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Firefly algorithm (FA) is a new random swarm search optimization algorithm that is modeled after movement and the mutual attraction of flashing fireflies. The number of fitness comparisons and attractions in the FA varies depending on the attraction model. A large number of attractions can induce search oscillations, while a small number of attractions can cause early convergence and a large number of fitness comparisons that can add to the computational time complexity. This study aims to offer H-GA–FA, a hybrid algorithm that combines two metaheuristic algorithms, the genetic algorithm (GA) and the FA, to overcome the flaws of the FA and combine the benefits of both algorithms to solve engineering design problems (EDPs). In this hybrid system, which blends the concepts of GA and FA, individuals are formed in the new generation not only by GA processes but also by FA mechanisms to prevent falling into local optima, introduce sufficient diversity of the solutions, and make equilibrium between exploration/exploitation trends. On the other hand, to deal with the violation of constraints, a chaotic process was utilized to keep the solutions feasible. The proposed hybrid algorithm H-GA–FA is tested by well-known test problems that contain a set of 17 unconstrained multimodal test functions and 7 constrained benchmark problems, where the results have confirmed the superiority of H-GA–FA overoptimization search methods. Finally, the performance of the H-GA–FA is also investigated on many EDPs. Computational results show that the H-GA–FA algorithm is competitive and better than other optimization algorithms that solve EDPs.

In this paper, the natural convection heat transfer of water/alumina nanofluid is investigated in a closed square cavity. An oblique magnetic field is applied on the cavity of angle $$\gamma$$ γ . There is also radiation heat transfer in the cavity. The cavity includes a high-temperature source of L-shape. A low-temperature source as a quadrant of a circle is placed at the corner of the cavity. All other walls are well insulated. The novelty of this work is a low-temperature obstacle embedded in the cavity. Simulations are conducted with a Fortran code based on the control volume method and simple algorithm. Entropy generation rate, Bejan number, and heat transfer are studied by changing different parameters. Results indicate that the highest rates of heat transfer and entropy generation have occurred for the perpendicular magnetic field at high values of the Rayleigh number. At these Rayleigh numbers, the minimum value of the Bejan number is obtained for 15° magnetic field. The magnetic field variation can lead to a change up to 53% in Nusselt number and up to 34% in generated entropy. In a weak magnetic field, the involvement of the radiation heat transfer mechanism leads to an increase in the heat transfer rate so that the Nusselt number can be increased by ten units considering the radiation heat transfer when there is no magnetic field. The maximum heat transfer rate occurs in the horizontal cavity and the minimum value in the cavity of 60° angle. For water, these values are 10.75 and 9.98 for 0 and 60 angles, respectively. Moreover, a weak magnetic field increases the heat transfer rate in the absence of the radiation mechanism, while it is reduced by considering a strong magnetic field.