en Optimal design Optimal design پژوهشي Research This paper presents the application of metaheuristic methods to the minimum crossing number problem for the first time. These algorithms including particle swarm optimization, improved ray optimization, colliding bodies optimization and enhanced colliding bodies optimization. For each method, a pseudo code is provided. The crossing number problem is NP-hard and has important applications in engineering. The proposed algorithms are tested on six complete graphs and eight complete bipartite graphs and their results are compared with some existing methods. graph layout, crossing number, particle swarm optimization, improved ray optimization, colliding bodies optimization, enhanced colliding bodies optimization. 67 77 http://ijoce.iust.ac.ir/browse.php?a_code=A-10-66-53&slc_lang=en&sid=1 A. Kaveh 18003194753284600780 18003194753284600780 Yes M. Ilchi Ghazaan 18003194753284600781 18003194753284600781 No