An improved equilibrium optimizer for numerical optimization: A case study on engineering design of the shell and tube heat exchanger

The accurate design of the shell-and-tube heat exchanger (STHE) model is crucially important for industries such as process industries, oil refining, thermic devices, and power plants. However, determining its parameters is a challenging task due to the multimodality and nonlinearity of the cost function. Due to such nature, performing algorithms face the challenges of sluggish convergence speed and the proneness to local optima, making it difficult to reach satisfactory solutions. In line with the above challenges, this paper aims to contribute by developing an improved optimization algorithm that can effectively deal with such optimization issues. The proposed algorithm, named Levy-Opposition- Equilibrium Optimizer (LOEO), combines the Equilibrium Optimizer (EO) and the strategies of opposition learning (OL) and Levy flight learning (LFL) to overcome the dilemma of early convergence and getting trapped in local optima. It does this by employing the OL and LFL strategies after performing the EO phase to explore the search space in opposite directions and exploit the search space by means of the Levy distribution pattern, respectively. The proposed LOEO approach aims to enhance the algorithm's ability (provides enhanced exploration capability, effective search space exploitation, and enhanced convergence speed) and reach higher quality solutions to optimization problems. To assess the performance of the suggested LOEO, three stages of analysis are carried out. In the first stage, the performance of the proposed algorithm is investigated on some benchmark problems, including CEC 2005 and CEC 2020, using statistical verifications, convergence behavior, and the boxplot paradigm. The second stage involves the non-parametric Friedman test to assess the significance of results, where the results indicate that the LOEO outperforms the best existing one (equilibrium whale optimization algorithm (EWOA)) by an average rank of the Friedman test greater than 28% for CEC 2005 benchmark functions while outperforming enhanced EO (EEO) by 24% for CEC 2020. Finally, the LOEO's applicability is realized through determining the optimal design structure of the STHE model. The reported results indicate that the LOEO algorithm offers accurate performance compared to competing algorithms as it saves the cost of the device by 39.81% compared to the best existing results, and thus, it is commended to adopt for new applications.