{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:41:11Z","timestamp":1760136071129,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,1,16]],"date-time":"2022-01-16T00:00:00Z","timestamp":1642291200000},"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>This paper proposes a modification of the imperialist competitive algorithm to solve multi-objective optimization problems with hybrid methods (MOHMICA) based on a modification of the imperialist competitive algorithm with hybrid methods (HMICA). The rationale for this is that there is an obvious disadvantage of HMICA in that it can only solve single-objective optimization problems but cannot solve multi-objective optimization problems. In order to adapt to the characteristics of multi-objective optimization problems, this paper improves the establishment of the initial empires and colony allocation mechanism and empire competition in HMICA, and introduces an external archiving strategy. A total of 12 benchmark functions are calculated, including 10 bi-objective and 2 tri-objective benchmarks. Four metrics are used to verify the quality of MOHMICA. Then, a new comprehensive evaluation method is proposed, called \u201cradar map method\u201d, which could comprehensively evaluate the convergence and distribution performance of multi-objective optimization algorithm. It can be seen from the four coordinate axes of the radar maps that this is a symmetrical evaluation method. For this evaluation method, the larger the radar map area is, the better the calculation result of the algorithm. Using this new evaluation method, the algorithm proposed in this paper is compared with seven other high-quality algorithms. The radar map area of MOHMICA is at least 14.06% larger than that of other algorithms. Therefore, it is proven that MOHMICA has advantages as a whole.<\/jats:p>","DOI":"10.3390\/sym14010173","type":"journal-article","created":{"date-parts":[[2022,1,16]],"date-time":"2022-01-16T20:46:06Z","timestamp":1642365966000},"page":"173","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Modification of the Imperialist Competitive Algorithm with Hybrid Methods for Multi-Objective Optimization Problems"],"prefix":"10.3390","volume":"14","author":[{"given":"Jianfu","family":"Luo","sequence":"first","affiliation":[{"name":"School of Economics and Management, China University of Geosciences (Beijing), Beijing 100083, China"}]},{"given":"Jinsheng","family":"Zhou","sequence":"additional","affiliation":[{"name":"School of Economics and Management, China University of Geosciences (Beijing), Beijing 100083, China"}]},{"given":"Xi","family":"Jiang","sequence":"additional","affiliation":[{"name":"School of Economics and Management, China University of Geosciences (Beijing), Beijing 100083, China"}]},{"given":"Haodong","family":"Lv","sequence":"additional","affiliation":[{"name":"School of Economics and Management, China University of Geosciences (Beijing), Beijing 100083, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,16]]},"reference":[{"key":"ref_1","unstructured":"Schaffer, J.D. (1985, January 1). Multiple objective optimization with vector evaluated genetic algorithms. Proceedings of the first international conference on genetic algorithms. In Proceedings of the 1st International Conference on Genetic Algorithms, Pittsburgh, PA, USA."},{"key":"ref_2","unstructured":"Fonseca, C., and Fleming, P. (1993, January 1). Genetic algorithms for multiobjective optimization: Formulation discussion and generalization. Proceedings of the Fifth International Conference on Genetic Algorithms, Urbana, IL, USA."},{"key":"ref_3","unstructured":"Corne, D.W., Jerram, N.R., and Knowles, J. (2001, January 7\u201311). PESA-II: Region-based selection in evolutionary multiobjective optimization. Proceedings of the Genetic and Evolutionary Computation Conference (GECCO), San Francisco, CA, USA."},{"key":"ref_4","first-page":"221","article-title":"Multiobjective optimization using non-dominated sorting in genetic algorithms","volume":"2","author":"Srinivas","year":"1994","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_5","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_6","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_7","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_8","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.swevo.2011.08.001","article-title":"A multi-objective artificial bee colony algorithm","volume":"2","author":"Akbari","year":"2012","journal-title":"Swarm Evol. Comput."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"805","DOI":"10.1007\/s10489-017-1019-8","article-title":"Grasshopper optimization algorithm for multi-objective optimization problems","volume":"48","author":"Mirjalili","year":"2018","journal-title":"Appl. Intell."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1007\/s10489-016-0825-8","article-title":"Multi-objective ant lion optimizer: A multi-objective optimization algorithm for solving engineering problems","volume":"46","author":"Mirjalili","year":"2017","journal-title":"Appl. Intell."},{"key":"ref_11","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":"2016","journal-title":"Expert Syst. Appl."},{"key":"ref_12","first-page":"722","article-title":"Multiobjective optimization Test Instances for the CEC 2009 Special Session and Competition","volume":"57","author":"Zhang","year":"2008","journal-title":"Mech. Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"113134","DOI":"10.1016\/j.eswa.2019.113134","article-title":"A multiple search strategies based grey wolf optimizer for solving multi-objective optimization problems","volume":"145","author":"Liu","year":"2020","journal-title":"Expert Syst. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3037","DOI":"10.1007\/s00500-019-04080-6","article-title":"Multi-Objective Stochastic Fractal Search: A powerful algorithm for solving complex multi-objective optimization problems","volume":"24","author":"Khalilpourazari","year":"2020","journal-title":"Soft. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"112972","DOI":"10.1016\/j.eswa.2019.112972","article-title":"A guided population archive whale optimization algorithm for solving multiobjective optimization problems","volume":"141","author":"Got","year":"2020","journal-title":"Expert Syst. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"107250","DOI":"10.1016\/j.cie.2021.107250","article-title":"Aquila Optimizer: A novel meta-heuristic optimization Algorithm","volume":"157","author":"Abualigah","year":"2021","journal-title":"Comput. Ind. Eng."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"116158","DOI":"10.1016\/j.eswa.2021.116158","article-title":"Reptile Search Algorithm (RSA): A nature-inspired meta-heuristic optimizer","volume":"191","author":"Abualigah","year":"2022","journal-title":"Expert Syst. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"113609","DOI":"10.1016\/j.cma.2020.113609","article-title":"The arithmetic optimization algorithm","volume":"376","author":"Abualigah","year":"2021","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Atashpaz-Gargari, E., and Lucas, C. (2007, January 25\u201328). Imperialist competitive algorithm: An algorithm for optimization inspired by imperialistic competition. Proceedings of the 2007 IEEE Congress on Evolutionary Computation, Singapore.","DOI":"10.1109\/CEC.2007.4425083"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"7549","DOI":"10.1007\/s00521-018-3587-x","article-title":"CB-ICA: A crossover-based imperialist competitive algorithm for large-scale problems and engineering design optimization","volume":"31","author":"Aliniya","year":"2019","journal-title":"Neural Comput. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"444","DOI":"10.1016\/j.engappai.2019.07.001","article-title":"Adaptive Range Genetic Algorithm: A hybrid optimization approach and its application in the design and economic optimization of Shell-and-Tube Heat Exchanger","volume":"85","author":"Iyer","year":"2019","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"5017","DOI":"10.1007\/s00521-018-03995-9","article-title":"Design of neural network predictive controller based on imperialist competitive algorithm for automatic voltage regulator","volume":"31","author":"Elsisi","year":"2019","journal-title":"Neural Comput. Appl."},{"key":"ref_23","first-page":"8765","article-title":"Impact of ultra-capacitor on automatic generation control of electric energy systems using an optimal FFOID controller","volume":"43","author":"Arya","year":"2019","journal-title":"Int. J. Energy Res."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1145","DOI":"10.1080\/17415977.2018.1503259","article-title":"An iterated IRS technique for cross-sectional damage modelling and identification in beams using limited sensors measurement","volume":"27","author":"Hosseinzadeh","year":"2019","journal-title":"Inverse Probl. Sci. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"5343","DOI":"10.1007\/s00521-018-3369-5","article-title":"Sustainable closed-loop supply chain network design with discount supposition","volume":"31","author":"Fard","year":"2019","journal-title":"Neural Comput. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1108\/AA-10-2017-127","article-title":"An integrated production-distribution planning with a routing problem and transportation cost discount in a supply chain","volume":"39","author":"Karimi","year":"2019","journal-title":"Assem. Autom."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"963","DOI":"10.1051\/ro\/2019018","article-title":"A Fuzzy Multi-Objective Programming Approach to Develop a Green Closed-Loop Supply Chain Network Design Problem under Uncertainty: Modifications of Imperialist Competitive Algorithm","volume":"53","author":"Fakhrzad","year":"2019","journal-title":"RAIRO Res. Oper."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1999","DOI":"10.1080\/0305215X.2020.1843164","article-title":"A dynamic dispatching problem to allocate relief vehicles after a disaster","volume":"53","author":"Gharib","year":"2020","journal-title":"Eng Optimiz."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"722","DOI":"10.1080\/00207543.2018.1481301","article-title":"Integrated multi-factory production and distribution scheduling applying vehicle routing approach","volume":"57","author":"Marandi","year":"2019","journal-title":"Int. J. Prod. Res."},{"key":"ref_30","first-page":"9320","article-title":"A hybrid discrete imperialist competition algorithm for fuzzy job-shop scheduling problems","volume":"7","author":"Wang","year":"2017","journal-title":"IEEE Access"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"78414","DOI":"10.1109\/ACCESS.2018.2885030","article-title":"Balancing Problem of Stochastic Large-Scale U-Type Assembly Lines Using a Modified Evolutionary Algorithm","volume":"6","author":"Zhang","year":"2018","journal-title":"IEEE Access"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1097","DOI":"10.1109\/TCYB.2018.2796119","article-title":"A two-phase meta-heuristic for multi-objective flexible job shop scheduling problem with total energy consumption threshold","volume":"49","author":"Lei","year":"2018","journal-title":"IEEE Trans. Cybern."},{"key":"ref_33","first-page":"8829","article-title":"MOICA: A novel multi-objective approach based on imperialist competitive algorithm","volume":"219","author":"Enayatifar","year":"2013","journal-title":"Appl. Math. Comput."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1016\/j.ins.2014.09.051","article-title":"Multi-objective optimal electric power planning in the power system using Gaussian bare-bones imperialist competitive algorithm","volume":"294","author":"Ghasemi","year":"2015","journal-title":"Inform. Sci."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1080\/03772063.2017.1391130","article-title":"Multi-Objective Modified Imperialist Competitive Algorithm for Brushless DC Motor Optimization","volume":"65","author":"Mohammad","year":"2019","journal-title":"IETE J. Res."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"805","DOI":"10.2991\/ijcis.11.1.62","article-title":"Reduction of carbon emission and total late work criterion in job shop scheduling by applying a multi-objective imperialist competitive algorithm","volume":"11","author":"Piroozfard","year":"2018","journal-title":"Int. J. Comput. Int. Syst."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"116342","DOI":"10.1016\/j.apenergy.2020.116342","article-title":"Multi-objective optimization of energy use and environmental emissions for walnut production using imperialist competitive algorithm","volume":"284","author":"Khanali","year":"2021","journal-title":"Appl. Energy"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"104307","DOI":"10.1016\/j.engappai.2021.104307","article-title":"An imperialist competitive algorithm with feedback for energy-efficient flexible job shop scheduling with transportation and sequence-dependent setup times","volume":"103","author":"Li","year":"2021","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Kaveh, A., Rahmani, P., and Eslamlou, D. (2021). An efficient hybrid approach based on Harris Hawks optimization and imperialist competitive algorithm for structural optimization. Eng. Comput., 1\u201329.","DOI":"10.1007\/s00366-020-01258-7"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"4545","DOI":"10.3233\/JIFS-201391","article-title":"A novel imperialist competitive algorithm for fuzzy distributed assembly flow shop scheduling","volume":"40","author":"Li","year":"2021","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1007\/s40747-020-00193-w","article-title":"Discrete imperialist competitive algorithm for the resource-constrained hybrid flowshop problem with energy consumption","volume":"7","author":"Tao","year":"2021","journal-title":"Complex Intell. Syst."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"161745","DOI":"10.1109\/ACCESS.2021.3133579","article-title":"A modification of the imperialist competitive algorithm with hybrid methods for constrained optimization problems","volume":"9","author":"Luo","year":"2021","journal-title":"IEEE Access"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/1\/173\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:02:13Z","timestamp":1760133733000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/1\/173"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,16]]},"references-count":42,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,1]]}},"alternative-id":["sym14010173"],"URL":"https:\/\/doi.org\/10.3390\/sym14010173","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,1,16]]}}}