The Grey Wolf Optimizer (GWO) algorithm is a very famous algorithm in the field of swarm intelligence for solving global optimization problems and real-life engineering design problems. The GWO algorithm is unique among swarm-based algorithms in that it depends on leadership hierarchy. In this paper, a Modified Grey Wolf Optimization Algorithm (MGWO) is proposed by modifying the position update equation of the original GWO algorithm. The leadership hierarchy is simulated using four different types of grey wolves: lambda (λ), mu (μ), nu (ν), and xi (ξ). The effectiveness of the proposed MGWO is tested using CEC 2005 benchmark functions, with sensitivity analysis and convergence analysis, and the statistical results are compared with six other meta-heuristic algorithms. According to the results and discussion, MGWO is a competitive algorithm for solving global optimization problems. In addition, the MGWO algorithm is applied to three real-life optimization design problems, such as tension/compression design, gear train design, and three-bar truss design. The proposed MGWO algorithm performed well compared to other algorithms.