The Whale Optimization Algorithm (WOA) is one of the recent meta-heuristic algorithms. WOA has advantages such as an exploration mechanism that leads towards the global optimum, a suitable balance between exploration and exploitation that avoids the local optimum, and a very good exploitation capability. In this study, five new hybrid algorithms are proposed to develop these advantages. Two of them are developed by combining WOA and Particle Swarm Optimization (PSO) algorithms, and three of them are developed by adding the Lévy flight algorithm to this combination in different ways. The proposed algorithms have been tested with 23 mathematical optimization problems, and in order to make a more accurate comparison, the average optimization results and corresponding standard deviation results are calculated by running these algorithms 30 times for each optimization problem. The proposed algorithms' performances were evaluated among themselves, and the WOALFVWPSO algorithm performed better among these algorithms. This proposed algorithm has been first compared with WOA and PSO, then with other algorithms in the literature. According to WOA and PSO, the proposed algorithm performs better in 19 of 23 mathematical optimization problems, and according to other literature, it performs better in 15 of 23 problems. Also, the proposed algorithm has been applied to the pressure vessel design engineering problem and achieved the best result compared to other algorithms in the literature. It has been proven that the WOALFVWPSO algorithm provides competitive solutions for most optimization problems when compared to meta-heuristic algorithms in the literature.