{"id":5387,"date":"2026-05-03T12:38:07","date_gmt":"2026-05-03T04:38:07","guid":{"rendered":"\/jase\/?post_type=tkuisotope&#038;p=5387"},"modified":"2026-07-01T12:30:10","modified_gmt":"2026-07-01T04:30:10","slug":"new-simple-trigonometric-algorithms-for-solving-optimization-problems","status":"publish","type":"tkuisotope","link":"\/jase\/?tkuisotope=new-simple-trigonometric-algorithms-for-solving-optimization-problems","title":{"rendered":"New Simple Trigonometric Algorithms for Solving Optimization Problems"},"content":{"rendered":"\n<div class=\"wp-block-tkuwpbs5-bs5-row row article-info\">\n<div class=\"wp-block-tkuwpbs5-bs5-column col-md-3 align-self-start\">\n<p><i class=\"fa fa-folder\" aria-hidden=\"true\"><\/i>&nbsp;<a href=\"\/jase\/?page_id=4854\" data-type=\"page\" data-id=\"807\">2022<\/a><\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-tkuwpbs5-bs5-column col-md-3 align-self-start\">\n<p><i class=\"fa fa-folder-open\" aria-hidden=\"true\"><\/i>&nbsp;<a href=\"\/jase\/?page_id=5365\" data-type=\"page\" data-id=\"4630\">Volume 25, Issue 6<\/a><\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-tkuwpbs5-bs5-column col-md-6 align-self-start\">\n<div class=\"wp-block-tkuwpbs5-bs5-div dv_publish\" data-aos=\"normal\"><div class=\"wp-block-post-date\"><time datetime=\"2026-05-03T12:38:07+08:00\">2026-05-03<\/time><\/div><\/div>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-tkuwpbs5-bs5-row row\">\n<div class=\"wp-block-tkuwpbs5-bs5-column col-md-5 align-self-start\">\n<div class=\"wp-block-tkuwpbs5-bs5-div au-ol\" data-aos=\"normal\">\n<p>Baskar A<sup>1<\/sup><a href=\"mailto:a.baaskar@gmail.com\"><i class=\"fa fa-envelope\"><\/i><\/a><\/p>\n\n\n\n<p style=\"font-size:14px\"><sup>1<\/sup>Panimalar Institute of Technology, Poonamallee, Tamil Nadu 600123, India<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-tkuwpbs5-bs5-div\" style=\"margin-top:var(--wp--preset--spacing--40)\" data-aos=\"normal\">\n<p>Received:\u00a0December 11, 2021<br>Accepted:\u00a0February 02, 2022<br>Publication Date:\u00a0May 03, 2026<\/p>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-tkuwpbs5-bs5-column col-md-7 align-self-start clk=\u5716\u7247\"><img decoding=\"async\" src=\"\/jase\/wp-content\/uploads\/2026\/05\/25_6_20.jpg\" class=\"img-fluid img-fluid mx-auto d-block\" alt=\"\u4e0a\u50b3\u5716\u7247\">\n\n\n<p class=\"has-text-align-center img_caption\">Values of Sine (2<sup>\u2217<\/sup>pi<sup>\u2217<\/sup>rand)) and Cosine (2<sup>\u2217<\/sup>pi<sup>\u2217<\/sup>rand))<\/p>\n<\/div>\n<\/div>\n\n\n\n<p class=\"has-small-font-size\"><i class=\"fab fa-creative-commons\"><\/i>&nbsp;<strong>Copyright&nbsp;<\/strong>The Author(s). This is an open access article distributed under the terms of the&nbsp;<a rel=\"noreferrer noopener\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\" target=\"_blank\">Creative Commons Attribution&nbsp;License (CC BY 4.0)<\/a>, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are cited.<\/p>\n\n\n\n<p>Download Citation:&nbsp; <a href=\"\/jase\/wp-content\/uploads\/2026\/05\/V256.0020.bib\" data-type=\"attachment\" data-id=\"5407\" target=\"_blank\" rel=\"noreferrer noopener\">BibTeX <\/a>| <a href=\"http:\/\/dx.doi.org\/10.6180\/jase.202212_25(6).0020\" target=\"_blank\" rel=\"noreferrer noopener\">http:\/\/dx.doi.org\/10.6180\/jase.202212_25(6).0020<\/a>&nbsp;&nbsp;<\/p>\n\n\n\n<p class=\"btn btn-primary article-btn\"><a href=\"\/jase\/wp-content\/uploads\/2026\/05\/20_2021_0792_V25i6.pdf\" data-type=\"attachment\" data-id=\"5446\" target=\"_blank\" rel=\"noreferrer noopener\">Download PDF<\/a><\/p>\n\n\n\n<div style=\"height:24px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<p>Heuristics play a key role in solving optimization problems with complex functions. They are popular as efficient heuristics are capable of providing results quickly with acceptable solution quality. Population-based heuristics are stochastic and hence, several iterations and trials are needed to achieve the expected accuracy and convergence to the global optimum. This article proposes two new, simple; population-based trigonometric algorithms, Sine (AB) and Cosine (AB). The algorithms are validated using forty well-known benchmark test functions available in the literature. The results are compared with a similar popular Sine Cosine Algorithm and the computational results show that the performance of Sine (AB) and Cosine (AB) are better than Sine Cosine Algorithm. Wilcoxon Signed-Rank and Friedman tests are carried out for statistical analyses. In addition to unconstrained functions, three real-world, constrained problems are solved to have a more intensive analysis of the proposed algorithms.<\/p>\n\n\n\n<p><em>Keywords:\u00a0Optimization, Population-based heuristic, Un-constrained optimization, Sine Cosine Algorithm, Trigonometric Algorithm<\/em><\/p>\n\n\n\n<div style=\"height:2rem\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<div class=\"wp-block-tkuwpbs5-bs5-div ref_ol\" data-aos=\"normal\">\n<ol>\n<li>[1] Website. <a class=\"ng-star-inserted\" href=\"https:\/\/nptel.ac.in\/content\/storage2\/\" target=\"_blank\" rel=\"noopener\" data-hveid=\"0\" data-ved=\"0CAAQ_4QMahgKEwj9jsniwrCVAxUAAAAAHQAAAAAQ2wE\">https:\/\/nptel.ac.in\/content\/storage2\/<\/a> courses\/105108127\/pdf\/Module_1.<\/li>\n<li>[2] E. Burke, M. Gendreau, M. Hyde, G. Kendall, G. Ochoa, E. \u00d6zcan<span class=\"citation-3849 citation-end-3849\">, and R. Qu, (2013) \u201cHyper-heuristics: A survey of the state of the art&#8221; Journal of the Operational Research Society 64(12): 1695\u20131724. DOI: 10.1057\/jors.2013.71.<\/span><\/li>\n<li><span class=\"citation-3848\">[3] V. Kumar, J. Chhabra, and D. Kumar, (2014) \u201cParameter adapti<\/span><span class=\"citation-3847 citation-3848 citation-end-3848\">ve harmony search algor<\/span><span class=\"citation-3847 citation-end-3847\">ithm for unimodal and multimodal optimization problems&#8221; Journal of Computational Science 5(2): 144\u2013155. DOI: 10.1016\/j.jocs. 2013.12.001.<\/span><\/li>\n<li><span class=\"citation-3846\">[4] L. Abualigah<\/span><span class=\"citation-3845 citation-3846 citation-end-3846\"> and A. Diabat, (2021) \u201cAdvances in Sine Cosine Algorithm: A comprehe<\/span><span class=\"citation-3845 citation-end-3845\">nsive survey&#8221; Artificial Intelligence Review 54(4): 2567\u20132608. DOI: 10.1007\/ s10462-020-09909-3.<\/span><\/li>\n<li><span class=\"citation-3844\">[5] S. Droste, T. Jansen, and I. Wegener, (2006<\/span><span class=\"citation-3843 citation-3844 citation-end-3844\">) \u201cUpper and lower bounds for randomized <\/span><span class=\"citation-3843 citation-end-3843\">search heuristics in black-box optimization&#8221; Theory of Computing Systems 39(4): 525\u2013544. DOI: 10.1007\/s00224-004-1177z.<\/span><\/li>\n<li><span class=\"citation-3842\">[6] P. K. Pal<\/span><span class=\"citation-3841 citation-3842 citation-end-3842\">, K. Deep, and A. K. Nagar. \u201cPerformance of Sine\u2013Cosine Algorithm on Large<\/span><span class=\"citation-3841 citation-end-3841\">-Scale Optimization Problems\u201d. In: Decision science in action. Springer, 2019, 139\u2013154.<\/span><\/li>\n<li><span class=\"citation-3840 citation-end-3840\">[7] R. Venkata Rao, (2016) \u201cJaya: A simple and new optimization algorithm for solvi<\/span>ng constrained and unconstrained optimization problems&#8221; International Journal of Industrial Engineering Computations 7(1): <span class=\"citation-3838 citation-3839 citation-end-3839\">19\u201334. DOI: 10.5267\/j.ijiec.2015.8.004.<\/span><\/li>\n<li><span class=\"citation-3836 citation-3837\">[8] R. Zitar, M. Al-Betar, M. Awadallah, I. Doush, and K. Assaleh, (2021) \u201cAn Intensive and C<\/span><span class=\"citation-3835 citation-3836 citation-3837 citation-end-3837\">omprehensive Overview of JAYA Algorithm, its Versions and Applications&#8221; Archives of Computationa<\/span><span class=\"citation-3835 citation-3836 citation-end-3836\">l Methods in Enginee<\/span><span class=\"citation-3835 citation-end-3835\">ring: DOI: 10.1007\/s11831-021-09585-8.<\/span><\/li>\n<li><span class=\"citation-3834\">[9] <\/span><span class=\"citation-3833 citation-3834 citation-end-3834\">S. Mirjalili, (2016) \u201cSCA: A Sine Cosine Algorithm for solving optimization problems<\/span><span class=\"citation-3833\">&#8221; Knowledge-Based Systems 96: 120\u20131<\/span><span class=\"citation-3831 citation-3832 citation-3833 citation-end-3833\">33. DOI: 10.1016\/j.knosys.2015.12.022.<\/span><\/li>\n<li><span class=\"citation-3828 citation-3829 citation-3830 citation-end-3830\">[10] R. Kommadath, J. Dondeti, and P. Kotecha. \u201cBenchmarking JAYA and sine cosine al<\/span><span class=\"citation-3828 citation-3829\">gor<\/span><span class=\"citation-3826 citation-3827 citation-3828 citation-3829 citation-end-3829\">ithm on real parameter bound constrained single objective optimization problems (CEC2016)\u201d. In: Part F127854. ci<\/span><span class=\"citation-3826 citation-3827 citation-3828 citation-end-3828\">ted By<\/span><span class=\"citation-3826 citation-3827 citation-end-3827\"> 6. 2017, 31\u201334. DOI: 10.1145\/3059336.3059363.<\/span><\/li>\n<li><span class=\"citation-3823 citation-3824 citation-3825 citation-end-3825\">[11] S. Mirjalili and A. Lewis, (2016) \u201cThe Whale Optimization Algo<\/span><span class=\"citation-3823 citation-3824 citation-end-3824\">rithm&#8221; Advances <\/span><span class=\"citation-3823 citation-end-3823\">in Engineering Software 95: 51\u201367. DOI: 10.1016\/j.advengsoft.2016.01.008.<\/span><\/li>\n<li><span class=\"citation-3822\">[12] A. Hussien, A. Hassanien, E. Houssein, M. Amin, and A. Azar, (2020) <\/span><span class=\"citation-3821 citation-3822 citation-end-3822\">\u201cNew binary whale<\/span><span class=\"citation-3821 citation-end-3821\"> optimization algorithm for discrete optimization problems&#8221; Engineering Optimization 52(6): 945\u2013959. DOI: 10.1080\/0305215X. 2019.1624740.<\/span><\/li>\n<li><span class=\"citation-3820 citation-end-3820\">[13] E. Atashpaz-Gargari and C. Lucas. \u201cImperialist competitive algorithm: An <\/span>algorithm for optimization inspired by imperialistic competition\u201d. In: cited By 1792. 2007, 4661<span class=\"citation-3818 citation-3819 citation-end-3819\">\u20134667. DOI: 10.1109\/CEC.2007. 4425083.<\/span><\/li>\n<li><span class=\"citation-3816 citation-3817\">[14] R. Rao, V. Savsani, and D. Vakharia, (2011) \u201cTeaching-learning-based opti<\/span><span class=\"citation-3815 citation-3816 citation-3817 citation-end-3817\">mization: A novel method for constrained mechanical design optimization problems&#8221; CAD Computer Aided Design 43<\/span><span class=\"citation-3815 citation-3816 citation-end-3816\">(3):<\/span><span class=\"citation-3815 citation-end-3815\"> 303\u2013315. DOI: 10. 1016\/j.cad.2010.12.015.<\/span><\/li>\n<li><span class=\"citation-3814\">[15] L. Abualigah, A. Diabat, S. Mirjalili, M. Abd Elaziz<\/span><span class=\"citation-3812 citation-3813 citation-3814 citation-end-3814\">, and A. Gandomi, (2021) \u201cThe Arit<\/span><span class=\"citation-3812 citation-3813 citation-end-3813\">hmetic Optimization Algorithm&#8221; Computer Methods in Applied Mechanics and Engineering 376: DOI: 10.1016\/j.cma.2020. 113609.<\/span><\/li>\n<li><span class=\"citation-3810 citation-3811\">[16] F. Hashim, K. Hussain, E. Houssein, M. Mabrou<\/span><span class=\"citation-3809 citation-3810 citation-3811 citation-end-3811\">k, and W. Al-Atabany, (2021) <\/span><span class=\"citation-3809 citation-3810 citation-end-3810\">\u201cArchimed<\/span><span class=\"citation-3809 citation-end-3809\">es optimization algorithm: a new metaheuristic algorithm for solving optimization problems&#8221; Applied Intelligence 51(3): 15311551. DOI: 10.1007\/s10489-020-01893-z.<\/span><\/li>\n<li><span class=\"citation-3808\">[17] J. Agushaka and A. Ezugwu, <\/span><span class=\"citation-3807 citation-3808 citation-end-3808\">(2021) \u201cAdvanced arithmetic optimization algorithm fo<\/span><span class=\"citation-3807 citation-end-3807\">r solving mechanical engineering design problems&#8221; PLoS ONE 16(8 August): DOI: 10.1371\/journal.pone.0255703.<\/span><\/li>\n<li><span class=\"citation-3806\">[18] Z. Zhang, C. Huang, H. Huang, S. Tang, and K. Dong, (201<\/span><span class=\"citation-3805 citation-3806 citation-end-3806\">8) \u201cAn optimization method:<\/span><span class=\"citation-3805 citation-end-3805\"> Hummingbirds optimization algorithm&#8221; Journal of Systems Engineering and Electronics 29(2): 386\u2013404. DOI: 10.21629\/JSEE. 2018.02.19.<\/span><\/li>\n<li><span class=\"citation-3804\">[19] W. Zhao, L. Wang, and S. Mirjalili, (2022) \u201cArtificial hummingbird algorith<\/span><span class=\"citation-3803 citation-3804 citation-end-3804\">m: A <\/span><span class=\"citation-3803 citation-end-3803\">new bio-inspired optimizer with its engineering applications&#8221; Computer Methods in Applied Mechanics and Engineering 388: DOI: 10.1016\/j.cma.2021.114194.<\/span><\/li>\n<li><span class=\"citation-3802 citation-end-3802\">[20] D. Wolpert and W. Macready, (1997) \u201cNo free lunch theorems f<\/span>or optimization&#8221; IEEE Transactions on Evolutionary Computation 1(1): 67\u201382. DOI: 10.1109\/ 4235.585893.<\/li>\n<li>[21] D. Bertsekas. Nonlinear Programming. 3rd ed. Athena Scientific, 2016.<\/li>\n<li>[22] A. Gabis, Y. Meraihi, S. Mirjalili, and A. Ramdane Cherif, (2021) \u201cA comprehensive survey of sine cosine algorithm: variants and applications&#8221; Artificial Intelligence Review <span class=\"citation-3799 citation-3800 citation-3801 citation-end-3801\">54(7): 5469\u20135540. DOI: 10.1007\/s10462021-10026-y.<\/span><\/li>\n<li><span class=\"citation-3796 citation-3797 citation-3798\">[23] Virtual Library of Simulation Experiments: Test Functions and Datasets. Website. <\/span><a class=\"ng-star-inserted\" href=\"https:\/\/www.sfu.ca\/\" target=\"_blank\" rel=\"noopener\" data-hveid=\"0\" data-ved=\"0CAAQ_4QMahgKEwj9jsniwrCVAxUAAAAAHQAAAAAQ3AE\"><span class=\"citation-3796 citation-3797 citation-3798\">https:\/\/www.sfu.ca\/<\/span><\/a><span class=\"citation-3796 citation-3797 citation-3798 citation-end-3798\"> ~ssurjano\/optimization.html. 2013.<\/span><\/li>\n<li><span class=\"citation-3793 citation-3794 citation-3795\">[24] Website. <\/span><a class=\"ng-star-inserted\" href=\"https:\/\/seyedalimirjalili.com\/sca\" target=\"_blank\" rel=\"noopener\" data-hveid=\"0\" data-ved=\"0CAAQ_4QMahgKEwj9jsniwrCVAxUAAAAAHQAAAAAQ3QE\"><span class=\"citation-3793 citation-3794 citation-3795\">https:\/\/seyedalimirjalili.com\/sca<\/span><\/a><span class=\"citation-3793 citation-3794 citation-3795 citation-end-3795\">..<\/span><\/li>\n<li><span class=\"citation-3790 citation-3791 citation-3792 citation-end-3792\">[25]<\/span><span class=\"citation-3790 citation-3791\"> R. Eberhart and J. Kennedy. \u201cNew optimizer using particle swarm theory\u201d. In<\/span><span class=\"citation-3788 citation-3789 citation-3790 citation-3791 citation-end-3791\">: cited By 12748. 1995, 3943.<\/span><\/li>\n<li><span class=\"citation-3784 citation-3785 citation-3786 citation-3787 citation-end-3787\">[26] X.-S. Ya<\/span><span class=\"citation-3784 citation-3785 citation-3786 citation-end-3786\">ng, (2010) <\/span><span class=\"citation-3784 citation-3785 citation-end-3785\">\u201cFirefly algorithm, stochastic test functions and design optimization&#8221; International Journal of Bio-Inspired Computation 2(2): 78\u201384. DOI: 10. 1504\/IJBIC.2010.032124.<\/span><\/li>\n<li><span class=\"citation-3782 citation-3783 citation-end-3783\">[27] G. Farr, (2015) \u201cCost-effectiveness o<\/span><span class=\"citation-3782\">f algorithms&#8221; Discrete Mathematics &amp;<\/span><span class=\"citation-3781 citation-3782 citation-end-3782\"> Theoretical Computer Science 17:<\/span><\/li>\n<li><span class=\"citation-3779 citation-3780 citation-end-3780\">[28] V. Beiranvand, W. Hare,<\/span><span class=\"citation-3779 citation-end-3779\"> and Y. Lucet, (2017) \u201cBest practices for comparing optimization algorithms&#8221; Optimization and Engineering 18(4): 815\u2013848. DOI: 10. 1007\/s11081-017-9366-1.<\/span><\/li>\n<li><span class=\"citation-3778 citation-end-3778\">[29] A. Sadollah, A. Bahreinine<\/span>jad, H. Eskandar, <span class=\"citation-3777 citation-end-3777\">and M. Hamdi, (2013) \u201cMine blast algorithm: A new population based algorithm for solving constrained engineering optimization problems&#8221; Applied Soft Computing Journal 13(5): 2592\u20132612. DOI: 10.1016\/j.asoc.2012.11.026.<\/span><\/li>\n<li><span class=\"citation-3776 citation-end-3776\">[30] B. Kannan and S. K<\/span>ramer, (1994) \u201cAn augmented lagr<span class=\"citation-3774 citation-3775 citation-end-3775\">ange multiplier based method for mixed integer discrete continuous optimization and its applications to mechanical design&#8221; Journal of Mechanical Design, Transactions of the ASME 116(2): 405\u2013411. DOI: 10.1115\/1. 2919393.<\/span><\/li>\n<li><span class=\"citation-3772 citation-3773 citation-end-3773\">[31]<\/span><span class=\"citation-3772 citation-end-3772\"> C. Coello Coel<\/span>lo, (2000) \u201cUse of a self-adaptive penalty approach for engineering optimization problems&#8221; Computers in Industry 41(2): 113\u2013127. DOI: 10.1016\/ S0166-3615(99)00046-9.<\/li>\n<\/ol>\n<\/div>\n\n\n\n<p><\/p>\n","protected":false},"author":3,"template":"wp-custom-template-detail-4-aricles","meta":{"_uag_custom_page_level_css":""},"categories":[936,6,942],"tags":[1072],"acf":[],"uagb_featured_image_src":[],"uagb_author_info":{"display_name":"\u6797\u923a\u6db5","author_link":"\/jase\/?author=3"},"uagb_comment_info":0,"uagb_excerpt":"&nbsp;Copyright&nbsp;The Author(s). This is an open access article distributed under the terms of the&nbsp;Creative Commons Attribution&nbsp;License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are cited. Download Citation:&nbsp; BibTeX | http:\/\/dx.doi.org\/10.6180\/jase.202212_25(6).0020&nbsp;&nbsp; Download PDF Heuristics play a key role in solving optimization problems with complex&hellip;","_links":{"self":[{"href":"\/jase\/index.php?rest_route=\/wp\/v2\/tkuisotope\/5387"}],"collection":[{"href":"\/jase\/index.php?rest_route=\/wp\/v2\/tkuisotope"}],"about":[{"href":"\/jase\/index.php?rest_route=\/wp\/v2\/types\/tkuisotope"}],"author":[{"embeddable":true,"href":"\/jase\/index.php?rest_route=\/wp\/v2\/users\/3"}],"wp:attachment":[{"href":"\/jase\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5387"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"\/jase\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5387"},{"taxonomy":"post_tag","embeddable":true,"href":"\/jase\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5387"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}