{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,22]],"date-time":"2025-12-22T04:27:46Z","timestamp":1766377666999,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"24","license":[{"start":{"date-parts":[[2021,12,7]],"date-time":"2021-12-07T00:00:00Z","timestamp":1638835200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e Tecnologia","doi-asserted-by":"publisher","award":["."],"award-info":[{"award-number":["."]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>This work presents a novel approach for multiobjective optimization problems, extending the concept of a Pareto front to a new idea of the Pareto region. This new concept provides all the points beyond the Pareto front, leading to the same optimal condition with statistical assurance. This region is built using a Fisher\u2013Snedecor test over an augmented Lagragian function, for which deductions are proposed here. This test is meant to provide an approximated depiction of the feasible operation region while using meta-heuristic optimization results to extract this information. To do so, a Constrained Sliding Particle Swarm Optimizer (CSPSO) was applied to solve a series of four benchmarks and a case study. The proposed test analyzed the CSPSO results, and the novel Pareto regions were estimated. Over this Pareto region, a clustering strategy was also developed and applied to define sub-regions that prioritize one of the objectives and an intermediary region that provides a balance between objectives. This is a valuable tool in the context of process optimization, aiming at assertive decision-making purposes. As this is a novel concept, the only way to compare it was to draw the entire regions of the benchmark functions and compare them with the methodology result. The benchmark results demonstrated that the proposed method could efficiently portray the Pareto regions. Then, the optimization of a Pressure Swing Adsorption unit was performed using the proposed approach to provide a practical application of the methodology developed here. It was possible to build the Pareto region and its respective sub-regions, where each process performance parameter is prioritized. The results demonstrated that this methodology could be helpful in processes optimization and operation. It provides more flexibility and more profound knowledge of the system under evaluation.<\/jats:p>","DOI":"10.3390\/math9243152","type":"journal-article","created":{"date-parts":[[2021,12,7]],"date-time":"2021-12-07T11:00:23Z","timestamp":1638874823000},"page":"3152","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["From a Pareto Front to Pareto Regions: A Novel Standpoint for Multiobjective Optimization"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0796-8116","authenticated-orcid":false,"given":"Carine M.","family":"Rebello","sequence":"first","affiliation":[{"name":"Departamento de Engenharia Qu\u00edmica, Escola Polit\u00e9cnica (Polytechnic Institute), Universidade Federal da Bahia, Salvador 40210-630, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2390-1525","authenticated-orcid":false,"given":"M\u00e1rcio A. F.","family":"Martins","sequence":"additional","affiliation":[{"name":"Departamento de Engenharia Qu\u00edmica, Escola Polit\u00e9cnica (Polytechnic Institute), Universidade Federal da Bahia, Salvador 40210-630, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2798-0283","authenticated-orcid":false,"given":"Daniel D.","family":"Santana","sequence":"additional","affiliation":[{"name":"Departamento de Engenharia Qu\u00edmica, Escola Polit\u00e9cnica (Polytechnic Institute), Universidade Federal da Bahia, Salvador 40210-630, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0715-4761","authenticated-orcid":false,"given":"Al\u00edrio E.","family":"Rodrigues","sequence":"additional","affiliation":[{"name":"Laboratory of Separation and Reaction Engineering, Associate Laboratory LSRE\/LCM, Department of Chemical Engineering, Faculty of Engineering, University of Porto, 4200-465 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6051-6039","authenticated-orcid":false,"given":"Jos\u00e9 M.","family":"Loureiro","sequence":"additional","affiliation":[{"name":"Laboratory of Separation and Reaction Engineering, Associate Laboratory LSRE\/LCM, Department of Chemical Engineering, Faculty of Engineering, University of Porto, 4200-465 Porto, Portugal"}]},{"given":"Ana M.","family":"Ribeiro","sequence":"additional","affiliation":[{"name":"Laboratory of Separation and Reaction Engineering, Associate Laboratory LSRE\/LCM, Department of Chemical Engineering, Faculty of Engineering, University of Porto, 4200-465 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0963-6449","authenticated-orcid":false,"given":"Idelfonso B. R.","family":"Nogueira","sequence":"additional","affiliation":[{"name":"Laboratory of Separation and Reaction Engineering, Associate Laboratory LSRE\/LCM, Department of Chemical Engineering, Faculty of Engineering, University of Porto, 4200-465 Porto, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,7]]},"reference":[{"key":"ref_1","first-page":"129","article-title":"Multi-objective Optimization with Combination of Particle Swarm and Extremal Optimization for Constrained Engineering Design","volume":"7","author":"Yu","year":"2012","journal-title":"WSEAS Trans. Syst. Control"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"9977","DOI":"10.1021\/acs.iecr.8b00207","article-title":"MO-MCS, a derivative-free algorithm for the multiobjective optimization of adsorption processes","volume":"57","author":"Capra","year":"2018","journal-title":"Ind. Eng. Chem. Res."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Rangaiah, G.P., Feng, Z., and Hoadley, A.F. (2020). Multi-objective optimization applications in chemical process engineering: Tutorial and review. Processes, 8.","DOI":"10.3390\/pr8050508"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"120078","DOI":"10.1016\/j.energy.2021.120078","article-title":"Surrogate-assisted multi-objective particle swarm optimization for the operation of CO2 capture using VPSA","volume":"224","author":"Alkebsi","year":"2021","journal-title":"Energy"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"368","DOI":"10.1016\/j.cie.2019.06.020","article-title":"Optimization of a True Moving Bed unit and determination of its feasible operating region using a novel Sliding Particle Swarm Optimization","volume":"135","author":"Nogueira","year":"2019","journal-title":"Comput. Ind. Eng."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Rebello, C.M., Martins, M.A.F., Loureiro, J.M., Rodrigues, A.E., Ribeiro, A.M., and Nogueira, I.B.R. (2021). From an Optimal Point to an Optimal Region: A Novel Methodology for Optimization of Multimodal Constrained Problems and a Novel Constrained Sliding Particle Swarm Optimization Strategy. Mathematics, 9.","DOI":"10.3390\/math9151808"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"107821","DOI":"10.1016\/j.cep.2020.107821","article-title":"Dynamics of a True Moving Bed Reactor: Synthesis of n-Propyl Propionate and an alternative optimization method","volume":"148","author":"Nogueira","year":"2020","journal-title":"Chem. Eng. Process.-Process Intensif."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Mnasri, S., Nasri, N., Van Den Bossche, A., and Val, T. (2018, January 25\u201329). 3D indoor redeployment in IoT collection networks: A real prototyping using a hybrid PI-NSGA-III-VF. Proceedings of the 2018 14th International Wireless Communications & Mobile Computing Conference (IWCMC), Limassol, Cyprus.","DOI":"10.1109\/IWCMC.2018.8450372"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1007\/s10732-020-09445-x","article-title":"IoT networks 3D deployment using hybrid many-objective optimization algorithms","volume":"26","author":"Mnasri","year":"2020","journal-title":"J. Heuristics"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"105069","DOI":"10.1016\/j.envsoft.2021.105069","article-title":"Quantifying uncertainty in Pareto fronts arising from spatial data","volume":"141","author":"Hildemann","year":"2021","journal-title":"Environ. Model. Softw."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"386","DOI":"10.1016\/j.ejor.2014.07.032","article-title":"Quantifying uncertainty on Pareto fronts with Gaussian process conditional simulations","volume":"243","author":"Binois","year":"2015","journal-title":"Eur. J. Oper. Res."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Bassi, M., de Cursi, E.S., Pagnacco, E., and Ellaia, R. (2018). Statistics of the Pareto front in multi-objective optimization under uncertainties. Lat. Am. J. Solids Struct., 15.","DOI":"10.1590\/1679-78255018"},{"key":"ref_13","first-page":"215","article-title":"Comparison of Three Evolutionary Algorithms: GA, PSO and DE","volume":"11","author":"Kachitvichyanukul","year":"2012","journal-title":"Ind. Eng. Manag. Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1542","DOI":"10.1016\/j.ces.2007.11.024","article-title":"Nonlinear parameter estimation through particle swarm optimization","volume":"63","author":"Schwaab","year":"2008","journal-title":"Chem. Eng. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1016\/j.ces.2012.12.013","article-title":"Emulsion copolymerization of styrene and butyl acrylate in the presence of a chain transfer agent. Part 2: Parameters estimability and confidence regions","volume":"90","author":"Benyahia","year":"2013","journal-title":"Chem. Eng. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"14037","DOI":"10.1021\/acs.iecr.0c01155","article-title":"Big Data-Based Optimization of a Pressure Swing Adsorption Unit for Syngas Purification: On Mapping Uncertainties from a Metaheuristic Technique","volume":"59","author":"Nogueira","year":"2020","journal-title":"Ind. Eng. Chem. Res."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"912","DOI":"10.1016\/j.camwa.2011.11.057","article-title":"Comparison of multi-objective optimization methodologies for engineering applications","volume":"63","author":"Chiandussi","year":"2012","journal-title":"Comput. Math. Appl."},{"key":"ref_18","first-page":"148","article-title":"Multiobjective optimization using nondominated sorting in genetic algorithms","volume":"148","author":"Srinivas","year":"1994","journal-title":"Evol. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"807","DOI":"10.1016\/j.applthermaleng.2009.12.010","article-title":"A grid based multi-objective evolutionary algorithm for the optimization of power plants","volume":"30","author":"Dipama","year":"2010","journal-title":"Appl. Therm. Eng."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.swevo.2011.11.005","article-title":"Local search based hybrid particle swarm optimization algorithm for multiobjective optimization Local search based hybrid particle swarm optimization algorithm for multiobjective optimization","volume":"3","author":"Mousa","year":"2018","journal-title":"Swarm Evol. Comput."},{"key":"ref_21","first-page":"1556","article-title":"GA-based decision support system for multicriteria optimization","volume":"2","author":"Tanaka","year":"1995","journal-title":"Ind. Manag. Eng. Fields"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1162\/106365600568202","article-title":"Comparison of Multiobjective Evolutionary Algorithms: Empirical Results","volume":"8","author":"Zitzler","year":"2000","journal-title":"Evol. Comput."},{"key":"ref_23","unstructured":"Coello, C.A., Lamont, G.B., and Van Veldhuizen, D.A. (2007). Evolutionary Algorithms for Solving Multi-Objective Problems, Springer."},{"key":"ref_24","first-page":"12","article-title":"The application of multi-objective charged system search algorithm for optimization problems","volume":"26","author":"Ranjbar","year":"2019","journal-title":"Sci. Iran."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"392","DOI":"10.1109\/TEVC.2005.859464","article-title":"Max-min surrogate-assisted evolutionary algorithm for robust design","volume":"10","author":"Ong","year":"2006","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_26","unstructured":"Hansen, M.P., and Jaszkiewicz, A. (1998). Evaluating the Quality of Approximations to the Non-dominated Set, IMM, Department of Mathematical Modelling, Technical University of Denmark."},{"key":"ref_27","first-page":"13","article-title":"Running Performance Metrics for Evolutionary Multi-Objective Optimization","volume":"2002004","author":"Deb","year":"2002","journal-title":"Kangal Rep."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1162\/106365600568158","article-title":"Multiobjective evolutionary algorithms: Analyzing the state-of-the-art","volume":"8","author":"Lamont","year":"2000","journal-title":"Evol. Comput."},{"key":"ref_29","first-page":"110","article-title":"Evolutionary multi-criterion optimization: 8th international conference, EMO 2015 Guimar\u00e3es, Portugal, 29 March\u20141 April 2015 proceedings, Part II","volume":"9019","author":"Antunes","year":"2015","journal-title":"Lect. Notes Comput. Sci."},{"key":"ref_30","first-page":"688","article-title":"A study of the parallelization of a coevolutionary multi-objective evolutionary algorithm","volume":"2972","author":"Coello","year":"2004","journal-title":"Lect. Notes Artif. Intell."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"292","DOI":"10.1007\/BFb0056872","article-title":"Multiobjective optimization using evolutionary algorithms\u2014A comparative case study","volume":"1498","author":"Zitzler","year":"1998","journal-title":"Lect. Notes Comput. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"4654","DOI":"10.1021\/acs.energyfuels.5b00975","article-title":"Syngas Purification by Porous Amino\u2014Functionalized Titanium Terephthalate MIL-125","volume":"29","author":"Regufe","year":"2015","journal-title":"Energy Fuels"}],"container-title":["Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2227-7390\/9\/24\/3152\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:42:35Z","timestamp":1760168555000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2227-7390\/9\/24\/3152"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,7]]},"references-count":32,"journal-issue":{"issue":"24","published-online":{"date-parts":[[2021,12]]}},"alternative-id":["math9243152"],"URL":"https:\/\/doi.org\/10.3390\/math9243152","relation":{},"ISSN":["2227-7390"],"issn-type":[{"type":"electronic","value":"2227-7390"}],"subject":[],"published":{"date-parts":[[2021,12,7]]}}}