated-orcid":false,"given":"Alhanouf","family":"Alburaikan","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science and Arts, Qassim University, Al-Badaya 51951, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9099-8422","authenticated-orcid":false,"given":"Harish","family":"Garg","sequence":"additional","affiliation":[{"name":"School of Mathematics, Thapar Institute of Engineering & Technology (Deemed University), Patiala 147004, Punjab, India"},{"name":"Department of Mathematics, Graphic Era Deemed to be University, Dehradun 248002, Uttarakhand, India"},{"name":"Applied Science Research Center, Applied Science Private University, Amman 11931, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8269-8822","authenticated-orcid":false,"given":"Hamiden Abd El-Wahed","family":"Khalifa","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science and Arts, Qassim University, Al-Badaya 51951, Saudi Arabia"},{"name":"Department of Operations and Management Research, Faculty of Graduate Studies for Statistical Research, Cairo University, Giza 12613, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1002\/nav.3800010110","article-title":"Optimal two and three stages production schedule with Set up times included","volume":"1","author":"Johnson","year":"1954","journal-title":"Nav. Res. Logist. Q."},{"key":"ref_2","first-page":"1335","article-title":"Scheduling parallel machine problem under general effects of deterioration and learning with past-sequence-dependent setup time: Heuristic and meta-heuristic approaches","volume":"24","author":"Mir","year":"2000","journal-title":"Soft Comput."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1007\/s10951-019-00622-w","article-title":"Single-machine scheduling with job-dependent machine deterioration","volume":"22","author":"Luo","year":"2019","journal-title":"J. Sched."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"595","DOI":"10.1007\/s10951-019-00615-9","article-title":"Unary NP-hardness of single-machine scheduling to minimize the total tardiness with deadlines","volume":"22","author":"Chen","year":"2019","journal-title":"J. Sched."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1016\/j.omega.2016.06.008","article-title":"On single-machine scheduling with workload-dependent maintenance duration","volume":"68","author":"Luo","year":"2017","journal-title":"Omega"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1007\/s10479-016-2251-z","article-title":"Minimizing value-at-risk in single-machine scheduling","volume":"248","author":"Atakan","year":"2017","journal-title":"Anal. Oper. Res."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"6826","DOI":"10.1080\/00207543.2019.1685708","article-title":"A new unrelated parallel machine scheduling problem with tool changes to minimize the total energy consumption","volume":"58","author":"Zhang","year":"2019","journal-title":"Int. J. Prod. Res."},{"key":"ref_8","unstructured":"Ueno, N., Sotojima, S., and Takeda, J. (1991, January 1). Multi-Stage Flow-Shop in Steel Works. Proceedings of the 24th Annual Simulation Symposium, New Orleans, LO, USA."},{"key":"ref_9","first-page":"1773","article-title":"Model and algorithm for two-stage flow shop group scheduling problem with special blocking constraint","volume":"35","author":"Yuan","year":"2020","journal-title":"Control. Decis."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1007\/s00170-014-5841-4","article-title":"A algorithm for flow shop scheduling problem with consideration of position-based learning effect and multiple availability constraints","volume":"73","author":"Fattahi","year":"2014","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"320140","DOI":"10.1155\/2015\/320140","article-title":"A local search algorithm for the flow-shop scheduling problem with release dates","volume":"2015","author":"Ren","year":"2015","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1478","DOI":"10.1016\/j.ifacol.2016.07.780","article-title":"Heuristics for solving flow shop scheduling problem under resources constraints","volume":"49","author":"Laribi","year":"2016","journal-title":"IFAC-Pap. Online"},{"key":"ref_13","first-page":"59","article-title":"Modeling and scheduling no-idle hybrid flow shop problem","volume":"10","author":"Yazdani","year":"2017","journal-title":"J. Optim. Ind. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1973604","DOI":"10.1155\/2018\/1973604","article-title":"Solutions to no wait flow-shop scheduling problem using the flower pollination algorithm based on the hormone modulation mechanism","volume":"2018","author":"Qu","year":"2018","journal-title":"Complexity"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control"},{"key":"ref_16","first-page":"75","article-title":"Triangular fuzzy matrices","volume":"4","author":"Shymala","year":"2007","journal-title":"Iran. J. Fuzzy Syst."},{"key":"ref_17","first-page":"60","article-title":"Generalized trapezoidal fuzzy number with its arithmetic operations and its application in fuzzy system reliability analysis","volume":"5","author":"Shaw","year":"2011","journal-title":"Int. J. Pure Appl. Sci. Technol."},{"key":"ref_18","first-page":"131","article-title":"A study on pentagonal fuzzy number and its corresponding matrices","volume":"1","author":"Panda","year":"2015","journal-title":"Pac. Sci. Rev. B Humanit. Soc. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Chakraborty, A., Mondal, S.P., Alam, S., Ahmadian, A., Senu, N., De, D., and Salahshour, S. (2019). The pentagonal fuzzy number: Its different representations, properties, ranking, defuzzification and application in game problems. Symmetry, 11.","DOI":"10.3390\/sym11020248"},{"key":"ref_20","first-page":"1003","article-title":"Pentagonal fuzzy number","volume":"3","author":"Pathinathan","year":"2014","journal-title":"Int. J. Comput. Algorithm"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1","DOI":"10.17485\/ijst\/2016\/v9i17\/76802","article-title":"Sieving out the poor using fuzzy decision-making tools","volume":"8","author":"Kumar","year":"2015","journal-title":"Indian J. Sci. Technol."},{"key":"ref_22","first-page":"1047","article-title":"Execution proportions of multi-server queuing model with pentagonal fuzzy number: DSW algorithm approach","volume":"8","author":"Prameela","year":"2019","journal-title":"Int. J. Innov. Technol. Explor. Eng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/0898-1221(90)90191-L","article-title":"Job sequencing with fuzzy processing times","volume":"19","author":"McCahon","year":"1990","journal-title":"Comput. Math. Appl."},{"key":"ref_24","first-page":"62","article-title":"Flow shop scheduling problem to minimize the rental cost under fuzzy environment","volume":"2","author":"Sathish","year":"2012","journal-title":"J. Nat. Sci. Res."},{"key":"ref_25","first-page":"272","article-title":"On single machine scheduling problem with distinct due dates under fuzzy environment","volume":"7","author":"Khalifa","year":"2020","journal-title":"Int. J. Supply Oper. Manag."},{"key":"ref_26","first-page":"6697060","article-title":"Solving constrained flow-shop scheduling problem through Multistage Fuzzy Binding Approach with Fuzzy Due Dates","volume":"2021","author":"Khalifa","year":"2021","journal-title":"Adv. Fuzzy Syst."},{"key":"ref_27","first-page":"6695174","article-title":"On a flow-shop scheduling problem with fuzzy pentagonal processing time","volume":"2021","author":"Alharbi","year":"2021","journal-title":"Hindawi J. Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1016\/j.jmsy.2021.03.020","article-title":"Maintenance costs and makespan minimization for assembly permutation flow shop scheduling by considering preventive and corrective maintenance","volume":"59","author":"Zhang","year":"2021","journal-title":"J. Manuf. Syst."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1016\/j.susoc.2021.09.002","article-title":"A parameter tuned hybrid algorithm for solving flow shop scheduling problems with parallel assembly stages","volume":"3","author":"Jabbari","year":"2022","journal-title":"Sustain. Oper. Comput."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Li, R., Gong, W., Lu, C., and Wang, A. (2022). Learning-based Memetic Algorithm for Energy-Efficient Flexible Job Shop Scheduling with Type-2 Fuzzy Processing Time. IEEE Trans. Evol. Comput.","DOI":"10.1109\/TEVC.2022.3175832"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"3495228","DOI":"10.1155\/2022\/3990534","article-title":"Minimizing the machine processing time in a flow shop scheduling problem under piecewise quadratic fuzzy numbers","volume":"2022","author":"Zhou","year":"2022","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2787","DOI":"10.1016\/j.aej.2021.01.030","article-title":"Solving flow-shop scheduling problem with a reinforcement-learning algorithm that generalizes the value function with neural network","volume":"60","author":"Ren","year":"2021","journal-title":"Alex. Eng. J."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"17","DOI":"10.32604\/csse.2022.016992","article-title":"Scheduling flexible flow shop in labeling companies to minimize the makespan","volume":"40","author":"Wang","year":"2022","journal-title":"Comput. Syst. Sci. Eng."},{"key":"ref_34","first-page":"3031","article-title":"TFlow-shop scheduling with transportation capacity and time consideration","volume":"70","author":"Wang","year":"2022","journal-title":"Comput. Mater. Contin."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"563","DOI":"10.32604\/csse.2023.022716","article-title":"Hybrid Flow Shop with Setup Times Scheduling Problem","volume":"44","author":"Jemmali","year":"2023","journal-title":"Comput. Syst. Sci. Eng."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"8519","DOI":"10.1109\/TII.2022.3165636","article-title":"Solving Multi-Objective Fuzzy Job-shop Scheduling Problem by a Hybrid Adaptive Differential Evolution Algorithm","volume":"18","author":"Wang","year":"2022","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_37","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_38","doi-asserted-by":"crossref","first-page":"26","DOI":"10.58496\/MJBD\/2021\/005","article-title":"Distributed Reduced Convolution Neural Networks","volume":"2021","author":"Alajanbi","year":"2021","journal-title":"Mesop. J. Big Data"},{"key":"ref_39","first-page":"14","article-title":"Character Recognition by Implementing FPGA-Based Artificial Neural Network","volume":"2021","author":"Ali","year":"2021","journal-title":"Mesop. J. Comput. Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/1\/130\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T17:56:11Z","timestamp":1760118971000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/1\/130"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,2]]},"references-count":39,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["sym15010130"],"URL":"https:\/\/doi.org\/10.3390\/sym15010130","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,1,2]]}}}